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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > finnzfsuppd | Structured version Visualization version GIF version |
Description: If a function is zero outside of a finite set, it has finite support. (Contributed by Rohan Ridenour, 13-May-2024.) |
Ref | Expression |
---|---|
finnzfsuppd.1 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
finnzfsuppd.2 | ⊢ (𝜑 → 𝐹 Fn 𝐷) |
finnzfsuppd.3 | ⊢ (𝜑 → 𝑍 ∈ 𝑈) |
finnzfsuppd.4 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
finnzfsuppd.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝐴 ∨ (𝐹‘𝑥) = 𝑍)) |
Ref | Expression |
---|---|
finnzfsuppd | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | finnzfsuppd.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | finnzfsuppd.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn 𝐷) | |
3 | finnzfsuppd.1 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
4 | 3, 2 | fndmexd 7908 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ V) |
5 | finnzfsuppd.3 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
6 | elsuppfn 8171 | . . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐷 ∈ V ∧ 𝑍 ∈ 𝑈) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) ≠ 𝑍))) | |
7 | 2, 4, 5, 6 | syl3anc 1368 | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) ≠ 𝑍))) |
8 | 7 | biimpa 475 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → (𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) ≠ 𝑍)) |
9 | 8 | simpld 493 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐷) |
10 | finnzfsuppd.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝐴 ∨ (𝐹‘𝑥) = 𝑍)) | |
11 | 9, 10 | syldan 589 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → (𝑥 ∈ 𝐴 ∨ (𝐹‘𝑥) = 𝑍)) |
12 | 8 | simprd 494 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → (𝐹‘𝑥) ≠ 𝑍) |
13 | 12 | neneqd 2935 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → ¬ (𝐹‘𝑥) = 𝑍) |
14 | 11, 13 | olcnd 875 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴) |
15 | 14 | ex 411 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) → 𝑥 ∈ 𝐴)) |
16 | 15 | ssrdv 3978 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝐴) |
17 | 1, 16 | ssfid 9288 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
18 | fnfun 6648 | . . . 4 ⊢ (𝐹 Fn 𝐷 → Fun 𝐹) | |
19 | 2, 18 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
20 | funisfsupp 9389 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) | |
21 | 19, 3, 5, 20 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) |
22 | 17, 21 | mpbird 256 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 Vcvv 3463 class class class wbr 5143 Fun wfun 6536 Fn wfn 6537 ‘cfv 6542 (class class class)co 7415 supp csupp 8161 Fincfn 8960 finSupp cfsupp 9383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-supp 8162 df-1o 8483 df-en 8961 df-fin 8964 df-fsupp 9384 |
This theorem is referenced by: mnringmulrcld 43729 |
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