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Theorem lmhmlin 20511
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 2733 . . . . . 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 20503 . . . . 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 498 . . . 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 1145 . . 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 7381 . . . . 5 (π‘Ž = 𝑋 β†’ (πΉβ€˜(π‘Ž Β· 𝑏)) = (πΉβ€˜(𝑋 Β· 𝑏)))
11 oveq1 7365 . . . . 5 (π‘Ž = 𝑋 β†’ (π‘Ž Γ— (πΉβ€˜π‘)) = (𝑋 Γ— (πΉβ€˜π‘)))
1210, 11eqeq12d 2749 . . . 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 7366 . . . . . 6 (𝑏 = π‘Œ β†’ (𝑋 Β· 𝑏) = (𝑋 Β· π‘Œ))
1413fveq2d 6847 . . . . 5 (𝑏 = π‘Œ β†’ (πΉβ€˜(𝑋 Β· 𝑏)) = (πΉβ€˜(𝑋 Β· π‘Œ)))
15 fveq2 6843 . . . . . 6 (𝑏 = π‘Œ β†’ (πΉβ€˜π‘) = (πΉβ€˜π‘Œ))
1615oveq2d 7374 . . . . 5 (𝑏 = π‘Œ β†’ (𝑋 Γ— (πΉβ€˜π‘)) = (𝑋 Γ— (πΉβ€˜π‘Œ)))
1714, 16eqeq12d 2749 . . . 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 3589 . . 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 1117 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Scalarcsca 17141   ·𝑠 cvsca 17142   GrpHom cghm 19010  LModclmod 20336   LMHom clmhm 20495
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 5257  ax-nul 5264  ax-pr 5385
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-lmhm 20498
This theorem is referenced by:  islmhm2  20514  lmhmco  20519  lmhmplusg  20520  lmhmvsca  20521  lmhmf1o  20522  lmhmima  20523  lmhmpreima  20524  reslmhm  20528  reslmhm2  20529  reslmhm2b  20530  lmhmeql  20531  ipass  21065  lindfmm  21249  nmoleub2lem3  24494  nmoleub3  24498  mendassa  41564
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