Theorem mndinvmod 13352

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 13343	. . . . . . . 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 2212	. . . . . 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 5965	. . . . . . . . 9 |- ( .0. = ( A .+ v ) -> ( w .+ .0. ) = ( w .+ ( A .+ v ) ) )
11	10	eqcoms 2209	. . . . . . . 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 13331	. . . . . . . . 9 |- ( ( G e. Mnd /\ ( w e. B /\ A e. B /\ v e. B ) ) -> ( ( w .+ A ) .+ v ) = ( w .+ ( A .+ v ) ) )
22	21	eqcomd 2212	. . . . . . . 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 1252	. . . . . . 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 5964	. . . . . . . . 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 13342	. . . . . . . 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 2243	. . . . 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 2243	. . . 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 2589	. 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 5964	. . . . 5 |- ( w = v -> ( w .+ A ) = ( v .+ A ) )
37	36	eqeq1d 2215	. . . 4 |- ( w = v -> ( ( w .+ A ) = .0. <-> ( v .+ A ) = .0. ) )
38		oveq2 5965	. . . . 5 |- ( w = v -> ( A .+ w ) = ( A .+ v ) )
39	38	eqeq1d 2215	. . . 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 2970	. 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 981   = wceq 1373   e. wcel 2177   A.wral 2485   E*wrmo 2488   ` cfv 5280  (class class class)co 5957   Basecbs 12907   +g cplusg 12984   0gc0g 13163   Mndcmnd 13323

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-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-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-iota 5241  df-fun 5282  df-fn 5283  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

This theorem is referenced by:  rinvmod  13720