Theorem lmhmlem 21309

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 2760	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2760	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
5		eqid 2760	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
6		eqid 2760	. . 3 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
7	1, 2, 3, 4, 5, 6	islmhm 21307	. 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 1569   ∈ wcel 2144  ∀wral 3076  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  Scalarcsca 17436   ·_𝑠 cvsca 17437   GrpHom cghm 19432  LModclmod 21140   LMHom clmhm 21299

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-lmhm 21302

This theorem is used by:  lmhmsca  21310  lmghm  21311  lmhmlmod2  21312  lmhmlmod1  21313