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Theorem lmhmlin 20790
Description: A homomorphism of left modules is 𝐾-linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
lmhmlin.k 𝐾 = (Scalarβ€˜π‘†)
lmhmlin.b 𝐡 = (Baseβ€˜πΎ)
lmhmlin.e 𝐸 = (Baseβ€˜π‘†)
lmhmlin.m Β· = ( ·𝑠 β€˜π‘†)
lmhmlin.n Γ— = ( ·𝑠 β€˜π‘‡)
Assertion
Ref Expression
lmhmlin ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐸) β†’ (πΉβ€˜(𝑋 Β· π‘Œ)) = (𝑋 Γ— (πΉβ€˜π‘Œ)))
Proof of Theorem lmhmlin
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmlin.k . . . . . 6 𝐾 = (Scalarβ€˜π‘†)
2 eqid 2730 . . . . . 6 (Scalarβ€˜π‘‡) = (Scalarβ€˜π‘‡)
3 lmhmlin.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
4 lmhmlin.e . . . . . 6 𝐸 = (Baseβ€˜π‘†)
5 lmhmlin.m . . . . . 6 Β· = ( ·𝑠 β€˜π‘†)
6 lmhmlin.n . . . . . 6 Γ— = ( ·𝑠 β€˜π‘‡)
71, 2, 3, 4, 5, 6islmhm 20782 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalarβ€˜π‘‡) = 𝐾 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐸 (πΉβ€˜(π‘Ž Β· 𝑏)) = (π‘Ž Γ— (πΉβ€˜π‘)))))
87simprbi 495 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalarβ€˜π‘‡) = 𝐾 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐸 (πΉβ€˜(π‘Ž Β· 𝑏)) = (π‘Ž Γ— (πΉβ€˜π‘))))
98simp3d 1142 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐸 (πΉβ€˜(π‘Ž Β· 𝑏)) = (π‘Ž Γ— (πΉβ€˜π‘)))
10 fvoveq1 7434 . . . . 5 (π‘Ž = 𝑋 β†’ (πΉβ€˜(π‘Ž Β· 𝑏)) = (πΉβ€˜(𝑋 Β· 𝑏)))
11 oveq1 7418 . . . . 5 (π‘Ž = 𝑋 β†’ (π‘Ž Γ— (πΉβ€˜π‘)) = (𝑋 Γ— (πΉβ€˜π‘)))
1210, 11eqeq12d 2746 . . . 4 (π‘Ž = 𝑋 β†’ ((πΉβ€˜(π‘Ž Β· 𝑏)) = (π‘Ž Γ— (πΉβ€˜π‘)) ↔ (πΉβ€˜(𝑋 Β· 𝑏)) = (𝑋 Γ— (πΉβ€˜π‘))))
13 oveq2 7419 . . . . . 6 (𝑏 = π‘Œ β†’ (𝑋 Β· 𝑏) = (𝑋 Β· π‘Œ))
1413fveq2d 6894 . . . . 5 (𝑏 = π‘Œ β†’ (πΉβ€˜(𝑋 Β· 𝑏)) = (πΉβ€˜(𝑋 Β· π‘Œ)))
15 fveq2 6890 . . . . . 6 (𝑏 = π‘Œ β†’ (πΉβ€˜π‘) = (πΉβ€˜π‘Œ))
1615oveq2d 7427 . . . . 5 (𝑏 = π‘Œ β†’ (𝑋 Γ— (πΉβ€˜π‘)) = (𝑋 Γ— (πΉβ€˜π‘Œ)))
1714, 16eqeq12d 2746 . . . 4 (𝑏 = π‘Œ β†’ ((πΉβ€˜(𝑋 Β· 𝑏)) = (𝑋 Γ— (πΉβ€˜π‘)) ↔ (πΉβ€˜(𝑋 Β· π‘Œ)) = (𝑋 Γ— (πΉβ€˜π‘Œ))))
1812, 17rspc2v 3621 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐸) β†’ (βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐸 (πΉβ€˜(π‘Ž Β· 𝑏)) = (π‘Ž Γ— (πΉβ€˜π‘)) β†’ (πΉβ€˜(𝑋 Β· π‘Œ)) = (𝑋 Γ— (πΉβ€˜π‘Œ))))
199, 18syl5com 31 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐸) β†’ (πΉβ€˜(𝑋 Β· π‘Œ)) = (𝑋 Γ— (πΉβ€˜π‘Œ))))
20193impib 1114 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐸) β†’ (πΉβ€˜(𝑋 Β· π‘Œ)) = (𝑋 Γ— (πΉβ€˜π‘Œ)))
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:  islmhm2  20793  lmhmco  20798  lmhmplusg  20799  lmhmvsca  20800  lmhmf1o  20801  lmhmima  20802  lmhmpreima  20803  reslmhm  20807  reslmhm2  20808  reslmhm2b  20809  lmhmeql  20810  ipass  21417  lindfmm  21601  nmoleub2lem3  24862  nmoleub3  24866  lmhmimasvsca  32467  lmhmqusker  32808  mendassa  42238
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