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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 7844 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ V) |
5 | finnzfsuppd.3 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
6 | elsuppfn 8103 | . . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐷 ∈ V ∧ 𝑍 ∈ 𝑈) → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) ≠ 𝑍))) | |
7 | 2, 4, 5, 6 | syl3anc 1372 | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) ↔ (𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) ≠ 𝑍))) |
8 | 7 | biimpa 478 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → (𝑥 ∈ 𝐷 ∧ (𝐹‘𝑥) ≠ 𝑍)) |
9 | 8 | simpld 496 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐷) |
10 | finnzfsuppd.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝐴 ∨ (𝐹‘𝑥) = 𝑍)) | |
11 | 9, 10 | syldan 592 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → (𝑥 ∈ 𝐴 ∨ (𝐹‘𝑥) = 𝑍)) |
12 | 8 | simprd 497 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → (𝐹‘𝑥) ≠ 𝑍) |
13 | 12 | neneqd 2949 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → ¬ (𝐹‘𝑥) = 𝑍) |
14 | 11, 13 | olcnd 876 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴) |
15 | 14 | ex 414 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐹 supp 𝑍) → 𝑥 ∈ 𝐴)) |
16 | 15 | ssrdv 3951 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝐴) |
17 | 1, 16 | ssfid 9212 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
18 | fnfun 6603 | . . . 4 ⊢ (𝐹 Fn 𝐷 → Fun 𝐹) | |
19 | 2, 18 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
20 | funisfsupp 9311 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝐹 finSupp 𝑍 ↔ (𝐹 supp 𝑍) ∈ Fin)) | |
21 | 19, 3, 5, 20 | syl3anc 1372 | . 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 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 Vcvv 3446 class class class wbr 5106 Fun wfun 6491 Fn wfn 6492 ‘cfv 6497 (class class class)co 7358 supp csupp 8093 Fincfn 8884 finSupp cfsupp 9306 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-supp 8094 df-1o 8413 df-en 8885 df-fin 8888 df-fsupp 9307 |
This theorem is referenced by: mnringmulrcld 42515 |
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