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Theorem lmhmlin 19736
Description: A homomorphism of left modules is 𝐾-linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
lmhmlin.k 𝐾 = (Scalar‘𝑆)
lmhmlin.b 𝐵 = (Base‘𝐾)
lmhmlin.e 𝐸 = (Base‘𝑆)
lmhmlin.m · = ( ·𝑠𝑆)
lmhmlin.n × = ( ·𝑠𝑇)
Assertion
Ref Expression
lmhmlin ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝐵𝑌𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌)))

Proof of Theorem lmhmlin
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmlin.k . . . . . 6 𝐾 = (Scalar‘𝑆)
2 eqid 2818 . . . . . 6 (Scalar‘𝑇) = (Scalar‘𝑇)
3 lmhmlin.b . . . . . 6 𝐵 = (Base‘𝐾)
4 lmhmlin.e . . . . . 6 𝐸 = (Base‘𝑆)
5 lmhmlin.m . . . . . 6 · = ( ·𝑠𝑆)
6 lmhmlin.n . . . . . 6 × = ( ·𝑠𝑇)
71, 2, 3, 4, 5, 6islmhm 19728 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = 𝐾 ∧ ∀𝑎𝐵𝑏𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏)))))
87simprbi 497 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = 𝐾 ∧ ∀𝑎𝐵𝑏𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏))))
98simp3d 1136 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → ∀𝑎𝐵𝑏𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏)))
10 fvoveq1 7168 . . . . 5 (𝑎 = 𝑋 → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑋 · 𝑏)))
11 oveq1 7152 . . . . 5 (𝑎 = 𝑋 → (𝑎 × (𝐹𝑏)) = (𝑋 × (𝐹𝑏)))
1210, 11eqeq12d 2834 . . . 4 (𝑎 = 𝑋 → ((𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏)) ↔ (𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹𝑏))))
13 oveq2 7153 . . . . . 6 (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
1413fveq2d 6667 . . . . 5 (𝑏 = 𝑌 → (𝐹‘(𝑋 · 𝑏)) = (𝐹‘(𝑋 · 𝑌)))
15 fveq2 6663 . . . . . 6 (𝑏 = 𝑌 → (𝐹𝑏) = (𝐹𝑌))
1615oveq2d 7161 . . . . 5 (𝑏 = 𝑌 → (𝑋 × (𝐹𝑏)) = (𝑋 × (𝐹𝑌)))
1714, 16eqeq12d 2834 . . . 4 (𝑏 = 𝑌 → ((𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹𝑏)) ↔ (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌))))
1812, 17rspc2v 3630 . . 3 ((𝑋𝐵𝑌𝐸) → (∀𝑎𝐵𝑏𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹𝑏)) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌))))
199, 18syl5com 31 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑋𝐵𝑌𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌))))
20193impib 1108 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝐵𝑌𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145  Basecbs 16471  Scalarcsca 16556   ·𝑠 cvsca 16557   GrpHom cghm 18293  LModclmod 19563   LMHom clmhm 19720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-lmhm 19723
This theorem is referenced by:  islmhm2  19739  lmhmco  19744  lmhmplusg  19745  lmhmvsca  19746  lmhmf1o  19747  lmhmima  19748  lmhmpreima  19749  reslmhm  19753  reslmhm2  19754  reslmhm2b  19755  lmhmeql  19756  ipass  20717  lindfmm  20899  nmoleub2lem3  23646  nmoleub3  23650  mendassa  39672
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