Theorem mndinvmod 13086

Description: Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.)

Hypotheses

Ref	Expression
mndinvmod.b	|- B = ( Base ` G )
mndinvmod.0	|- .0. = ( 0g ` G )
mndinvmod.p	|- .+ = ( +g ` G )
mndinvmod.m	|- ( ph -> G e. Mnd )
mndinvmod.a	|- ( ph -> A e. B )

Assertion

Ref	Expression
mndinvmod	|- ( ph -> E* w e. B ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) )

Distinct variable groups:   w, A   w, B   w, .0.   w, .+   ph, w

Allowed substitution hint:   G( w)

Proof of Theorem mndinvmod

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mndinvmod.m	. . . . . . . 8 |- ( ph -> G e. Mnd )
2		simpl 109	. . . . . . . 8 |- ( ( w e. B /\ v e. B ) -> w e. B )
3		mndinvmod.b	. . . . . . . . 9 |- B = ( Base ` G )
4		mndinvmod.p	. . . . . . . . 9 |- .+ = ( +g ` G )
5		mndinvmod.0	. . . . . . . . 9 |- .0. = ( 0g ` G )
6	3, 4, 5	mndrid 13077	. . . . . . . 8 |- ( ( G e. Mnd /\ w e. B ) -> ( w .+ .0. ) = w )
7	1, 2, 6	syl2an 289	. . . . . . 7 |- ( ( ph /\ ( w e. B /\ v e. B ) ) -> ( w .+ .0. ) = w )
8	7	eqcomd 2202	. . . . . 6 |- ( ( ph /\ ( w e. B /\ v e. B ) ) -> w = ( w .+ .0. ) )
9	8	adantr 276	. . . . 5 |- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> w = ( w .+ .0. ) )
10		oveq2 5930	. . . . . . . . 9 |- ( .0. = ( A .+ v ) -> ( w .+ .0. ) = ( w .+ ( A .+ v ) ) )
11	10	eqcoms 2199	. . . . . . . 8 |- ( ( A .+ v ) = .0. -> ( w .+ .0. ) = ( w .+ ( A .+ v ) ) )
12	11	adantl 277	. . . . . . 7 |- ( ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) -> ( w .+ .0. ) = ( w .+ ( A .+ v ) ) )
13	12	adantl 277	. . . . . 6 |- ( ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) -> ( w .+ .0. ) = ( w .+ ( A .+ v ) ) )
14	13	adantl 277	. . . . 5 |- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> ( w .+ .0. ) = ( w .+ ( A .+ v ) ) )
15	1	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( w e. B /\ v e. B ) ) -> G e. Mnd )
16	2	adantl 277	. . . . . . . 8 |- ( ( ph /\ ( w e. B /\ v e. B ) ) -> w e. B )
17		mndinvmod.a	. . . . . . . . 9 |- ( ph -> A e. B )
18	17	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( w e. B /\ v e. B ) ) -> A e. B )
19		simpr 110	. . . . . . . . 9 |- ( ( w e. B /\ v e. B ) -> v e. B )
20	19	adantl 277	. . . . . . . 8 |- ( ( ph /\ ( w e. B /\ v e. B ) ) -> v e. B )
21	3, 4	mndass 13065	. . . . . . . . 9 |- ( ( G e. Mnd /\ ( w e. B /\ A e. B /\ v e. B ) ) -> ( ( w .+ A ) .+ v ) = ( w .+ ( A .+ v ) ) )
22	21	eqcomd 2202	. . . . . . . 8 |- ( ( G e. Mnd /\ ( w e. B /\ A e. B /\ v e. B ) ) -> ( w .+ ( A .+ v ) ) = ( ( w .+ A ) .+ v ) )
23	15, 16, 18, 20, 22	syl13anc 1251	. . . . . . 7 |- ( ( ph /\ ( w e. B /\ v e. B ) ) -> ( w .+ ( A .+ v ) ) = ( ( w .+ A ) .+ v ) )
24	23	adantr 276	. . . . . 6 |- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> ( w .+ ( A .+ v ) ) = ( ( w .+ A ) .+ v ) )
25		oveq1 5929	. . . . . . . . 9 |- ( ( w .+ A ) = .0. -> ( ( w .+ A ) .+ v ) = ( .0. .+ v ) )
26	25	adantr 276	. . . . . . . 8 |- ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) -> ( ( w .+ A ) .+ v ) = ( .0. .+ v ) )
27	26	adantr 276	. . . . . . 7 |- ( ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) -> ( ( w .+ A ) .+ v ) = ( .0. .+ v ) )
28	27	adantl 277	. . . . . 6 |- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> ( ( w .+ A ) .+ v ) = ( .0. .+ v ) )
29	3, 4, 5	mndlid 13076	. . . . . . . 8 |- ( ( G e. Mnd /\ v e. B ) -> ( .0. .+ v ) = v )
30	1, 19, 29	syl2an 289	. . . . . . 7 |- ( ( ph /\ ( w e. B /\ v e. B ) ) -> ( .0. .+ v ) = v )
31	30	adantr 276	. . . . . 6 |- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> ( .0. .+ v ) = v )
32	24, 28, 31	3eqtrd 2233	. . . . 5 |- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> ( w .+ ( A .+ v ) ) = v )
33	9, 14, 32	3eqtrd 2233	. . . 4 |- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> w = v )
34	33	ex 115	. . 3 |- ( ( ph /\ ( w e. B /\ v e. B ) ) -> ( ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) -> w = v ) )
35	34	ralrimivva 2579	. 2 |- ( ph -> A. w e. B A. v e. B ( ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) -> w = v ) )
36		oveq1 5929	. . . . 5 |- ( w = v -> ( w .+ A ) = ( v .+ A ) )
37	36	eqeq1d 2205	. . . 4 |- ( w = v -> ( ( w .+ A ) = .0. <-> ( v .+ A ) = .0. ) )
38		oveq2 5930	. . . . 5 |- ( w = v -> ( A .+ w ) = ( A .+ v ) )
39	38	eqeq1d 2205	. . . 4 |- ( w = v -> ( ( A .+ w ) = .0. <-> ( A .+ v ) = .0. ) )
40	37, 39	anbi12d 473	. . 3 |- ( w = v -> ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) <-> ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) )
41	40	rmo4 2957	. 2 |- ( E* w e. B ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) <-> A. w e. B A. v e. B ( ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) -> w = v ) )
42	35, 41	sylibr 134	1 |- ( ph -> E* w e. B ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   A.wral 2475   E*wrmo 2478   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   Mndcmnd 13057

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058

This theorem is referenced by:  rinvmod  13439