Theorem rinvmod 14026

Description: Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6248. (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 14019	. . . . . . . 8 |- ( ( G e. CMnd /\ w e. B /\ A e. B ) -> ( w .+ A ) = ( A .+ w ) )
9	2, 3, 5, 8	syl3anc 1274	. . . . . . 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 2265	. . . . 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 2615	. 2 |- ( ph -> A. w e. B ( ( A .+ w ) = .0. -> ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) ) )
16		rinvmod.0	. . 3 |- .0. = ( 0g ` G )
17		cmnmnd 14018	. . . 4 |- ( G e. CMnd -> G e. Mnd )
18	1, 17	syl 14	. . 3 |- ( ph -> G e. Mnd )
19	6, 16, 7, 18, 4	mndinvmod 13658	. 2 |- ( ph -> E* w e. B ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) )
20		rmoim 3018	. 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 1398   e. wcel 2203   A.wral 2520   E*wrmo 2523   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   0gc0g 13469   Mndcmnd 13629  CMndccmn 14001

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-riota 6003  df-ov 6053  df-inn 9238  df-2 9296  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-cmn 14003

This theorem is referenced by: (None)