Theorem grpidinv2 13465

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 13438	. 2 |- ( ( G e. Grp /\ A e. B ) -> ( .0. .+ A ) = A )
5	1, 2, 3	grprid 13439	. 2 |- ( ( G e. Grp /\ A e. B ) -> ( A .+ .0. ) = A )
6	1, 2, 3	grplrinv 13464	. . 3 |- ( G e. Grp -> A. z e. B E. y e. B ( ( y .+ z ) = .0. /\ ( z .+ y ) = .0. ) )
7		oveq2 5965	. . . . . . 7 |- ( z = A -> ( y .+ z ) = ( y .+ A ) )
8	7	eqeq1d 2215	. . . . . 6 |- ( z = A -> ( ( y .+ z ) = .0. <-> ( y .+ A ) = .0. ) )
9		oveq1 5964	. . . . . . 7 |- ( z = A -> ( z .+ y ) = ( A .+ y ) )
10	9	eqeq1d 2215	. . . . . 6 |- ( z = A -> ( ( z .+ y ) = .0. <-> ( A .+ y ) = .0. ) )
11	8, 10	anbi12d 473	. . . . 5 |- ( z = A -> ( ( ( y .+ z ) = .0. /\ ( z .+ y ) = .0. ) <-> ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) ) )
12	11	rexbidv 2508	. . . 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 2877	. . 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 281	. 2 |- ( ( G e. Grp /\ A e. B ) -> E. y e. B ( ( y .+ A ) = .0. /\ ( A .+ y ) = .0. ) )
15	4, 5, 14	jca31 309	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 104   = wceq 1373   e. wcel 2177   A.wral 2485   E.wrex 2486   ` cfv 5280  (class class class)co 5957   Basecbs 12907   +g cplusg 12984   0gc0g 13163   Grpcgrp 13407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-cnex 8036  ax-resscn 8037  ax-1re 8039  ax-addrcl 8042

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-inn 9057  df-2 9115  df-ndx 12910  df-slot 12911  df-base 12913  df-plusg 12997  df-0g 13165  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411

This theorem is referenced by:  grpidinv  13466