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Theorem lmhmlem 20784
Description: Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
lmhmlem.k 𝐾 = (Scalarβ€˜π‘†)
lmhmlem.l 𝐿 = (Scalarβ€˜π‘‡)
Assertion
Ref Expression
lmhmlem (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)))

Proof of Theorem lmhmlem
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmlem.k . . 3 𝐾 = (Scalarβ€˜π‘†)
2 lmhmlem.l . . 3 𝐿 = (Scalarβ€˜π‘‡)
3 eqid 2730 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4 eqid 2730 . . 3 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
5 eqid 2730 . . 3 ( ·𝑠 β€˜π‘†) = ( ·𝑠 β€˜π‘†)
6 eqid 2730 . . 3 ( ·𝑠 β€˜π‘‡) = ( ·𝑠 β€˜π‘‡)
71, 2, 3, 4, 5, 6islmhm 20782 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ βˆ€π‘Ž ∈ (Baseβ€˜πΎ)βˆ€π‘ ∈ (Baseβ€˜π‘†)(πΉβ€˜(π‘Ž( ·𝑠 β€˜π‘†)𝑏)) = (π‘Ž( ·𝑠 β€˜π‘‡)(πΉβ€˜π‘)))))
8 3simpa 1146 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ βˆ€π‘Ž ∈ (Baseβ€˜πΎ)βˆ€π‘ ∈ (Baseβ€˜π‘†)(πΉβ€˜(π‘Ž( ·𝑠 β€˜π‘†)𝑏)) = (π‘Ž( ·𝑠 β€˜π‘‡)(πΉβ€˜π‘))) β†’ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾))
98anim2i 615 . 2 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ βˆ€π‘Ž ∈ (Baseβ€˜πΎ)βˆ€π‘ ∈ (Baseβ€˜π‘†)(πΉβ€˜(π‘Ž( ·𝑠 β€˜π‘†)𝑏)) = (π‘Ž( ·𝑠 β€˜π‘‡)(πΉβ€˜π‘)))) β†’ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)))
107, 9sylbi 216 1 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  Scalarcsca 17204   ·𝑠 cvsca 17205   GrpHom cghm 19127  LModclmod 20614   LMHom clmhm 20774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-lmhm 20777
This theorem is referenced by:  lmhmsca  20785  lmghm  20786  lmhmlmod2  20787  lmhmlmod1  20788
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