Theorem invfval 8261

Description: Function for the negative of a vector on a normed complex vector space, in terms of the underlying addition group inverse. (We currently do not have a separate notation for the negative of a vector.)

Hypotheses

Ref	Expression
invfval.2	|- G = (+v` U)
invfval.4	|- S = (.s` U)
invfval.3	|- N = (S o. `'(2nd |` ({-u1} X. V)))

Assertion

Ref	Expression
invfval	|- (U e. NrmCVec -> N = (inv` G))

Proof of Theorem invfval

Step	Hyp	Ref	Expression
1		ax1cn 5269	. . . . . . . 8 |- 1 e. CC
2	1	negcl 5369	. . . . . . 7 |- -u1 e. CC
3		invfval.3	. . . . . . . 8 |- N = (S o. `'(2nd |` ({-u1} X. V)))
4	3	curry1val 4100	. . . . . . 7 |- ((S Fn (CC X. (Base` U)) /\ -u1 e. CC /\ x e. (Base` U)) -> (N` x) = (-u1Sx))
5	2, 4	mp3an2 904	. . . . . 6 |- ((S Fn (CC X. (Base` U)) /\ x e. (Base` U)) -> (N` x) = (-u1Sx))
6		eqid 1475	. . . . . . . 8 |- (Base` U) = (Base` U)
7		invfval.4	. . . . . . . 8 |- S = (.s` U)
8	6, 7	nvsf 8238	. . . . . . 7 |- (U e. NrmCVec -> S:(CC X. (Base` U))-->(Base` U))
9		ffn 3627	. . . . . . 7 |- (S:(CC X. (Base` U))-->(Base` U) -> S Fn (CC X. (Base` U)))
10	8, 9	syl 10	. . . . . 6 |- (U e. NrmCVec -> S Fn (CC X. (Base` U)))
11	5, 10	sylan 448	. . . . 5 |- ((U e. NrmCVec /\ x e. (Base` U)) -> (N` x) = (-u1Sx))
12		invfval.2	. . . . . 6 |- G = (+v` U)
13		eqid 1475	. . . . . 6 |- (inv` G) = (inv` G)
14	6, 12, 7, 13	nvinv 8260	. . . . 5 |- ((U e. NrmCVec /\ x e. (Base` U)) -> (-u1Sx) = ((inv` G)` x))
15	11, 14	eqtrd 1507	. . . 4 |- ((U e. NrmCVec /\ x e. (Base` U)) -> (N` x) = ((inv` G)` x))
16	15	r19.21aiva 1714	. . 3 |- (U e. NrmCVec -> A.x e. (Base` U)(N` x) = ((inv` G)` x))
17	16, 6	jctil 292	. 2 |- (U e. NrmCVec -> ((Base` U) = (Base` U) /\ A.x e. (Base` U)(N` x) = ((inv` G)` x)))
18		eqfnfv 3797	. . 3 |- ((N Fn (Base` U) /\ (inv` G) Fn (Base` U)) -> (N = (inv` G) <-> ((Base` U) = (Base` U) /\ A.x e. (Base` U)(N` x) = ((inv` G)` x))))
19	3	curry1f 4099	. . . . . 6 |- ((S:(CC X. (Base` U))-->(Base` U) /\ -u1 e. CC) -> N:(Base` U)-->(Base` U))
20	2, 19	mpan2 696	. . . . 5 |- (S:(CC X. (Base` U))-->(Base` U) -> N:(Base` U)-->(Base` U))
21	8, 20	syl 10	. . . 4 |- (U e. NrmCVec -> N:(Base` U)-->(Base` U))
22		ffn 3627	. . . 4 |- (N:(Base` U)-->(Base` U) -> N Fn (Base` U))
23	21, 22	syl 10	. . 3 |- (U e. NrmCVec -> N Fn (Base` U))
24	12	nvgrp 8236	. . . . 5 |- (U e. NrmCVec -> G e. Grp)
25	6, 12	bafval 8223	. . . . . 6 |- (Base` U) = ran G
26	25, 13	grpinvf 8079	. . . . 5 |- (G e. Grp -> (inv` G):(Base` U)-1-1-onto->(Base` U))
27	24, 26	syl 10	. . . 4 |- (U e. NrmCVec -> (inv` G):(Base` U)-1-1-onto->(Base` U))
28		f1ofn 3690	. . . 4 |- ((inv` G):(Base` U)-1-1-onto->(Base` U) -> (inv` G) Fn (Base` U))
29	27, 28	syl 10	. . 3 |- (U e. NrmCVec -> (inv` G) Fn (Base` U))
30	18, 23, 29	sylanc 471	. 2 |- (U e. NrmCVec -> (N = (inv` G) <-> ((Base` U) = (Base` U) /\ A.x e. (Base` U)(N` x) = ((inv` G)` x))))
31	17, 30	mpbird 196	1 |- (U e. NrmCVec -> N = (inv` G))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811  {csn 2409   X. cxp 3168  `'ccnv 3169   |` cres 3172   o. ccom 3174   Fn wfn 3177  -->wf 3178  -1-1-onto->wf1o 3181  ` cfv 3182  (class class class)co 3963  2ndc2nd 4078  CCcc 5232  1c1 5235  -ucneg 5293  Grpcgr 8033  invcgn 8035  NrmCVeccnv 8203  +vcpv 8204  Basecba 8205  .scns 8206

This theorem is referenced by:  hhssabl 9132

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-f1 3195  df-fo 3196  df-f1o 3197  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-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  df-nv 8211  df-va 8214  df-ba 8215  df-sm 8216  df-0v 8217  df-nm 8219

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