Theorem invfval 8257

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 5281	. . . . . . . 8 |- 1 e. CC
2	1	negcl 5381	. . . . . . 7 |- -u1 e. CC
3		invfval.3	. . . . . . . 8 |- N = (S o. `'(2nd |` ({-u1} X. V)))
4	3	curry1val 4106	. . . . . . 7 |- ((S Fn (CC X. (Base` U)) /\ -u1 e. CC /\ x e. (Base` U)) -> (N` x) = (-u1Sx))
5	2, 4	mp3an2 906	. . . . . 6 |- ((S Fn (CC X. (Base` U)) /\ x e. (Base` U)) -> (N` x) = (-u1Sx))
6		eqid 1478	. . . . . . . 8 |- (Base` U) = (Base` U)
7		invfval.4	. . . . . . . 8 |- S = (.s` U)
8	6, 7	nvsf 8234	. . . . . . 7 |- (U e. NrmCVec -> S:(CC X. (Base` U))-->(Base` U))
9		ffn 3633	. . . . . . 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 450	. . . . 5 |- ((U e. NrmCVec /\ x e. (Base` U)) -> (N` x) = (-u1Sx))
12		invfval.2	. . . . . 6 |- G = (+v` U)
13		eqid 1478	. . . . . 6 |- (inv` G) = (inv` G)
14	6, 12, 7, 13	nvinv 8256	. . . . 5 |- ((U e. NrmCVec /\ x e. (Base` U)) -> (-u1Sx) = ((inv` G)` x))
15	11, 14	eqtrd 1510	. . . 4 |- ((U e. NrmCVec /\ x e. (Base` U)) -> (N` x) = ((inv` G)` x))
16	15	r19.21aiva 1717	. . 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 3803	. . 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 4105	. . . . . 6 |- ((S:(CC X. (Base` U))-->(Base` U) /\ -u1 e. CC) -> N:(Base` U)-->(Base` U))
20	2, 19	mpan2 698	. . . . 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 3633	. . . 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 8232	. . . . 5 |- (U e. NrmCVec -> G e. Grp)
25	6, 12	bafval 8219	. . . . . 6 |- (Base` U) = ran G
26	25, 13	grpinvf 8075	. . . . 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 3696	. . . 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 473	. 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 958   e. wcel 960  A.wral 1648  Vcvv 1814  {csn 2413   X. cxp 3174  `'ccnv 3175   |` cres 3178   o. ccom 3180   Fn wfn 3183  -->wf 3184  -1-1-onto->wf1o 3187  ` cfv 3188  (class class class)co 3969  2ndc2nd 4084  CCcc 5244  1c1 5247  -ucneg 5305  Grpcgr 8030  invcgn 8032  NrmCVeccnv 8199  +vcpv 8200  Basecba 8201  .scns 8202

This theorem is referenced by:  hhssabl 9127

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-sub 5368  df-neg 5370  df-grp 8034  df-gid 8035  df-ginv 8036  df-abl 8096  df-vc 8161  df-nv 8207  df-va 8210  df-ba 8211  df-sm 8212  df-0v 8213  df-nm 8215

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