Theorem islmhmd 19039

Description: Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)

Hypotheses

Ref	Expression
islmhmd.x	|- X = ( Base ` S )
islmhmd.a	|- .x. = ( .s ` S )
islmhmd.b	|- .X. = ( .s ` T )
islmhmd.k	|- K = (Scalar ` S )
islmhmd.j	|- J = (Scalar ` T )
islmhmd.n	|- N = ( Base ` K )
islmhmd.s	|- ( ph -> S e. LMod )
islmhmd.t	|- ( ph -> T e. LMod )
islmhmd.c	|- ( ph -> J = K )
islmhmd.f	|- ( ph -> F e. ( S GrpHom T ) )
islmhmd.l	|- ( ( ph /\ ( x e. N /\ y e. X ) ) -> ( F ` ( x .x. y ) ) = ( x .X. ( F ` y ) ) )

Assertion

Ref	Expression
islmhmd	|- ( ph -> F e. ( S LMHom T ) )

Distinct variable groups:   ph, x, y   x, F, y   x, S, y   x, T, y   x, X, y   x, J, y   x, N, y   x, K, y

Allowed substitution hints:   .x. ( x, y)   .X. ( x, y)

Proof of Theorem islmhmd

Step	Hyp	Ref	Expression
1		islmhmd.s	. . 3 |- ( ph -> S e. LMod )
2		islmhmd.t	. . 3 |- ( ph -> T e. LMod )
3	1, 2	jca 554	. 2 |- ( ph -> ( S e. LMod /\ T e. LMod ) )
4		islmhmd.f	. . 3 |- ( ph -> F e. ( S GrpHom T ) )
5		islmhmd.c	. . 3 |- ( ph -> J = K )
6		islmhmd.l	. . . 4 |- ( ( ph /\ ( x e. N /\ y e. X ) ) -> ( F ` ( x .x. y ) ) = ( x .X. ( F ` y ) ) )
7	6	ralrimivva 2971	. . 3 |- ( ph -> A. x e. N A. y e. X ( F ` ( x .x. y ) ) = ( x .X. ( F ` y ) ) )
8	4, 5, 7	3jca 1242	. 2 |- ( ph -> ( F e. ( S GrpHom T ) /\ J = K /\ A. x e. N A. y e. X ( F ` ( x .x. y ) ) = ( x .X. ( F ` y ) ) ) )
9		islmhmd.k	. . 3 |- K = (Scalar ` S )
10		islmhmd.j	. . 3 |- J = (Scalar ` T )
11		islmhmd.n	. . 3 |- N = ( Base ` K )
12		islmhmd.x	. . 3 |- X = ( Base ` S )
13		islmhmd.a	. . 3 |- .x. = ( .s ` S )
14		islmhmd.b	. . 3 |- .X. = ( .s ` T )
15	9, 10, 11, 12, 13, 14	islmhm 19027	. 2 |- ( F e. ( S LMHom T ) <-> ( ( S e. LMod /\ T e. LMod ) /\ ( F e. ( S GrpHom T ) /\ J = K /\ A. x e. N A. y e. X ( F ` ( x .x. y ) ) = ( x .X. ( F ` y ) ) ) ) )
16	3, 8, 15	sylanbrc 698	1 |- ( ph -> F e. ( S LMHom T ) )

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:  0lmhm  19040  idlmhm  19041  invlmhm  19042  lmhmco  19043  lmhmplusg  19044  lmhmvsca  19045  lmhmf1o  19046  reslmhm2  19053  reslmhm2b  19054  pwsdiaglmhm  19057  pwssplit3  19061  frlmup1  20137