Theorem mgmhmlin 41786

Description: A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.)

Hypotheses

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

Assertion

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

Proof of Theorem mgmhmlin

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

Step	Hyp	Ref	Expression
1		mgmhmlin.b	. . . 4 |- B = ( Base ` S )
2		eqid 2622	. . . 4 |- ( Base ` T ) = ( Base ` T )
3		mgmhmlin.p	. . . 4 |- .+ = ( +g ` S )
4		mgmhmlin.q	. . . 4 |- .+^ = ( +g ` T )
5	1, 2, 3, 4	ismgmhm 41783	. . 3 |- ( F e. ( S MgmHom T ) <-> ( ( S e. Mgm /\ T e. Mgm ) /\ ( F : B --> ( Base ` T ) /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) ) ) )
6		oveq1 6657	. . . . . . . 8 |- ( x = X -> ( x .+ y ) = ( X .+ y ) )
7	6	fveq2d 6195	. . . . . . 7 |- ( x = X -> ( F ` ( x .+ y ) ) = ( F ` ( X .+ y ) ) )
8		fveq2 6191	. . . . . . . 8 |- ( x = X -> ( F ` x ) = ( F ` X ) )
9	8	oveq1d 6665	. . . . . . 7 |- ( x = X -> ( ( F ` x ) .+^ ( F ` y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) )
10	7, 9	eqeq12d 2637	. . . . . 6 |- ( x = X -> ( ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) <-> ( F ` ( X .+ y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) ) )
11		oveq2 6658	. . . . . . . 8 |- ( y = Y -> ( X .+ y ) = ( X .+ Y ) )
12	11	fveq2d 6195	. . . . . . 7 |- ( y = Y -> ( F ` ( X .+ y ) ) = ( F ` ( X .+ Y ) ) )
13		fveq2 6191	. . . . . . . 8 |- ( y = Y -> ( F ` y ) = ( F ` Y ) )
14	13	oveq2d 6666	. . . . . . 7 |- ( y = Y -> ( ( F ` X ) .+^ ( F ` y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) )
15	12, 14	eqeq12d 2637	. . . . . 6 |- ( y = Y -> ( ( F ` ( X .+ y ) ) = ( ( F ` X ) .+^ ( F ` y ) ) <-> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) ) )
16	10, 15	rspc2v 3322	. . . . 5 |- ( ( 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 ) ) ) )
17	16	com12 32	. . . 4 |- ( A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) -> ( ( X e. B /\ Y e. B ) -> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) ) )
18	17	ad2antll 765	. . 3 |- ( ( ( S e. Mgm /\ T e. Mgm ) /\ ( F : B --> ( Base ` T ) /\ A. x e. B A. y e. B ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) ) ) -> ( ( X e. B /\ Y e. B ) -> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) ) )
19	5, 18	sylbi 207	. 2 |- ( F e. ( S MgmHom T ) -> ( ( X e. B /\ Y e. B ) -> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) ) )
20	19	3impib 1262	1 |- ( ( F e. ( S MgmHom T ) /\ X e. B /\ Y e. B ) -> ( F ` ( X .+ Y ) ) = ( ( F ` X ) .+^ ( F ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Mgmcmgm 17240   MgmHom cmgmhm 41777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-mgmhm 41779

This theorem is referenced by:  mgmhmf1o  41787  resmgmhm  41798  resmgmhm2  41799  resmgmhm2b  41800  mgmhmco  41801  mgmhmima  41802  mgmhmeql  41803