Theorem isvci 8139

Description: Properties that determine a complex vector space.

Hypotheses

Ref	Expression
isvci.1	|- G e. Abel
isvci.2	|- dom G = (X X. X)
isvci.3	|- S:(CC X. X)-->X
isvci.4	|- (x e. X -> (1Sx) = x)
isvci.5	|- ((y e. CC /\ x e. X /\ z e. X) -> (yS(xGz)) = ((ySx)G(ySz)))
isvci.6	|- ((y e. CC /\ z e. CC /\ x e. X) -> ((y + z)Sx) = ((ySx)G(zSx)))
isvci.7	|- ((y e. CC /\ z e. CC /\ x e. X) -> ((y x. z)Sx) = (yS(zSx)))
isvci.8	|- W = <.G, S>.

Assertion

Ref	Expression
isvci	|- W e. CVec

Distinct variable groups:   x,y,z,G   x,S,y,z   x,X,y,z

Proof of Theorem isvci

Step	Hyp	Ref	Expression
1		isvci.8	. 2 |- W = <.G, S>.
2		isvci.1	. . . 4 |- G e. Abel
3		isvci.3	. . . 4 |- S:(CC X. X)-->X
4		isvci.4	. . . . . 6 |- (x e. X -> (1Sx) = x)
5		isvci.5	. . . . . . . . . . 11 |- ((y e. CC /\ x e. X /\ z e. X) -> (yS(xGz)) = ((ySx)G(ySz)))
6	5	3com12 835	. . . . . . . . . 10 |- ((x e. X /\ y e. CC /\ z e. X) -> (yS(xGz)) = ((ySx)G(ySz)))
7	6	3expa 831	. . . . . . . . 9 |- (((x e. X /\ y e. CC) /\ z e. X) -> (yS(xGz)) = ((ySx)G(ySz)))
8	7	r19.21aiva 1706	. . . . . . . 8 |- ((x e. X /\ y e. CC) -> A.z e. X (yS(xGz)) = ((ySx)G(ySz)))
9		isvci.6	. . . . . . . . . . . 12 |- ((y e. CC /\ z e. CC /\ x e. X) -> ((y + z)Sx) = ((ySx)G(zSx)))
10		isvci.7	. . . . . . . . . . . 12 |- ((y e. CC /\ z e. CC /\ x e. X) -> ((y x. z)Sx) = (yS(zSx)))
11	9, 10	jca 288	. . . . . . . . . . 11 |- ((y e. CC /\ z e. CC /\ x e. X) -> (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))
12	11	3comr 839	. . . . . . . . . 10 |- ((x e. X /\ y e. CC /\ z e. CC) -> (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))
13	12	3expa 831	. . . . . . . . 9 |- (((x e. X /\ y e. CC) /\ z e. CC) -> (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))
14	13	r19.21aiva 1706	. . . . . . . 8 |- ((x e. X /\ y e. CC) -> A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))
15	8, 14	jca 288	. . . . . . 7 |- ((x e. X /\ y e. CC) -> (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx)))))
16	15	r19.21aiva 1706	. . . . . 6 |- (x e. X -> A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx)))))
17	4, 16	jca 288	. . . . 5 |- (x e. X -> ((1Sx) = x /\ A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))))
18	17	rgen 1690	. . . 4 |- A.x e. X ((1Sx) = x /\ A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx)))))
19	2, 3, 18	3pm3.2i 816	. . 3 |- (G e. Abel /\ S:(CC X. X)-->X /\ A.x e. X ((1Sx) = x /\ A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))))
20		ablgrp 8038	. . . . . 6 |- (G e. Abel -> G e. Grp)
21	2, 20	ax-mp 7	. . . . 5 |- G e. Grp
22		isvci.2	. . . . 5 |- dom G = (X X. X)
23	21, 22	grprn 7990	. . . 4 |- X = ran G
24	23	isvc 8138	. . 3 |- (<.G, S>. e. CVec <-> (G e. Abel /\ S:(CC X. X)-->X /\ A.x e. X ((1Sx) = x /\ A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx)))))))
25	19, 24	mpbir 190	. 2 |- <.G, S>. e. CVec
26	1, 25	eqeltr 1536	1 |- W e. CVec

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  A.wral 1637  <.cop 2401   X. cxp 3158  dom cdm 3160  -->wf 3168  (class class class)co 3948  CCcc 5204  1c1 5207   + caddc 5209   x. cmul 5211  Grpcgr 7967  Abelcabl 8035  CVeccvc 8101

This theorem is referenced by:  cnvc 8140  hilvc 8950  hhssnv 9054

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fo 3186  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-sub 5328  df-neg 5330  df-grp 7971  df-gid 7972  df-ginv 7973  df-abl 8036  df-vc 8102

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