Theorem mhmlin 13042

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 2193	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
3		mhmlin.p	. . . . . 6 |- .+ = ( +g ` S )
4		mhmlin.q	. . . . . 6 |- .+^ = ( +g ` T )
5		eqid 2193	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
6		eqid 2193	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
7	1, 2, 3, 4, 5, 6	ismhm 13036	. . . . 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 1012	. . 3 |- ( F e. ( S MndHom T ) -> A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
10		fvoveq1 5942	. . . . 5 |- ( x = X -> ( F ` ( x .+ y ) ) = ( F ` ( X .+ y ) ) )
11		fveq2 5555	. . . . . 6 |- ( x = X -> ( F ` x ) = ( F ` X ) )
12	11	oveq1d 5934	. . . . 5 |- ( x = X -> ( ( F ` x ) .+^ ( F ` y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) )
13	10, 12	eqeq12d 2208	. . . 4 |- ( x = X -> ( ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) <-> ( F ` ( X .+ y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) ) )
14		oveq2 5927	. . . . . 6 |- ( y = Y -> ( X .+ y ) = ( X .+ Y ) )
15	14	fveq2d 5559	. . . . 5 |- ( y = Y -> ( F ` ( X .+ y ) ) = ( F ` ( X .+ Y ) ) )
16		fveq2 5555	. . . . . 6 |- ( y = Y -> ( F ` y ) = ( F ` Y ) )
17	16	oveq2d 5935	. . . . 5 |- ( y = Y -> ( ( F ` X ) .+^ ( F ` y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) )
18	15, 17	eqeq12d 2208	. . . 4 |- ( y = Y -> ( ( F ` ( X .+ y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) <-> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) ) )
19	13, 18	rspc2v 2878	. . 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 1203	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 980   = wceq 1364   e. wcel 2164   A.wral 2472   -->wf 5251   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   0gc0g 12870   Mndcmnd 13000   MndHom cmhm 13032

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-inn 8985  df-ndx 12624  df-slot 12625  df-base 12627  df-mhm 13034

This theorem is referenced by:  mhmf1o  13045  resmhm  13062  resmhm2  13063  resmhm2b  13064  mhmco  13065  mhmima  13066  mhmeql  13067  gsumwmhm  13073  mhmmulg  13236  ghmmhmb  13327  gsumfzmhm  13416  rhmmul  13663