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Mirrors > Home > MPE Home > Th. List > lcomfsupp | Structured version Visualization version GIF version |
Description: A linear-combination sum is finitely supported if the coefficients are. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by AV, 15-Jul-2019.) |
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) |
lcomfsupp.z | ⊢ 0 = (0g‘𝑊) |
lcomfsupp.y | ⊢ 𝑌 = (0g‘𝐹) |
lcomfsupp.j | ⊢ (𝜑 → 𝐺 finSupp 𝑌) |
Ref | Expression |
---|---|
lcomfsupp | ⊢ (𝜑 → (𝐺 ∘f · 𝐻) finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcomfsupp.j | . . . 4 ⊢ (𝜑 → 𝐺 finSupp 𝑌) | |
2 | 1 | fsuppimpd 9407 | . . 3 ⊢ (𝜑 → (𝐺 supp 𝑌) ∈ Fin) |
3 | lcomf.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | lcomf.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
5 | lcomf.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
6 | lcomf.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
7 | lcomf.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
8 | lcomf.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝐼⟶𝐾) | |
9 | lcomf.h | . . . . 5 ⊢ (𝜑 → 𝐻:𝐼⟶𝐵) | |
10 | lcomf.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | lcomf 20916 | . . . 4 ⊢ (𝜑 → (𝐺 ∘f · 𝐻):𝐼⟶𝐵) |
12 | eldifi 4141 | . . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌)) → 𝑥 ∈ 𝐼) | |
13 | 8 | ffnd 6738 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
14 | 13 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 Fn 𝐼) |
15 | 9 | ffnd 6738 | . . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn 𝐼) |
16 | 15 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐻 Fn 𝐼) |
17 | 10 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑉) |
18 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) | |
19 | fnfvof 7714 | . . . . . . 7 ⊢ (((𝐺 Fn 𝐼 ∧ 𝐻 Fn 𝐼) ∧ (𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼)) → ((𝐺 ∘f · 𝐻)‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑥))) | |
20 | 14, 16, 17, 18, 19 | syl22anc 839 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐺 ∘f · 𝐻)‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑥))) |
21 | 12, 20 | sylan2 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → ((𝐺 ∘f · 𝐻)‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑥))) |
22 | ssidd 4019 | . . . . . . 7 ⊢ (𝜑 → (𝐺 supp 𝑌) ⊆ (𝐺 supp 𝑌)) | |
23 | lcomfsupp.y | . . . . . . . . 9 ⊢ 𝑌 = (0g‘𝐹) | |
24 | 23 | fvexi 6921 | . . . . . . . 8 ⊢ 𝑌 ∈ V |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V) |
26 | 8, 22, 10, 25 | suppssr 8219 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → (𝐺‘𝑥) = 𝑌) |
27 | 26 | oveq1d 7446 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → ((𝐺‘𝑥) · (𝐻‘𝑥)) = (𝑌 · (𝐻‘𝑥))) |
28 | 9 | ffvelcdmda 7104 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐻‘𝑥) ∈ 𝐵) |
29 | lcomfsupp.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑊) | |
30 | 6, 3, 5, 23, 29 | lmod0vs 20910 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝐻‘𝑥) ∈ 𝐵) → (𝑌 · (𝐻‘𝑥)) = 0 ) |
31 | 7, 28, 30 | syl2an2r 685 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑌 · (𝐻‘𝑥)) = 0 ) |
32 | 12, 31 | sylan2 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → (𝑌 · (𝐻‘𝑥)) = 0 ) |
33 | 21, 27, 32 | 3eqtrd 2779 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → ((𝐺 ∘f · 𝐻)‘𝑥) = 0 ) |
34 | 11, 33 | suppss 8218 | . . 3 ⊢ (𝜑 → ((𝐺 ∘f · 𝐻) supp 0 ) ⊆ (𝐺 supp 𝑌)) |
35 | 2, 34 | ssfid 9299 | . 2 ⊢ (𝜑 → ((𝐺 ∘f · 𝐻) supp 0 ) ∈ Fin) |
36 | 13, 15, 10, 10 | offun 7711 | . . 3 ⊢ (𝜑 → Fun (𝐺 ∘f · 𝐻)) |
37 | ovexd 7466 | . . 3 ⊢ (𝜑 → (𝐺 ∘f · 𝐻) ∈ V) | |
38 | 29 | fvexi 6921 | . . . 4 ⊢ 0 ∈ V |
39 | 38 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
40 | funisfsupp 9405 | . . 3 ⊢ ((Fun (𝐺 ∘f · 𝐻) ∧ (𝐺 ∘f · 𝐻) ∈ V ∧ 0 ∈ V) → ((𝐺 ∘f · 𝐻) finSupp 0 ↔ ((𝐺 ∘f · 𝐻) supp 0 ) ∈ Fin)) | |
41 | 36, 37, 39, 40 | syl3anc 1370 | . 2 ⊢ (𝜑 → ((𝐺 ∘f · 𝐻) finSupp 0 ↔ ((𝐺 ∘f · 𝐻) supp 0 ) ∈ Fin)) |
42 | 35, 41 | mpbird 257 | 1 ⊢ (𝜑 → (𝐺 ∘f · 𝐻) finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 class class class wbr 5148 Fun wfun 6557 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 supp csupp 8184 Fincfn 8984 finSupp cfsupp 9399 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 LModclmod 20875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-supp 8185 df-1o 8505 df-en 8985 df-fin 8988 df-fsupp 9400 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-ring 20253 df-lmod 20877 |
This theorem is referenced by: islindf4 21876 fedgmullem2 33658 |
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