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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 9409 | . . 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 20899 | . . . 4 ⊢ (𝜑 → (𝐺 ∘f · 𝐻):𝐼⟶𝐵) |
| 12 | eldifi 4131 | . . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌)) → 𝑥 ∈ 𝐼) | |
| 13 | 8 | ffnd 6737 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
| 14 | 13 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 Fn 𝐼) |
| 15 | 9 | ffnd 6737 | . . . . . . . 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 4007 | . . . . . . 7 ⊢ (𝜑 → (𝐺 supp 𝑌) ⊆ (𝐺 supp 𝑌)) | |
| 23 | lcomfsupp.y | . . . . . . . . 9 ⊢ 𝑌 = (0g‘𝐹) | |
| 24 | 23 | fvexi 6920 | . . . . . . . 8 ⊢ 𝑌 ∈ V |
| 25 | 24 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V) |
| 26 | 8, 22, 10, 25 | suppssr 8220 | . . . . . 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 20893 | . . . . . . 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 2781 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ (𝐺 supp 𝑌))) → ((𝐺 ∘f · 𝐻)‘𝑥) = 0 ) |
| 34 | 11, 33 | suppss 8219 | . . 3 ⊢ (𝜑 → ((𝐺 ∘f · 𝐻) supp 0 ) ⊆ (𝐺 supp 𝑌)) |
| 35 | 2, 34 | ssfid 9301 | . 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 6920 | . . . 4 ⊢ 0 ∈ V |
| 39 | 38 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
| 40 | funisfsupp 9407 | . . 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 1373 | . 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 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 class class class wbr 5143 Fun wfun 6555 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 supp csupp 8185 Fincfn 8985 finSupp cfsupp 9401 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 LModclmod 20858 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-supp 8186 df-1o 8506 df-en 8986 df-fin 8989 df-fsupp 9402 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-ring 20232 df-lmod 20860 |
| This theorem is referenced by: islindf4 21858 fedgmullem2 33681 |
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