Theorem mhmlin 12721

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 2175	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
3		mhmlin.p	. . . . . 6 |- .+ = ( +g ` S )
4		mhmlin.q	. . . . . 6 |- .+^ = ( +g ` T )
5		eqid 2175	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
6		eqid 2175	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
7	1, 2, 3, 4, 5, 6	ismhm 12716	. . . . 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 1010	. . 3 |- ( F e. ( S MndHom T ) -> A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
10		fvoveq1 5888	. . . . 5 |- ( x = X -> ( F ` ( x .+ y ) ) = ( F ` ( X .+ y ) ) )
11		fveq2 5507	. . . . . 6 |- ( x = X -> ( F ` x ) = ( F ` X ) )
12	11	oveq1d 5880	. . . . 5 |- ( x = X -> ( ( F ` x ) .+^ ( F ` y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) )
13	10, 12	eqeq12d 2190	. . . 4 |- ( x = X -> ( ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) <-> ( F ` ( X .+ y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) ) )
14		oveq2 5873	. . . . . 6 |- ( y = Y -> ( X .+ y ) = ( X .+ Y ) )
15	14	fveq2d 5511	. . . . 5 |- ( y = Y -> ( F ` ( X .+ y ) ) = ( F ` ( X .+ Y ) ) )
16		fveq2 5507	. . . . . 6 |- ( y = Y -> ( F ` y ) = ( F ` Y ) )
17	16	oveq2d 5881	. . . . 5 |- ( y = Y -> ( ( F ` X ) .+^ ( F ` y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) )
18	15, 17	eqeq12d 2190	. . . 4 |- ( y = Y -> ( ( F ` ( X .+ y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) <-> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) ) )
19	13, 18	rspc2v 2852	. . 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 1201	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 978   = wceq 1353   e. wcel 2146   A.wral 2453   -->wf 5204   ` cfv 5208  (class class class)co 5865   Basecbs 12429   +g cplusg 12493   0gc0g 12627   Mndcmnd 12683   MndHom cmhm 12712

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-map 6640  df-inn 8893  df-ndx 12432  df-slot 12433  df-base 12435  df-mhm 12714

This theorem is referenced by:  mhmf1o  12724  mhmco  12736  mhmima  12737  mhmeql  12738  mhmmulg  12884