Theorem grpidinv2 12758

Description: A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by AV, 1-Sep-2021.)

Hypotheses

Ref	Expression
grplrinv.b	|- B = ( Base ` G )
grplrinv.p	|- .+ = ( +g ` G )
grplrinv.i	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
grpidinv2	|- ( ( G e. Grp /\ A e. B ) -> ( ( ( .0. .+ A ) = A /\ ( A .+ .0. ) = A ) /\ E. y e. B ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) ) )

Distinct variable groups:   y, B   y, G   y, .+   y, .0.   y, A

Proof of Theorem grpidinv2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grplrinv.b	. . 3 |- B = ( Base ` G )
2		grplrinv.p	. . 3 |- .+ = ( +g ` G )
3		grplrinv.i	. . 3 |- .0. = ( 0g ` G )
4	1, 2, 3	grplid 12736	. 2 |- ( ( G e. Grp /\ A e. B ) -> ( .0. .+ A ) = A )
5	1, 2, 3	grprid 12737	. 2 |- ( ( G e. Grp /\ A e. B ) -> ( A .+ .0. ) = A )
6	1, 2, 3	grplrinv 12757	. . 3 |- ( G e. Grp -> A. z e. B E. y e. B ( ( y .+ z ) = .0. /\ ( z .+ y ) = .0. ) )
7		oveq2 5861	. . . . . . 7 |- ( z = A -> ( y .+ z ) = ( y .+ A ) )
8	7	eqeq1d 2179	. . . . . 6 |- ( z = A -> ( ( y .+ z ) = .0. <-> ( y .+ A ) = .0. ) )
9		oveq1 5860	. . . . . . 7 |- ( z = A -> ( z .+ y ) = ( A .+ y ) )
10	9	eqeq1d 2179	. . . . . 6 |- ( z = A -> ( ( z .+ y ) = .0. <-> ( A .+ y ) = .0. ) )
11	8, 10	anbi12d 470	. . . . 5 |- ( z = A -> ( ( ( y .+ z ) = .0. /\ ( z .+ y ) = .0. ) <-> ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) ) )
12	11	rexbidv 2471	. . . 4 |- ( z = A -> ( E. y e. B ( ( y .+ z ) = .0. /\ ( z .+ y ) = .0. ) <-> E. y e. B ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) ) )
13	12	rspcv 2830	. . 3 |- ( A e. B -> ( A. z e. B E. y e. B ( ( y .+ z ) = .0. /\ ( z .+ y ) = .0. ) -> E. y e. B ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) ) )
14	6, 13	mpan9 279	. 2 |- ( ( G e. Grp /\ A e. B ) -> E. y e. B ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) )
15	4, 5, 14	jca31 307	1 |- ( ( G e. Grp /\ A e. B ) -> ( ( ( .0. .+ A ) = A /\ ( A .+ .0. ) = A ) /\ E. y e. B ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   ` cfv 5198  (class class class)co 5853   Basecbs 12416   +g cplusg 12480   0gc0g 12596   Grpcgrp 12708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-inn 8879  df-2 8937  df-ndx 12419  df-slot 12420  df-base 12422  df-plusg 12493  df-0g 12598  df-mgm 12610  df-sgrp 12643  df-mnd 12653  df-grp 12711  df-minusg 12712

This theorem is referenced by:  grpidinv  12759