Theorem mhmlin 12858

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 2177	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
3		mhmlin.p	. . . . . 6 |- .+ = ( +g ` S )
4		mhmlin.q	. . . . . 6 |- .+^ = ( +g ` T )
5		eqid 2177	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
6		eqid 2177	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
7	1, 2, 3, 4, 5, 6	ismhm 12853	. . . . 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 5898	. . . . 5 |- ( x = X -> ( F ` ( x .+ y ) ) = ( F ` ( X .+ y ) ) )
11		fveq2 5516	. . . . . 6 |- ( x = X -> ( F ` x ) = ( F ` X ) )
12	11	oveq1d 5890	. . . . 5 |- ( x = X -> ( ( F ` x ) .+^ ( F ` y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) )
13	10, 12	eqeq12d 2192	. . . 4 |- ( x = X -> ( ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) <-> ( F ` ( X .+ y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) ) )
14		oveq2 5883	. . . . . 6 |- ( y = Y -> ( X .+ y ) = ( X .+ Y ) )
15	14	fveq2d 5520	. . . . 5 |- ( y = Y -> ( F ` ( X .+ y ) ) = ( F ` ( X .+ Y ) ) )
16		fveq2 5516	. . . . . 6 |- ( y = Y -> ( F ` y ) = ( F ` Y ) )
17	16	oveq2d 5891	. . . . 5 |- ( y = Y -> ( ( F ` X ) .+^ ( F ` y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) )
18	15, 17	eqeq12d 2192	. . . 4 |- ( y = Y -> ( ( F ` ( X .+ y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) <-> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) ) )
19	13, 18	rspc2v 2855	. . 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 2148   A.wral 2455   -->wf 5213   ` cfv 5217  (class class class)co 5875   Basecbs 12462   +g cplusg 12536   0gc0g 12705   Mndcmnd 12817   MndHom cmhm 12849

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-map 6650  df-inn 8920  df-ndx 12465  df-slot 12466  df-base 12468  df-mhm 12851

This theorem is referenced by:  mhmf1o  12861  mhmco  12874  mhmima  12875  mhmeql  12876  mhmmulg  13024