Theorem rinvmod 13382

Description: Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6114. (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 13375	. . . . . . . 8 |- ( ( G e. CMnd /\ w e. B /\ A e. B ) -> ( w .+ A ) = ( A .+ w ) )
9	2, 3, 5, 8	syl3anc 1249	. . . . . . 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 2226	. . . . 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 2567	. 2 |- ( ph -> A. w e. B ( ( A .+ w ) = .0. -> ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) ) )
16		rinvmod.0	. . 3 |- .0. = ( 0g ` G )
17		cmnmnd 13374	. . . 4 |- ( G e. CMnd -> G e. Mnd )
18	1, 17	syl 14	. . 3 |- ( ph -> G e. Mnd )
19	6, 16, 7, 18, 4	mndinvmod 13029	. 2 |- ( ph -> E* w e. B ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) )
20		rmoim 2962	. 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 1364   e. wcel 2164   A.wral 2472   E*wrmo 2475   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   0gc0g 12870   Mndcmnd 13000  CMndccmn 13357

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-cmn 13359

This theorem is referenced by: (None)