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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 7901 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ V) |
5 | finnzfsuppd.3 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
6 | elsuppfn 8161 | . . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐷 ∈ V ∧ 𝑍 ∈ 𝑈) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) ≠ 𝑍))) | |
7 | 2, 4, 5, 6 | syl3anc 1370 | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) ≠ 𝑍))) |
8 | 7 | biimpa 476 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → (𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) ≠ 𝑍)) |
9 | 8 | simpld 494 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐷) |
10 | finnzfsuppd.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝐴 ∨ (𝐹‘𝑥) = 𝑍)) | |
11 | 9, 10 | syldan 590 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → (𝑥 ∈ 𝐴 ∨ (𝐹‘𝑥) = 𝑍)) |
12 | 8 | simprd 495 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → (𝐹‘𝑥) ≠ 𝑍) |
13 | 12 | neneqd 2944 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → ¬ (𝐹‘𝑥) = 𝑍) |
14 | 11, 13 | olcnd 874 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴) |
15 | 14 | ex 412 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) → 𝑥 ∈ 𝐴)) |
16 | 15 | ssrdv 3988 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝐴) |
17 | 1, 16 | ssfid 9273 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
18 | fnfun 6649 | . . . 4 ⊢ (𝐹 Fn 𝐷 → Fun 𝐹) | |
19 | 2, 18 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
20 | funisfsupp 9373 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) | |
21 | 19, 3, 5, 20 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) |
22 | 17, 21 | mpbird 257 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 Vcvv 3473 class class class wbr 5148 Fun wfun 6537 Fn wfn 6538 ‘cfv 6543 (class class class)co 7412 supp csupp 8151 Fincfn 8945 finSupp cfsupp 9367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-supp 8152 df-1o 8472 df-en 8946 df-fin 8949 df-fsupp 9368 |
This theorem is referenced by: mnringmulrcld 43450 |
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