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Mirrors > Home > MPE Home > Th. List > lcomf | Structured version Visualization version GIF version |
Description: A linear-combination sum is a function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Ref | Expression |
---|---|
lcomf.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lcomf.k | ⊢ 𝐾 = (Base‘𝐹) |
lcomf.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lcomf.b | ⊢ 𝐵 = (Base‘𝑊) |
lcomf.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lcomf.g | ⊢ (𝜑 → 𝐺:𝐼⟶𝐾) |
lcomf.h | ⊢ (𝜑 → 𝐻:𝐼⟶𝐵) |
lcomf.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
Ref | Expression |
---|---|
lcomf | ⊢ (𝜑 → (𝐺 ∘f · 𝐻):𝐼⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcomf.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lcomf.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
3 | lcomf.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | lcomf.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
5 | lcomf.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
6 | 2, 3, 4, 5 | lmodvscl 19870 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵) |
7 | 6 | 3expb 1122 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵) |
8 | 1, 7 | sylan 583 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵) |
9 | lcomf.g | . 2 ⊢ (𝜑 → 𝐺:𝐼⟶𝐾) | |
10 | lcomf.h | . 2 ⊢ (𝜑 → 𝐻:𝐼⟶𝐵) | |
11 | lcomf.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
12 | inidm 4119 | . 2 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
13 | 8, 9, 10, 11, 11, 12 | off 7464 | 1 ⊢ (𝜑 → (𝐺 ∘f · 𝐻):𝐼⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ∘f cof 7445 Basecbs 16666 Scalarcsca 16752 ·𝑠 cvsca 16753 LModclmod 19853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-lmod 19855 |
This theorem is referenced by: lcomfsupp 19893 frlmup2 20715 islindf4 20754 fedgmullem2 31379 |
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