Theorem subcmnd 13640

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 2205	. 2 |- ( ph -> ( Base ` H ) = ( Base ` H ) )
2		subcmnd.h	. . 3 |- ( ph -> H = ( G ↾s S ) )
3		eqidd 2205	. . 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 12932	. 2 |- ( ph -> ( +g ` G ) = ( +g ` H ) )
7		subcmnd.m	. 2 |- ( ph -> H e. Mnd )
8	5	3ad2ant1 1020	. . 3 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> G e. CMnd )
9		eqidd 2205	. . . . . 6 |- ( ph -> ( Base ` G ) = ( Base ` G ) )
10	2, 9, 5, 4	ressbasssd 12872	. . . . 5 |- ( ph -> ( Base ` H ) C_ ( Base ` G ) )
11	10	sselda 3192	. . . 4 |- ( ( ph /\ x e. ( Base ` H ) ) -> x e. ( Base ` G ) )
12	11	3adant3 1019	. . 3 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> x e. ( Base ` G ) )
13	10	sselda 3192	. . . 4 |- ( ( ph /\ y e. ( Base ` H ) ) -> y e. ( Base ` G ) )
14	13	3adant2 1018	. . 3 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> y e. ( Base ` G ) )
15		eqid 2204	. . . 4 |- ( Base ` G ) = ( Base ` G )
16		eqid 2204	. . . 4 |- ( +g ` G ) = ( +g ` G )
17	15, 16	cmncom 13609	. . 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 1249	. 2 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
19	1, 6, 7, 18	iscmnd 13605	1 |- ( ph -> H e. CMnd )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1372   e. wcel 2175   ` cfv 5270  (class class class)co 5943   Basecbs 12803   ↾_s cress 12804   +g cplusg 12880   Mndcmnd 13219  CMndccmn 13591

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-ndx 12806  df-slot 12807  df-base 12809  df-sets 12810  df-iress 12811  df-plusg 12893  df-cmn 13593

This theorem is referenced by:  unitabl  13850  subrgcrng  13958