Theorem lmhmlem 19029

Description: Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Hypotheses

Ref	Expression
lmhmlem.k	|- K = (Scalar ` S )
lmhmlem.l	|- L = (Scalar ` T )

Assertion

Ref	Expression
lmhmlem	|- ( F e. ( S LMHom T ) -> ( ( S e. LMod /\ T e. LMod ) /\ ( F e. ( S GrpHom T ) /\ L = K ) ) )

Proof of Theorem lmhmlem

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

Step	Hyp	Ref	Expression
1		lmhmlem.k	. . 3 |- K = (Scalar ` S )
2		lmhmlem.l	. . 3 |- L = (Scalar ` T )
3		eqid 2622	. . 3 |- ( Base ` K ) = ( Base ` K )
4		eqid 2622	. . 3 |- ( Base ` S ) = ( Base ` S )
5		eqid 2622	. . 3 |- ( .s ` S ) = ( .s ` S )
6		eqid 2622	. . 3 |- ( .s ` T ) = ( .s ` T )
7	1, 2, 3, 4, 5, 6	islmhm 19027	. 2 |- ( F e. ( S LMHom T ) <-> ( ( S e. LMod /\ T e. LMod ) /\ ( F e. ( S GrpHom T ) /\ L = K /\ A. a e. ( Base ` K ) A. b e. ( Base ` S ) ( F ` ( a ( .s ` S ) b ) ) = ( a ( .s ` T ) ( F ` b ) ) ) ) )
8		3simpa 1058	. . 3 |- ( ( F e. ( S GrpHom T ) /\ L = K /\ A. a e. ( Base ` K ) A. b e. ( Base ` S ) ( F ` ( a ( .s ` S ) b ) ) = ( a ( .s ` T ) ( F ` b ) ) ) -> ( F e. ( S GrpHom T ) /\ L = K ) )
9	8	anim2i 593	. 2 |- ( ( ( S e. LMod /\ T e. LMod ) /\ ( F e. ( S GrpHom T ) /\ L = K /\ A. a e. ( Base ` K ) A. b e. ( Base ` S ) ( F ` ( a ( .s ` S ) b ) ) = ( a ( .s ` T ) ( F ` b ) ) ) ) -> ( ( S e. LMod /\ T e. LMod ) /\ ( F e. ( S GrpHom T ) /\ L = K ) ) )
10	7, 9	sylbi 207	1 |- ( F e. ( S LMHom T ) -> ( ( S e. LMod /\ T e. LMod ) /\ ( F e. ( S GrpHom T ) /\ L = K ) ) )

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:  lmhmsca  19030  lmghm  19031  lmhmlmod2  19032  lmhmlmod1  19033