Theorem rinvmod 13841

Description: Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6198. (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 13834	. . . . . . . 8 |- ( ( G e. CMnd /\ w e. B /\ A e. B ) -> ( w .+ A ) = ( A .+ w ) )
9	2, 3, 5, 8	syl3anc 1271	. . . . . . 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 2262	. . . . 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 2603	. 2 |- ( ph -> A. w e. B ( ( A .+ w ) = .0. -> ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) ) )
16		rinvmod.0	. . 3 |- .0. = ( 0g ` G )
17		cmnmnd 13833	. . . 4 |- ( G e. CMnd -> G e. Mnd )
18	1, 17	syl 14	. . 3 |- ( ph -> G e. Mnd )
19	6, 16, 7, 18, 4	mndinvmod 13473	. 2 |- ( ph -> E* w e. B ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) )
20		rmoim 3004	. 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 1395   e. wcel 2200   A.wral 2508   E*wrmo 2511   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   0gc0g 13284   Mndcmnd 13444  CMndccmn 13816

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-riota 5953  df-ov 6003  df-inn 9107  df-2 9165  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-cmn 13818

This theorem is referenced by: (None)