Theorem lmhmlin 19035

Description: A homomorphism of left modules is K-linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Hypotheses

Ref	Expression
lmhmlin.k	|- K = (Scalar ` S )
lmhmlin.b	|- B = ( Base ` K )
lmhmlin.e	|- E = ( Base ` S )
lmhmlin.m	|- .x. = ( .s ` S )
lmhmlin.n	|- .X. = ( .s ` T )

Assertion

Ref	Expression
lmhmlin	|- ( ( F e. ( S LMHom T ) /\ X e. B /\ Y e. E ) -> ( F ` ( X .x. Y ) ) = ( X .X. ( F ` Y ) ) )

Proof of Theorem lmhmlin

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmhmlin.k	. . . . . 6 |- K = (Scalar ` S )
2		eqid 2622	. . . . . 6 |- (Scalar ` T ) = (Scalar ` T )
3		lmhmlin.b	. . . . . 6 |- B = ( Base ` K )
4		lmhmlin.e	. . . . . 6 |- E = ( Base ` S )
5		lmhmlin.m	. . . . . 6 |- .x. = ( .s ` S )
6		lmhmlin.n	. . . . . 6 |- .X. = ( .s ` T )
7	1, 2, 3, 4, 5, 6	islmhm 19027	. . . . 5 |- ( F e. ( S LMHom T ) <-> ( ( S e. LMod /\ T e. LMod ) /\ ( F e. ( S GrpHom T ) /\ (Scalar ` T ) = K /\ A. a e. B A. b e. E ( F ` ( a .x. b ) ) = ( a .X. ( F ` b ) ) ) ) )
8	7	simprbi 480	. . . 4 |- ( F e. ( S LMHom T ) -> ( F e. ( S GrpHom T ) /\ (Scalar ` T ) = K /\ A. a e. B A. b e. E ( F ` ( a .x. b ) ) = ( a .X. ( F ` b ) ) ) )
9	8	simp3d 1075	. . 3 |- ( F e. ( S LMHom T ) -> A. a e. B A. b e. E ( F ` ( a .x. b ) ) = ( a .X. ( F ` b ) ) )
10		oveq1 6657	. . . . . 6 |- ( a = X -> ( a .x. b ) = ( X .x. b ) )
11	10	fveq2d 6195	. . . . 5 |- ( a = X -> ( F ` ( a .x. b ) ) = ( F ` ( X .x. b ) ) )
12		oveq1 6657	. . . . 5 |- ( a = X -> ( a .X. ( F ` b ) ) = ( X .X. ( F ` b ) ) )
13	11, 12	eqeq12d 2637	. . . 4 |- ( a = X -> ( ( F ` ( a .x. b ) ) = ( a .X. ( F ` b ) ) <-> ( F ` ( X .x. b ) ) = ( X .X. ( F ` b ) ) ) )
14		oveq2 6658	. . . . . 6 |- ( b = Y -> ( X .x. b ) = ( X .x. Y ) )
15	14	fveq2d 6195	. . . . 5 |- ( b = Y -> ( F ` ( X .x. b ) ) = ( F ` ( X .x. Y ) ) )
16		fveq2 6191	. . . . . 6 |- ( b = Y -> ( F ` b ) = ( F ` Y ) )
17	16	oveq2d 6666	. . . . 5 |- ( b = Y -> ( X .X. ( F ` b ) ) = ( X .X. ( F ` Y ) ) )
18	15, 17	eqeq12d 2637	. . . 4 |- ( b = Y -> ( ( F ` ( X .x. b ) ) = ( X .X. ( F ` b ) ) <-> ( F ` ( X .x. Y ) ) = ( X .X. ( F ` Y ) ) ) )
19	13, 18	rspc2v 3322	. . 3 |- ( ( X e. B /\ Y e. E ) -> ( A. a e. B A. b e. E ( F ` ( a .x. b ) ) = ( a .X. ( F ` b ) ) -> ( F ` ( X .x. Y ) ) = ( X .X. ( F ` Y ) ) ) )
20	9, 19	syl5com 31	. 2 |- ( F e. ( S LMHom T ) -> ( ( X e. B /\ Y e. E ) -> ( F ` ( X .x. Y ) ) = ( X .X. ( F ` Y ) ) ) )
21	20	3impib 1262	1 |- ( ( F e. ( S LMHom T ) /\ X e. B /\ Y e. E ) -> ( F ` ( X .x. Y ) ) = ( X .X. ( F ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   GrpHom cghm 17657   LModclmod 18863   LMHom clmhm 19019

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

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-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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-lmhm 19022

This theorem is referenced by:  islmhm2  19038  lmhmco  19043  lmhmplusg  19044  lmhmvsca  19045  lmhmf1o  19046  lmhmima  19047  lmhmpreima  19048  reslmhm  19052  reslmhm2  19053  reslmhm2b  19054  lmhmeql  19055  ipass  19990  lindfmm  20166  nmoleub2lem3  22915  nmoleub3  22919  mendassa  37764