Theorem lmhmlem 21187

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 2766	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2766	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
5		eqid 2766	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
6		eqid 2766	. . 3 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
7	1, 2, 3, 4, 5, 6	islmhm 21185	. 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 1166	. . 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 629	. 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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  Scalarcsca 17338   ·_𝑠 cvsca 17339   GrpHom cghm 19314  LModclmod 21018   LMHom clmhm 21177

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-lmhm 21180

This theorem is used by:  lmhmsca  21188  lmghm  21189  lmhmlmod2  21190  lmhmlmod1  21191