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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 19363 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵) |
7 | 6 | 3expb 1100 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵) |
8 | 1, 7 | sylan 572 | . 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 4077 | . 2 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
13 | 8, 9, 10, 11, 11, 12 | off 7236 | 1 ⊢ (𝜑 → (𝐺 ∘𝑓 · 𝐻):𝐼⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 ∘𝑓 cof 7219 Basecbs 16329 Scalarcsca 16414 ·𝑠 cvsca 16415 LModclmod 19346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-lmod 19348 |
This theorem is referenced by: lcomfsupp 19386 frlmup2 20635 islindf4 20674 fedgmullem2 30611 |
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