Theorem mndinvmod 13527

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 13518	. . . . . . . 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 2237	. . . . . 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 6025	. . . . . . . . 9 |- ( .0. = ( A .+ v ) -> ( w .+ .0. ) = ( w .+ ( A .+ v ) ) )
11	10	eqcoms 2234	. . . . . . . 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 13506	. . . . . . . . 9 |- ( ( G e. Mnd /\ ( w e. B /\ A e. B /\ v e. B ) ) -> ( ( w .+ A ) .+ v ) = ( w .+ ( A .+ v ) ) )
22	21	eqcomd 2237	. . . . . . . 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 1275	. . . . . . 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 6024	. . . . . . . . 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 13517	. . . . . . . 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 2268	. . . . 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 2268	. . . 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 2614	. 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 6024	. . . . 5 |- ( w = v -> ( w .+ A ) = ( v .+ A ) )
37	36	eqeq1d 2240	. . . 4 |- ( w = v -> ( ( w .+ A ) = .0. <-> ( v .+ A ) = .0. ) )
38		oveq2 6025	. . . . 5 |- ( w = v -> ( A .+ w ) = ( A .+ v ) )
39	38	eqeq1d 2240	. . . 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 2999	. 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 1004   = wceq 1397   e. wcel 2202   A.wral 2510   E*wrmo 2513   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338   Mndcmnd 13498

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499

This theorem is referenced by:  rinvmod  13895