Theorem subcmnd 14092

Description: A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)

Hypotheses

Ref	Expression
subcmnd.h	|- ( ph -> H = ( G ↾s S ) )
subcmnd.g	|- ( ph -> G e. CMnd )
subcmnd.m	|- ( ph -> H e. Mnd )
subcmnd.s	|- ( ph -> S e. V )

Assertion

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

Proof of Theorem subcmnd

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

Step	Hyp	Ref	Expression
1		eqidd 2235	. 2 |- ( ph -> ( Base ` H ) = ( Base ` H ) )
2		subcmnd.h	. . 3 |- ( ph -> H = ( G ↾s S ) )
3		eqidd 2235	. . 3 |- ( ph -> ( +g ` G ) = ( +g ` G ) )
4		subcmnd.s	. . 3 |- ( ph -> S e. V )
5		subcmnd.g	. . 3 |- ( ph -> G e. CMnd )
6	2, 3, 4, 5	ressplusgd 13432	. 2 |- ( ph -> ( +g ` G ) = ( +g ` H ) )
7		subcmnd.m	. 2 |- ( ph -> H e. Mnd )
8	5	3ad2ant1 1045	. . 3 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> G e. CMnd )
9		eqidd 2235	. . . . . 6 |- ( ph -> ( Base ` G ) = ( Base ` G ) )
10	2, 9, 5, 4	ressbasssd 13372	. . . . 5 |- ( ph -> ( Base ` H ) C_ ( Base ` G ) )
11	10	sselda 3242	. . . 4 |- ( ( ph /\ x e. ( Base ` H ) ) -> x e. ( Base ` G ) )
12	11	3adant3 1044	. . 3 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> x e. ( Base ` G ) )
13	10	sselda 3242	. . . 4 |- ( ( ph /\ y e. ( Base ` H ) ) -> y e. ( Base ` G ) )
14	13	3adant2 1043	. . 3 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> y e. ( Base ` G ) )
15		eqid 2234	. . . 4 |- ( Base ` G ) = ( Base ` G )
16		eqid 2234	. . . 4 |- ( +g ` G ) = ( +g ` G )
17	15, 16	cmncom 14061	. . 3 |- ( ( G e. CMnd /\ x e. ( Base ` G ) /\ y e. ( Base ` G ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
18	8, 12, 14, 17	syl3anc 1274	. 2 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
19	1, 6, 7, 18	iscmnd 14057	1 |- ( ph -> H e. CMnd )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2205   ` cfv 5359  (class class class)co 6060   Basecbs 13302   ↾_s cress 13303   +g cplusg 13380   Mndcmnd 13683  CMndccmn 14043

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 619  ax-in2 620  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-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-cnex 8236  ax-resscn 8237  ax-1cn 8238  ax-1re 8239  ax-icn 8240  ax-addcl 8241  ax-addrcl 8242  ax-mulcl 8243  ax-addcom 8245  ax-addass 8247  ax-i2m1 8250  ax-0lt1 8251  ax-0id 8253  ax-rnegex 8254  ax-pre-ltirr 8257  ax-pre-ltadd 8261

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-iota 5319  df-fun 5361  df-fv 5367  df-ov 6063  df-oprab 6064  df-mpo 6065  df-pnf 8328  df-mnf 8329  df-ltxr 8331  df-inn 9260  df-2 9318  df-ndx 13305  df-slot 13306  df-base 13308  df-sets 13309  df-iress 13310  df-plusg 13393  df-cmn 14045

This theorem is referenced by:  unitabl  14368  subrgcrng  14477