Theorem rinvmod 12908

Description: Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6058. (Contributed by AV, 31-Dec-2023.)

Hypotheses

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

Assertion

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

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

Allowed substitution hint:   G( w)

Proof of Theorem rinvmod

Step	Hyp	Ref	Expression
1		rinvmod.m	. . . . . . . . 9 |- ( ph -> G e. CMnd )
2	1	adantr 276	. . . . . . . 8 |- ( ( ph /\ w e. B ) -> G e. CMnd )
3		simpr 110	. . . . . . . 8 |- ( ( ph /\ w e. B ) -> w e. B )
4		rinvmod.a	. . . . . . . . 9 |- ( ph -> A e. B )
5	4	adantr 276	. . . . . . . 8 |- ( ( ph /\ w e. B ) -> A e. B )
6		rinvmod.b	. . . . . . . . 9 |- B = ( Base ` G )
7		rinvmod.p	. . . . . . . . 9 |- .+ = ( +g ` G )
8	6, 7	cmncom 12901	. . . . . . . 8 |- ( ( G e. CMnd /\ w e. B /\ A e. B ) -> ( w .+ A ) = ( A .+ w ) )
9	2, 3, 5, 8	syl3anc 1238	. . . . . . 7 |- ( ( ph /\ w e. B ) -> ( w .+ A ) = ( A .+ w ) )
10	9	adantr 276	. . . . . 6 |- ( ( ( ph /\ w e. B ) /\ ( A .+ w ) = .0. ) -> ( w .+ A ) = ( A .+ w ) )
11		simpr 110	. . . . . 6 |- ( ( ( ph /\ w e. B ) /\ ( A .+ w ) = .0. ) -> ( A .+ w ) = .0. )
12	10, 11	eqtrd 2208	. . . . 5 |- ( ( ( ph /\ w e. B ) /\ ( A .+ w ) = .0. ) -> ( w .+ A ) = .0. )
13	12, 11	jca 306	. . . 4 |- ( ( ( ph /\ w e. B ) /\ ( A .+ w ) = .0. ) -> ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) )
14	13	ex 115	. . 3 |- ( ( ph /\ w e. B ) -> ( ( A .+ w ) = .0. -> ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) ) )
15	14	ralrimiva 2548	. 2 |- ( ph -> A. w e. B ( ( A .+ w ) = .0. -> ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) ) )
16		rinvmod.0	. . 3 |- .0. = ( 0g ` G )
17		cmnmnd 12900	. . . 4 |- ( G e. CMnd -> G e. Mnd )
18	1, 17	syl 14	. . 3 |- ( ph -> G e. Mnd )
19	6, 16, 7, 18, 4	mndinvmod 12708	. 2 |- ( ph -> E* w e. B ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) )
20		rmoim 2936	. 2 |- ( A. w e. B ( ( A .+ w ) = .0. -> ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) ) -> ( E* w e. B ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) -> E* w e. B ( A .+ w ) = .0. ) )
21	15, 19, 20	sylc 62	1 |- ( ph -> E* w e. B ( A .+ w ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146   A.wral 2453   E*wrmo 2456   ` cfv 5208  (class class class)co 5865   Basecbs 12428   +g cplusg 12492   0gc0g 12626   Mndcmnd 12682  CMndccmn 12884

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-riota 5821  df-ov 5868  df-inn 8891  df-2 8949  df-ndx 12431  df-slot 12432  df-base 12434  df-plusg 12505  df-0g 12628  df-mgm 12640  df-sgrp 12673  df-mnd 12683  df-cmn 12886

This theorem is referenced by: (None)