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Theorem lmhmlem 20640
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 2733 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4 eqid 2733 . . 3 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
5 eqid 2733 . . 3 ( ·𝑠 β€˜π‘†) = ( ·𝑠 β€˜π‘†)
6 eqid 2733 . . 3 ( ·𝑠 β€˜π‘‡) = ( ·𝑠 β€˜π‘‡)
71, 2, 3, 4, 5, 6islmhm 20638 . 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 1149 . . 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 618 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  Scalarcsca 17200   ·𝑠 cvsca 17201   GrpHom cghm 19089  LModclmod 20471   LMHom clmhm 20630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-lmhm 20633
This theorem is referenced by:  lmhmsca  20641  lmghm  20642  lmhmlmod2  20643  lmhmlmod1  20644
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