Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 18880 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵) |
7 | 6 | 3expb 1266 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥 · 𝑦) ∈ 𝐵) |
8 | 1, 7 | sylan 488 | . 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 3822 | . 2 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
13 | 8, 9, 10, 11, 11, 12 | off 6912 | 1 ⊢ (𝜑 → (𝐺 ∘𝑓 · 𝐻):𝐼⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 Basecbs 15857 Scalarcsca 15944 ·𝑠 cvsca 15945 LModclmod 18863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-lmod 18865 |
This theorem is referenced by: lcomfsupp 18903 frlmup2 20138 islindf4 20177 |
Copyright terms: Public domain | W3C validator |