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Theorem lcomf 20743
Description: A linear-combination sum is a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Hypotheses
Ref Expression
lcomf.f 𝐹 = (Scalarβ€˜π‘Š)
lcomf.k 𝐾 = (Baseβ€˜πΉ)
lcomf.s Β· = ( ·𝑠 β€˜π‘Š)
lcomf.b 𝐡 = (Baseβ€˜π‘Š)
lcomf.w (πœ‘ β†’ π‘Š ∈ LMod)
lcomf.g (πœ‘ β†’ 𝐺:𝐼⟢𝐾)
lcomf.h (πœ‘ β†’ 𝐻:𝐼⟢𝐡)
lcomf.i (πœ‘ β†’ 𝐼 ∈ 𝑉)
Assertion
Ref Expression
lcomf (πœ‘ β†’ (𝐺 ∘f Β· 𝐻):𝐼⟢𝐡)

Proof of Theorem lcomf
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lcomf.w . . 3 (πœ‘ β†’ π‘Š ∈ LMod)
2 lcomf.b . . . . 5 𝐡 = (Baseβ€˜π‘Š)
3 lcomf.f . . . . 5 𝐹 = (Scalarβ€˜π‘Š)
4 lcomf.s . . . . 5 Β· = ( ·𝑠 β€˜π‘Š)
5 lcomf.k . . . . 5 𝐾 = (Baseβ€˜πΉ)
62, 3, 4, 5lmodvscl 20720 . . . 4 ((π‘Š ∈ LMod ∧ π‘₯ ∈ 𝐾 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ Β· 𝑦) ∈ 𝐡)
763expb 1117 . . 3 ((π‘Š ∈ LMod ∧ (π‘₯ ∈ 𝐾 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯ Β· 𝑦) ∈ 𝐡)
81, 7sylan 579 . 2 ((πœ‘ ∧ (π‘₯ ∈ 𝐾 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯ Β· 𝑦) ∈ 𝐡)
9 lcomf.g . 2 (πœ‘ β†’ 𝐺:𝐼⟢𝐾)
10 lcomf.h . 2 (πœ‘ β†’ 𝐻:𝐼⟢𝐡)
11 lcomf.i . 2 (πœ‘ β†’ 𝐼 ∈ 𝑉)
12 inidm 4211 . 2 (𝐼 ∩ 𝐼) = 𝐼
138, 9, 10, 11, 11, 12off 7682 1 (πœ‘ β†’ (𝐺 ∘f Β· 𝐻):𝐼⟢𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402   ∘f cof 7662  Basecbs 17149  Scalarcsca 17205   ·𝑠 cvsca 17206  LModclmod 20702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-of 7664  df-lmod 20704
This theorem is referenced by:  lcomfsupp  20744  frlmup2  21683  islindf4  21722  fedgmullem2  33223
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