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Theorem lcomf 20784
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 20761 . . . 4 ((π‘Š ∈ LMod ∧ π‘₯ ∈ 𝐾 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ Β· 𝑦) ∈ 𝐡)
763expb 1118 . . 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 4219 . 2 (𝐼 ∩ 𝐼) = 𝐼
138, 9, 10, 11, 11, 12off 7703 1 (πœ‘ β†’ (𝐺 ∘f Β· 𝐻):𝐼⟢𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420   ∘f cof 7683  Basecbs 17180  Scalarcsca 17236   ·𝑠 cvsca 17237  LModclmod 20743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7685  df-lmod 20745
This theorem is referenced by:  lcomfsupp  20785  frlmup2  21733  islindf4  21772  fedgmullem2  33328
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