Theorem cmnsubm 14089

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 13764	. . 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 13765	. . . . . . . . 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 13760	. . . . . . . . 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 14082	. . . . 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 13460	. . . . . . 7 |- ( ph -> ( +g ` G ) = ( +g ` H ) )
24	23	oveqd 6092	. . . . . 6 |- ( ph -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )
25	23	oveqd 6092	. . . . . 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 14073	. 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 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   ` cfv 5372  (class class class)co 6075   Basecbs 13330   ↾_s cress 13331   +g cplusg 13408   Mndcmnd 13706  SubMndcsubmnd 13742  CMndccmn 14064

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-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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-submnd 13744  df-cmn 14066

This theorem is referenced by:  gsumsubmfi  14145