Theorem mhmlin 13537

Description: A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.)

Hypotheses

Ref	Expression
mhmlin.b	|- B = ( Base ` S )
mhmlin.p	|- .+ = ( +g ` S )
mhmlin.q	|- .+^ = ( +g ` T )

Assertion

Ref	Expression
mhmlin	|- ( ( F e. ( S MndHom T ) /\ X e. B /\ Y e. B ) -> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) )

Proof of Theorem mhmlin

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

Step	Hyp	Ref	Expression
1		mhmlin.b	. . . . . 6 |- B = ( Base ` S )
2		eqid 2229	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
3		mhmlin.p	. . . . . 6 |- .+ = ( +g ` S )
4		mhmlin.q	. . . . . 6 |- .+^ = ( +g ` T )
5		eqid 2229	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
6		eqid 2229	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
7	1, 2, 3, 4, 5, 6	ismhm 13531	. . . . 5 |- ( F e. ( S MndHom T ) <-> ( ( S e. Mnd /\ T e. Mnd ) /\ ( F : B --> ( Base ` T ) /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` T ) ) ) )
8	7	simprbi 275	. . . 4 |- ( F e. ( S MndHom T ) -> ( F : B --> ( Base ` T ) /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` T ) ) )
9	8	simp2d 1034	. . 3 |- ( F e. ( S MndHom T ) -> A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
10		fvoveq1 6034	. . . . 5 |- ( x = X -> ( F ` ( x .+ y ) ) = ( F ` ( X .+ y ) ) )
11		fveq2 5633	. . . . . 6 |- ( x = X -> ( F ` x ) = ( F ` X ) )
12	11	oveq1d 6026	. . . . 5 |- ( x = X -> ( ( F ` x ) .+^ ( F ` y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) )
13	10, 12	eqeq12d 2244	. . . 4 |- ( x = X -> ( ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) <-> ( F ` ( X .+ y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) ) )
14		oveq2 6019	. . . . . 6 |- ( y = Y -> ( X .+ y ) = ( X .+ Y ) )
15	14	fveq2d 5637	. . . . 5 |- ( y = Y -> ( F ` ( X .+ y ) ) = ( F ` ( X .+ Y ) ) )
16		fveq2 5633	. . . . . 6 |- ( y = Y -> ( F ` y ) = ( F ` Y ) )
17	16	oveq2d 6027	. . . . 5 |- ( y = Y -> ( ( F ` X ) .+^ ( F ` y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) )
18	15, 17	eqeq12d 2244	. . . 4 |- ( y = Y -> ( ( F ` ( X .+ y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) <-> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) ) )
19	13, 18	rspc2v 2921	. . 3 |- ( ( X e. B /\ Y e. B ) -> ( A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) -> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) ) )
20	9, 19	syl5com 29	. 2 |- ( F e. ( S MndHom T ) -> ( ( X e. B /\ Y e. B ) -> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) ) )
21	20	3impib 1225	1 |- ( ( F e. ( S MndHom T ) /\ X e. B /\ Y e. B ) -> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   -->wf 5318   ` cfv 5322  (class class class)co 6011   Basecbs 13069   +g cplusg 13147   0gc0g 13326   Mndcmnd 13486   MndHom cmhm 13527

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 4203  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-cnex 8111  ax-resscn 8112  ax-1re 8114  ax-addrcl 8117

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-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-fv 5330  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-map 6812  df-inn 9132  df-ndx 13072  df-slot 13073  df-base 13075  df-mhm 13529

This theorem is referenced by:  mhmf1o  13540  resmhm  13557  resmhm2  13558  resmhm2b  13559  mhmco  13560  mhmima  13561  mhmeql  13562  gsumwmhm  13568  mhmmulg  13737  ghmmhmb  13828  gsumfzmhm  13917  rhmmul  14165