Theorem nvinvfval 27495

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.) (Contributed by NM, 27-Mar-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvinvfval.2	|- G = ( +v ` U )
nvinvfval.4	|- S = ( .sOLD ` U )
nvinvfval.3	|- N = ( S o. `' ( 2nd |` ( { -u 1 } X. _V ) ) )

Assertion

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

Proof of Theorem nvinvfval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( BaseSet ` U ) = ( BaseSet ` U )
2		nvinvfval.4	. . . . 5 |- S = ( .sOLD ` U )
3	1, 2	nvsf 27474	. . . 4 |- ( U e. NrmCVec -> S : ( CC X. ( BaseSet ` U ) ) --> ( BaseSet ` U ) )
4		neg1cn 11124	. . . 4 |- -u 1 e. CC
5		nvinvfval.3	. . . . 5 |- N = ( S o. `' ( 2nd |` ( { -u 1 } X. _V ) ) )
6	5	curry1f 7271	. . . 4 |- ( ( S : ( CC X. ( BaseSet ` U ) ) --> ( BaseSet ` U ) /\ -u 1 e. CC ) -> N : ( BaseSet ` U ) --> ( BaseSet ` U ) )
7	3, 4, 6	sylancl 694	. . 3 |- ( U e. NrmCVec -> N : ( BaseSet ` U ) --> ( BaseSet ` U ) )
8		ffn 6045	. . 3 |- ( N : ( BaseSet ` U ) --> ( BaseSet ` U ) -> N Fn ( BaseSet ` U ) )
9	7, 8	syl 17	. 2 |- ( U e. NrmCVec -> N Fn ( BaseSet ` U ) )
10		nvinvfval.2	. . . 4 |- G = ( +v ` U )
11	10	nvgrp 27472	. . 3 |- ( U e. NrmCVec -> G e. GrpOp )
12	1, 10	bafval 27459	. . . 4 |- ( BaseSet ` U ) = ran G
13		eqid 2622	. . . 4 |- ( inv ` G ) = ( inv ` G )
14	12, 13	grpoinvf 27386	. . 3 |- ( G e. GrpOp -> ( inv ` G ) : ( BaseSet ` U ) -1-1-onto-> ( BaseSet ` U ) )
15		f1ofn 6138	. . 3 |- ( ( inv ` G ) : ( BaseSet ` U ) -1-1-onto-> ( BaseSet ` U ) -> ( inv ` G ) Fn ( BaseSet ` U ) )
16	11, 14, 15	3syl 18	. 2 |- ( U e. NrmCVec -> ( inv ` G ) Fn ( BaseSet ` U ) )
17		ffn 6045	. . . . . 6 |- ( S : ( CC X. ( BaseSet ` U ) ) --> ( BaseSet ` U ) -> S Fn ( CC X. ( BaseSet ` U ) ) )
18	3, 17	syl 17	. . . . 5 |- ( U e. NrmCVec -> S Fn ( CC X. ( BaseSet ` U ) ) )
19	18	adantr 481	. . . 4 |- ( ( U e. NrmCVec /\ x e. ( BaseSet ` U ) ) -> S Fn ( CC X. ( BaseSet ` U ) ) )
20	5	curry1val 7270	. . . 4 |- ( ( S Fn ( CC X. ( BaseSet ` U ) ) /\ -u 1 e. CC ) -> ( N ` x ) = ( -u 1 S x ) )
21	19, 4, 20	sylancl 694	. . 3 |- ( ( U e. NrmCVec /\ x e. ( BaseSet ` U ) ) -> ( N ` x ) = ( -u 1 S x ) )
22	1, 10, 2, 13	nvinv 27494	. . 3 |- ( ( U e. NrmCVec /\ x e. ( BaseSet ` U ) ) -> ( -u 1 S x ) = ( ( inv ` G ) ` x ) )
23	21, 22	eqtrd 2656	. 2 |- ( ( U e. NrmCVec /\ x e. ( BaseSet ` U ) ) -> ( N ` x ) = ( ( inv ` G ) ` x ) )
24	9, 16, 23	eqfnfvd 6314	1 |- ( U e. NrmCVec -> N = ( inv ` G ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   2ndc2nd 7167   CCcc 9934   1c1 9937   -ucneg 10267   GrpOpcgr 27343   invcgn 27345   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455

This theorem is referenced by:  hhssabloilem  28118