Theorem vcoprne 8198

Description: The operations of a complex vector space cannot be identical.

Assertion

Ref	Expression
vcoprne	|- (<.G, S>. e. CVec -> G =/= S)

Proof of Theorem vcoprne

Step	Hyp	Ref	Expression
1		ax1ne0 5280	. . . . 5 |- 1 =/= 0
2		df-ne 1587	. . . . 5 |- (1 =/= 0 <-> -. 1 = 0)
3	1, 2	mpbi 189	. . . 4 |- -. 1 = 0
4		vcoprnelem 8197	. . . . . . . . 9 |- (<.G, G>. e. CVec -> G:(CC X. CC)-->CC)
5		axcnex 5267	. . . . . . . . . . 11 |- CC e. V
6	5, 5	xpex 3260	. . . . . . . . . 10 |- (CC X. CC) e. V
7		fex 3652	. . . . . . . . . 10 |- ((G:(CC X. CC)-->CC /\ (CC X. CC) e. V) -> G e. V)
8	6, 7	mpan2 696	. . . . . . . . 9 |- (G:(CC X. CC)-->CC -> G e. V)
9	4, 8	syl 10	. . . . . . . 8 |- (<.G, G>. e. CVec -> G e. V)
10		op1stg 4087	. . . . . . . 8 |- (G e. V -> (1st` <.G, G>.) = G)
11	9, 10	syl 10	. . . . . . 7 |- (<.G, G>. e. CVec -> (1st` <.G, G>.) = G)
12	11	opreqd 3977	. . . . . 6 |- (<.G, G>. e. CVec -> (0(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = (0G(Id` (1st` <.G, G>.))))
13	11	rneqd 3341	. . . . . . . . . . . 12 |- (<.G, G>. e. CVec -> ran (1st` <.G, G>.) = ran G)
14		eqid 1475	. . . . . . . . . . . . . . 15 |- (1st` <.G, G>.) = (1st` <.G, G>.)
15	14	vcgrp 8177	. . . . . . . . . . . . . 14 |- (<.G, G>. e. CVec -> (1st` <.G, G>.) e. Grp)
16	11, 15	eqeltrrd 1549	. . . . . . . . . . . . 13 |- (<.G, G>. e. CVec -> G e. Grp)
17		grprndm 8054	. . . . . . . . . . . . 13 |- (G e. Grp -> ran G = dom dom G)
18	16, 17	syl 10	. . . . . . . . . . . 12 |- (<.G, G>. e. CVec -> ran G = dom dom G)
19		fdm 3631	. . . . . . . . . . . . . . 15 |- (G:(CC X. CC)-->CC -> dom G = (CC X. CC))
20	4, 19	syl 10	. . . . . . . . . . . . . 14 |- (<.G, G>. e. CVec -> dom G = (CC X. CC))
21	20	dmeqd 3313	. . . . . . . . . . . . 13 |- (<.G, G>. e. CVec -> dom dom G = dom (CC X. CC))
22		dmxpid 3333	. . . . . . . . . . . . 13 |- dom (CC X. CC) = CC
23	21, 22	syl6eq 1523	. . . . . . . . . . . 12 |- (<.G, G>. e. CVec -> dom dom G = CC)
24	13, 18, 23	3eqtrd 1511	. . . . . . . . . . 11 |- (<.G, G>. e. CVec -> ran (1st` <.G, G>.) = CC)
25		ax1cn 5269	. . . . . . . . . . 11 |- 1 e. CC
26	24, 25	syl5eleqr 1555	. . . . . . . . . 10 |- (<.G, G>. e. CVec -> 1 e. ran (1st` <.G, G>.))
27		eqid 1475	. . . . . . . . . . 11 |- ran (1st` <.G, G>.) = ran (1st` <.G, G>.)
28		eqid 1475	. . . . . . . . . . 11 |- (Id` (1st` <.G, G>.)) = (Id` (1st` <.G, G>.))
29	14, 27, 28	vc0rid 8186	. . . . . . . . . 10 |- ((<.G, G>. e. CVec /\ 1 e. ran (1st` <.G, G>.)) -> (1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = 1)
30	26, 29	mpdan 704	. . . . . . . . 9 |- (<.G, G>. e. CVec -> (1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = 1)
31		eqid 1475	. . . . . . . . . . 11 |- (2nd` <.G, G>.) = (2nd` <.G, G>.)
32	14, 31, 27	vcid 8170	. . . . . . . . . 10 |- ((<.G, G>. e. CVec /\ 1 e. ran (1st` <.G, G>.)) -> (1(2nd` <.G, G>.)1) = 1)
33	26, 32	mpdan 704	. . . . . . . . 9 |- (<.G, G>. e. CVec -> (1(2nd` <.G, G>.)1) = 1)
34		op2ndg 4088	. . . . . . . . . . . . 13 |- ((G e. V /\ G e. V) -> (2nd` <.G, G>.) = G)
35	34	anidms 434	. . . . . . . . . . . 12 |- (G e. V -> (2nd` <.G, G>.) = G)
36	9, 35	syl 10	. . . . . . . . . . 11 |- (<.G, G>. e. CVec -> (2nd` <.G, G>.) = G)
37	36, 11	eqtr4d 1510	. . . . . . . . . 10 |- (<.G, G>. e. CVec -> (2nd` <.G, G>.) = (1st` <.G, G>.))
38	37	opreqd 3977	. . . . . . . . 9 |- (<.G, G>. e. CVec -> (1(2nd` <.G, G>.)1) = (1(1st` <.G, G>.)1))
39	30, 33, 38	3eqtr2d 1513	. . . . . . . 8 |- (<.G, G>. e. CVec -> (1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = (1(1st` <.G, G>.)1))
40	14, 27, 28	vczcl 8185	. . . . . . . . . 10 |- (<.G, G>. e. CVec -> (Id` (1st` <.G, G>.)) e. ran (1st` <.G, G>.))
41	40, 26, 26	3jca 819	. . . . . . . . 9 |- (<.G, G>. e. CVec -> ((Id` (1st` <.G, G>.)) e. ran (1st` <.G, G>.) /\ 1 e. ran (1st` <.G, G>.) /\ 1 e. ran (1st` <.G, G>.)))
42	14, 27	vclcan 8184	. . . . . . . . 9 |- ((<.G, G>. e. CVec /\ ((Id` (1st` <.G, G>.)) e. ran (1st` <.G, G>.) /\ 1 e. ran (1st` <.G, G>.) /\ 1 e. ran (1st` <.G, G>.))) -> ((1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = (1(1st` <.G, G>.)1) <-> (Id` (1st` <.G, G>.)) = 1))
43	41, 42	mpdan 704	. . . . . . . 8 |- (<.G, G>. e. CVec -> ((1(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = (1(1st` <.G, G>.)1) <-> (Id` (1st` <.G, G>.)) = 1))
44	39, 43	mpbid 195	. . . . . . 7 |- (<.G, G>. e. CVec -> (Id` (1st` <.G, G>.)) = 1)
45	44	opreq2d 3976	. . . . . 6 |- (<.G, G>. e. CVec -> (0G(Id` (1st` <.G, G>.))) = (0G1))
46	12, 45	eqtr2d 1508	. . . . 5 |- (<.G, G>. e. CVec -> (0G1) = (0(1st` <.G, G>.)(Id` (1st` <.G, G>.))))
47		0cn 5328	. . . . . . 7 |- 0 e. CC
48	14, 31, 27, 28	vcz 8189	. . . . . . 7 |- ((<.G, G>. e. CVec /\ 0 e. CC) -> (0(2nd` <.G, G>.)(Id` (1st` <.G, G>.))) = (Id` (1st` <.G, G>.)))
49	47, 48	mpan2 696	. . . . . 6 |- (<.G, G>. e. CVec -> (0(2nd` <.G, G>.)(Id` (1st` <.G, G>.))) = (Id` (1st` <.G, G>.)))
50	36	opreqd 3977	. . . . . . 7 |- (<.G, G>. e. CVec -> (0(2nd` <.G, G>.)(Id` (1st` <.G, G>.))) = (0G(Id` (1st` <.G, G>.))))
51	50, 45	eqtrd 1507	. . . . . 6 |- (<.G, G>. e. CVec -> (0(2nd` <.G, G>.)(Id` (1st` <.G, G>.))) = (0G1))
52	49, 51, 44	3eqtr3d 1515	. . . . 5 |- (<.G, G>. e. CVec -> (0G1) = 1)
53	24, 47	syl5eleqr 1555	. . . . . 6 |- (<.G, G>. e. CVec -> 0 e. ran (1st` <.G, G>.))
54	14, 27, 28	vc0rid 8186	. . . . . 6 |- ((<.G, G>. e. CVec /\ 0 e. ran (1st` <.G, G>.)) -> (0(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = 0)
55	53, 54	mpdan 704	. . . . 5 |- (<.G, G>. e. CVec -> (0(1st` <.G, G>.)(Id` (1st` <.G, G>.))) = 0)
56	46, 52, 55	3eqtr3d 1515	. . . 4 |- (<.G, G>. e. CVec -> 1 = 0)
57	3, 56	mto 106	. . 3 |- -. <.G, G>. e. CVec
58		opeq2 2488	. . . 4 |- (G = S -> <.G, G>. = <.G, S>.)
59	58	eleq1d 1540	. . 3 |- (G = S -> (<.G, G>. e. CVec <-> <.G, S>. e. CVec))
60	57, 59	mtbii 716	. 2 |- (G = S -> -. <.G, S>. e. CVec)
61	60	necon2ai 1611	1 |- (<.G, S>. e. CVec -> G =/= S)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ w3a 775   = wceq 956   e. wcel 958   =/= wne 1585  Vcvv 1811  <.cop 2411   X. cxp 3168  dom cdm 3170  ran crn 3171  -->wf 3178  ` cfv 3182  (class class class)co 3963  1stc1st 4077  2ndc2nd 4078  CCcc 5232  0cc0 5234  1c1 5235  Grpcgr 8033  Idcgi 8034  CVeccvc 8164

This theorem is referenced by:  vcex 8199  nvex 8230  nvoprne 8306

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-sub 5356  df-neg 5358  df-grp 8037  df-gid 8038  df-ginv 8039  df-abl 8100  df-vc 8165

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