Theorem subcmnd 13669

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 2206	. 2 |- ( ph -> ( Base ` H ) = ( Base ` H ) )
2		subcmnd.h	. . 3 |- ( ph -> H = ( G ↾s S ) )
3		eqidd 2206	. . 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 12961	. 2 |- ( ph -> ( +g ` G ) = ( +g ` H ) )
7		subcmnd.m	. 2 |- ( ph -> H e. Mnd )
8	5	3ad2ant1 1021	. . 3 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> G e. CMnd )
9		eqidd 2206	. . . . . 6 |- ( ph -> ( Base ` G ) = ( Base ` G ) )
10	2, 9, 5, 4	ressbasssd 12901	. . . . 5 |- ( ph -> ( Base ` H ) C_ ( Base ` G ) )
11	10	sselda 3193	. . . 4 |- ( ( ph /\ x e. ( Base ` H ) ) -> x e. ( Base ` G ) )
12	11	3adant3 1020	. . 3 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> x e. ( Base ` G ) )
13	10	sselda 3193	. . . 4 |- ( ( ph /\ y e. ( Base ` H ) ) -> y e. ( Base ` G ) )
14	13	3adant2 1019	. . 3 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> y e. ( Base ` G ) )
15		eqid 2205	. . . 4 |- ( Base ` G ) = ( Base ` G )
16		eqid 2205	. . . 4 |- ( +g ` G ) = ( +g ` G )
17	15, 16	cmncom 13638	. . 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 1250	. 2 |- ( ( ph /\ x e. ( Base ` H ) /\ y e. ( Base ` H ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
19	1, 6, 7, 18	iscmnd 13634	1 |- ( ph -> H e. CMnd )

Syntax hints:   -> wi 4   /\ w3a 981   = wceq 1373   e. wcel 2176   ` cfv 5271  (class class class)co 5944   Basecbs 12832   ↾_s cress 12833   +g cplusg 12909   Mndcmnd 13248  CMndccmn 13620

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-cmn 13622

This theorem is referenced by:  unitabl  13879  subrgcrng  13987