Theorem mndinvmod 13706

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 13697	. . . . . . . 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 2240	. . . . . 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 6066	. . . . . . . . 9 |- ( .0. = ( A .+ v ) -> ( w .+ .0. ) = ( w .+ ( A .+ v ) ) )
11	10	eqcoms 2237	. . . . . . . 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 13685	. . . . . . . . 9 |- ( ( G e. Mnd /\ ( w e. B /\ A e. B /\ v e. B ) ) -> ( ( w .+ A ) .+ v ) = ( w .+ ( A .+ v ) ) )
22	21	eqcomd 2240	. . . . . . . 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 1276	. . . . . . 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 6065	. . . . . . . . 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 13696	. . . . . . . 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 2271	. . . . 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 2271	. . . 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 2626	. 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 6065	. . . . 5 |- ( w = v -> ( w .+ A ) = ( v .+ A ) )
37	36	eqeq1d 2243	. . . 4 |- ( w = v -> ( ( w .+ A ) = .0. <-> ( v .+ A ) = .0. ) )
38		oveq2 6066	. . . . 5 |- ( w = v -> ( A .+ w ) = ( A .+ v ) )
39	38	eqeq1d 2243	. . . 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 3013	. 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 1005   = wceq 1398   e. wcel 2205   A.wral 2522   E*wrmo 2525   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   0gc0g 13553   Mndcmnd 13677

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678

This theorem is referenced by:  rinvmod  14062