Theorem subcmnd 13865

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 2230	. 2 |- ( ph -> ( Base ` H ) = ( Base ` H ) )
2		subcmnd.h	. . 3 |- ( ph -> H = ( G ↾s S ) )
3		eqidd 2230	. . 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 13157	. 2 |- ( ph -> ( +g ` G ) = ( +g ` H ) )
7		subcmnd.m	. 2 |- ( ph -> H e. Mnd )
8	5	3ad2ant1 1042	. . 3 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> G e. CMnd )
9		eqidd 2230	. . . . . 6 |- ( ph -> ( Base ` G ) = ( Base ` G ) )
10	2, 9, 5, 4	ressbasssd 13097	. . . . 5 |- ( ph -> ( Base ` H ) C_ ( Base ` G ) )
11	10	sselda 3224	. . . 4 |- ( ( ph /\ x e. ( Base ` H ) ) -> x e. ( Base ` G ) )
12	11	3adant3 1041	. . 3 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> x e. ( Base ` G ) )
13	10	sselda 3224	. . . 4 |- ( ( ph /\ y e. ( Base ` H ) ) -> y e. ( Base ` G ) )
14	13	3adant2 1040	. . 3 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> y e. ( Base ` G ) )
15		eqid 2229	. . . 4 |- ( Base ` G ) = ( Base ` G )
16		eqid 2229	. . . 4 |- ( +g ` G ) = ( +g ` G )
17	15, 16	cmncom 13834	. . 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 1271	. 2 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
19	1, 6, 7, 18	iscmnd 13830	1 |- ( ph -> H e. CMnd )

Syntax hints:   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5317  (class class class)co 6000   Basecbs 13027   ↾_s cress 13028   +g cplusg 13105   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-in1 617  ax-in2 618  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-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  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-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  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-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-iress 13035  df-plusg 13118  df-cmn 13818

This theorem is referenced by:  unitabl  14075  subrgcrng  14183