Theorem grpvlinv 20201

Description: Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Hypotheses

Ref	Expression
grpvlinv.b	|- B = ( Base ` G )
grpvlinv.p	|- .+ = ( +g ` G )
grpvlinv.n	|- N = ( invg ` G )
grpvlinv.z	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
grpvlinv	|- ( ( G e. Grp /\ X e. ( B ^m I ) ) -> ( ( N o. X ) oF .+ X ) = ( I X. { .0. } ) )

Proof of Theorem grpvlinv

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapex 7878	. . . 4 |- ( X e. ( B ^m I ) -> ( B e. _V /\ I e. _V ) )
2	1	simprd 479	. . 3 |- ( X e. ( B ^m I ) -> I e. _V )
3	2	adantl 482	. 2 |- ( ( G e. Grp /\ X e. ( B ^m I ) ) -> I e. _V )
4		elmapi 7879	. . 3 |- ( X e. ( B ^m I ) -> X : I --> B )
5	4	adantl 482	. 2 |- ( ( G e. Grp /\ X e. ( B ^m I ) ) -> X : I --> B )
6		grpvlinv.b	. . . 4 |- B = ( Base ` G )
7		grpvlinv.z	. . . 4 |- .0. = ( 0g ` G )
8	6, 7	grpidcl 17450	. . 3 |- ( G e. Grp -> .0. e. B )
9	8	adantr 481	. 2 |- ( ( G e. Grp /\ X e. ( B ^m I ) ) -> .0. e. B )
10		grpvlinv.n	. . . 4 |- N = ( invg ` G )
11	6, 10	grpinvf 17466	. . 3 |- ( G e. Grp -> N : B --> B )
12	11	adantr 481	. 2 |- ( ( G e. Grp /\ X e. ( B ^m I ) ) -> N : B --> B )
13		fcompt 6400	. . 3 |- ( ( N : B --> B /\ X : I --> B ) -> ( N o. X ) = ( x e. I |-> ( N ` ( X ` x ) ) ) )
14	11, 4, 13	syl2an 494	. 2 |- ( ( G e. Grp /\ X e. ( B ^m I ) ) -> ( N o. X ) = ( x e. I |-> ( N ` ( X ` x ) ) ) )
15		grpvlinv.p	. . . 4 |- .+ = ( +g ` G )
16	6, 15, 7, 10	grplinv 17468	. . 3 |- ( ( G e. Grp /\ y e. B ) -> ( ( N ` y ) .+ y ) = .0. )
17	16	adantlr 751	. 2 |- ( ( ( G e. Grp /\ X e. ( B ^m I ) ) /\ y e. B ) -> ( ( N ` y ) .+ y ) = .0. )
18	3, 5, 9, 12, 14, 17	caofinvl 6924	1 |- ( ( G e. Grp /\ X e. ( B ^m I ) ) -> ( ( N o. X ) oF .+ X ) = ( I X. { .0. } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   ^m cmap 7857   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-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-of 6897  df-1st 7168  df-2nd 7169  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426

This theorem is referenced by:  mendring  37762