Theorem cmnsubm 14112

Description: A submonoid of a commutative monoid is commutative. (Contributed by Jim Kingdon, 7-Jul-2026.)

Hypotheses

Ref	Expression
cmnsubm.s	|- ( ph -> S e. (SubMnd ` G ) )
cmnsubm.g	|- ( ph -> G e. CMnd )
cmnsubm.h	|- H = ( G ↾s S )

Assertion

Ref	Expression
cmnsubm	|- ( ph -> H e. CMnd )

Proof of Theorem cmnsubm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmnsubm.s	. . 3 |- ( ph -> S e. (SubMnd ` G ) )
2		cmnsubm.h	. . . 4 |- H = ( G ↾s S )
3	2	submmnd 13787	. . 3 |- ( S e. (SubMnd ` G ) -> H e. Mnd )
4	1, 3	syl 14	. 2 |- ( ph -> H e. Mnd )
5		cmnsubm.g	. . . . . 6 |- ( ph -> G e. CMnd )
6	5	adantr 276	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` H ) /\ y e. ( Base ` H ) ) ) -> G e. CMnd )
7	2	submbas 13788	. . . . . . . . 9 |- ( S e. (SubMnd ` G ) -> S = ( Base ` H ) )
8	1, 7	syl 14	. . . . . . . 8 |- ( ph -> S = ( Base ` H ) )
9		eqid 2238	. . . . . . . . . 10 |- ( Base ` G ) = ( Base ` G )
10	9	submss 13783	. . . . . . . . 9 |- ( S e. (SubMnd ` G ) -> S C_ ( Base ` G ) )
11	1, 10	syl 14	. . . . . . . 8 |- ( ph -> S C_ ( Base ` G ) )
12	8, 11	eqsstrrd 3285	. . . . . . 7 |- ( ph -> ( Base ` H ) C_ ( Base ` G ) )
13	12	adantr 276	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` H ) /\ y e. ( Base ` H ) ) ) -> ( Base ` H ) C_ ( Base ` G ) )
14		simprl 535	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` H ) /\ y e. ( Base ` H ) ) ) -> x e. ( Base ` H ) )
15	13, 14	sseldd 3249	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` H ) /\ y e. ( Base ` H ) ) ) -> x e. ( Base ` G ) )
16		simprr 537	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` H ) /\ y e. ( Base ` H ) ) ) -> y e. ( Base ` H ) )
17	13, 16	sseldd 3249	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` H ) /\ y e. ( Base ` H ) ) ) -> y e. ( Base ` G ) )
18		eqid 2238	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
19	9, 18	cmncom 14105	. . . . 5 |- ( ( G e. CMnd /\ x e. ( Base ` G ) /\ y e. ( Base ` G ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
20	6, 15, 17, 19	syl3anc 1278	. . . 4 |- ( ( ph /\ ( x e. ( Base ` H ) /\ y e. ( Base ` H ) ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
21	2	a1i 9	. . . . . . . 8 |- ( ph -> H = ( G ↾s S ) )
22		eqidd 2239	. . . . . . . 8 |- ( ph -> ( +g ` G ) = ( +g ` G ) )
23	21, 22, 1, 5	ressplusgd 13483	. . . . . . 7 |- ( ph -> ( +g ` G ) = ( +g ` H ) )
24	23	oveqd 6102	. . . . . 6 |- ( ph -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )
25	23	oveqd 6102	. . . . . 6 |- ( ph -> ( y ( +g ` G ) x ) = ( y ( +g ` H ) x ) )
26	24, 25	eqeq12d 2253	. . . . 5 |- ( ph -> ( ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) <-> ( x ( +g ` H ) y ) = ( y ( +g ` H ) x ) ) )
27	26	adantr 276	. . . 4 |- ( ( ph /\ ( x e. ( Base ` H ) /\ y e. ( Base ` H ) ) ) -> ( ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) <-> ( x ( +g ` H ) y ) = ( y ( +g ` H ) x ) ) )
28	20, 27	mpbid 147	. . 3 |- ( ( ph /\ ( x e. ( Base ` H ) /\ y e. ( Base ` H ) ) ) -> ( x ( +g ` H ) y ) = ( y ( +g ` H ) x ) )
29	28	ralrimivva 2632	. 2 |- ( ph -> A. x e. ( Base ` H ) A. y e. ( Base ` H ) ( x ( +g ` H ) y ) = ( y ( +g ` H ) x ) )
30		eqid 2238	. . 3 |- ( Base ` H ) = ( Base ` H )
31		eqid 2238	. . 3 |- ( +g ` H ) = ( +g ` H )
32	30, 31	iscmn 14096	. 2 |- ( H e. CMnd <-> ( H e. Mnd /\ A. x e. ( Base ` H ) A. y e. ( Base ` H ) ( x ( +g ` H ) y ) = ( y ( +g ` H ) x ) ) )
33	4, 29, 32	sylanbrc 421	1 |- ( ph -> H e. CMnd )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   ` cfv 5377  (class class class)co 6085   Basecbs 13352   ↾_s cress 13353   +g cplusg 13431   Mndcmnd 13729  SubMndcsubmnd 13765  CMndccmn 14087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-submnd 13767  df-cmn 14089

This theorem is used by:  gsumsubmfi  14168