Theorem lmhmlem 19874

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 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (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 2758	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2758	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
5		eqid 2758	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
6		eqid 2758	. . 3 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
7	1, 2, 3, 4, 5, 6	islmhm 19872	. 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑎 ∈ (Base‘𝐾)∀𝑏 ∈ (Base‘𝑆)(𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))))
8		3simpa 1145	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑎 ∈ (Base‘𝐾)∀𝑏 ∈ (Base‘𝑆)(𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏))) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾))
9	8	anim2i 619	. 2 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑎 ∈ (Base‘𝐾)∀𝑏 ∈ (Base‘𝑆)(𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)))
10	7, 9	sylbi 220	1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ‘cfv 6339  (class class class)co 7155  Basecbs 16546  Scalarcsca 16631   ·_𝑠 cvsca 16632   GrpHom cghm 18427  LModclmod 19707   LMHom clmhm 19864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-iota 6298  df-fun 6341  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-lmhm 19867

This theorem is referenced by:  lmhmsca  19875  lmghm  19876  lmhmlmod2  19877  lmhmlmod1  19878