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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 9271 | . . 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 20314 | . . . 4 ⊢ (𝜑 → (𝐺 ∘f · 𝐻):𝐼⟶𝐵) |
12 | eldifi 4085 | . . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌)) → 𝑥 ∈ 𝐼) | |
13 | 8 | ffnd 6667 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
14 | 13 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 Fn 𝐼) |
15 | 9 | ffnd 6667 | . . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn 𝐼) |
16 | 15 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐻 Fn 𝐼) |
17 | 10 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑉) |
18 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) | |
19 | fnfvof 7627 | . . . . . . 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 838 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐺 ∘f · 𝐻)‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑥))) |
21 | 12, 20 | sylan2 594 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → ((𝐺 ∘f · 𝐻)‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑥))) |
22 | ssidd 3966 | . . . . . . 7 ⊢ (𝜑 → (𝐺 supp 𝑌) ⊆ (𝐺 supp 𝑌)) | |
23 | lcomfsupp.y | . . . . . . . . 9 ⊢ 𝑌 = (0g‘𝐹) | |
24 | 23 | fvexi 6854 | . . . . . . . 8 ⊢ 𝑌 ∈ V |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V) |
26 | 8, 22, 10, 25 | suppssr 8120 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → (𝐺‘𝑥) = 𝑌) |
27 | 26 | oveq1d 7367 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → ((𝐺‘𝑥) · (𝐻‘𝑥)) = (𝑌 · (𝐻‘𝑥))) |
28 | 9 | ffvelcdmda 7032 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐻‘𝑥) ∈ 𝐵) |
29 | lcomfsupp.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑊) | |
30 | 6, 3, 5, 23, 29 | lmod0vs 20308 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝐻‘𝑥) ∈ 𝐵) → (𝑌 · (𝐻‘𝑥)) = 0 ) |
31 | 7, 28, 30 | syl2an2r 684 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑌 · (𝐻‘𝑥)) = 0 ) |
32 | 12, 31 | sylan2 594 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → (𝑌 · (𝐻‘𝑥)) = 0 ) |
33 | 21, 27, 32 | 3eqtrd 2782 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → ((𝐺 ∘f · 𝐻)‘𝑥) = 0 ) |
34 | 11, 33 | suppss 8118 | . . 3 ⊢ (𝜑 → ((𝐺 ∘f · 𝐻) supp 0 ) ⊆ (𝐺 supp 𝑌)) |
35 | 2, 34 | ssfid 9170 | . 2 ⊢ (𝜑 → ((𝐺 ∘f · 𝐻) supp 0 ) ∈ Fin) |
36 | 13, 15, 10, 10 | offun 7624 | . . 3 ⊢ (𝜑 → Fun (𝐺 ∘f · 𝐻)) |
37 | ovexd 7387 | . . 3 ⊢ (𝜑 → (𝐺 ∘f · 𝐻) ∈ V) | |
38 | 29 | fvexi 6854 | . . . 4 ⊢ 0 ∈ V |
39 | 38 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
40 | funisfsupp 9269 | . . 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 1372 | . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∖ cdif 3906 class class class wbr 5104 Fun wfun 6488 Fn wfn 6489 ⟶wf 6490 ‘cfv 6494 (class class class)co 7352 ∘f cof 7608 supp csupp 8085 Fincfn 8842 finSupp cfsupp 9264 Basecbs 17043 Scalarcsca 17096 ·𝑠 cvsca 17097 0gc0g 17281 LModclmod 20275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7665 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7610 df-om 7796 df-supp 8086 df-1o 8405 df-en 8843 df-fin 8846 df-fsupp 9265 df-0g 17283 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-grp 18711 df-ring 19920 df-lmod 20277 |
This theorem is referenced by: islindf4 21197 fedgmullem2 32139 |
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