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Mirrors > Home > MPE Home > Th. List > fczfsuppd | Structured version Visualization version GIF version |
Description: A constant function with value zero is finitely supported. (Contributed by AV, 30-Jun-2019.) |
Ref | Expression |
---|---|
fczfsuppd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
fczfsuppd.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
Ref | Expression |
---|---|
fczfsuppd | ⊢ (𝜑 → (𝐵 × {𝑍}) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fczfsuppd.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
2 | fnconstg 6607 | . . 3 ⊢ (𝑍 ∈ 𝑊 → (𝐵 × {𝑍}) Fn 𝐵) | |
3 | fnfun 6479 | . . 3 ⊢ ((𝐵 × {𝑍}) Fn 𝐵 → Fun (𝐵 × {𝑍})) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → Fun (𝐵 × {𝑍})) |
5 | fczsupp0 7935 | . . . 4 ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅ | |
6 | 0fin 8849 | . . . 4 ⊢ ∅ ∈ Fin | |
7 | 5, 6 | eqeltri 2834 | . . 3 ⊢ ((𝐵 × {𝑍}) supp 𝑍) ∈ Fin |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → ((𝐵 × {𝑍}) supp 𝑍) ∈ Fin) |
9 | fczfsuppd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
10 | snex 5324 | . . . 4 ⊢ {𝑍} ∈ V | |
11 | xpexg 7535 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑍} ∈ V) → (𝐵 × {𝑍}) ∈ V) | |
12 | 9, 10, 11 | sylancl 589 | . . 3 ⊢ (𝜑 → (𝐵 × {𝑍}) ∈ V) |
13 | isfsupp 8989 | . . 3 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐵 × {𝑍}) finSupp 𝑍 ↔ (Fun (𝐵 × {𝑍}) ∧ ((𝐵 × {𝑍}) supp 𝑍) ∈ Fin))) | |
14 | 12, 1, 13 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝐵 × {𝑍}) finSupp 𝑍 ↔ (Fun (𝐵 × {𝑍}) ∧ ((𝐵 × {𝑍}) supp 𝑍) ∈ Fin))) |
15 | 4, 8, 14 | mpbir2and 713 | 1 ⊢ (𝜑 → (𝐵 × {𝑍}) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2110 Vcvv 3408 ∅c0 4237 {csn 4541 class class class wbr 5053 × cxp 5549 Fun wfun 6374 Fn wfn 6375 (class class class)co 7213 supp csupp 7903 Fincfn 8626 finSupp cfsupp 8985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-supp 7904 df-en 8627 df-fin 8630 df-fsupp 8986 |
This theorem is referenced by: cantnf0 9290 cantnf 9308 dprdsubg 19411 tsms0 23039 tgptsmscls 23047 dchrptlem3 26147 elrspunidl 31320 |
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