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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 6797 | . . 3 ⊢ (𝑍 ∈ 𝑊 → (𝐵 × {𝑍}) Fn 𝐵) | |
3 | fnfun 6669 | . . 3 ⊢ ((𝐵 × {𝑍}) Fn 𝐵 → Fun (𝐵 × {𝑍})) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → Fun (𝐵 × {𝑍})) |
5 | fczsupp0 8217 | . . . 4 ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅ | |
6 | 0fi 9081 | . . . 4 ⊢ ∅ ∈ Fin | |
7 | 5, 6 | eqeltri 2835 | . . 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 5442 | . . . 4 ⊢ {𝑍} ∈ V | |
11 | xpexg 7769 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑍} ∈ V) → (𝐵 × {𝑍}) ∈ V) | |
12 | 9, 10, 11 | sylancl 586 | . . 3 ⊢ (𝜑 → (𝐵 × {𝑍}) ∈ V) |
13 | isfsupp 9403 | . . 3 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐵 × {𝑍}) finSupp 𝑍 ↔ (Fun (𝐵 × {𝑍}) ∧ ((𝐵 × {𝑍}) supp 𝑍) ∈ Fin))) | |
14 | 12, 1, 13 | syl2anc 584 | . 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 206 ∧ wa 395 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 class class class wbr 5148 × cxp 5687 Fun wfun 6557 Fn wfn 6558 (class class class)co 7431 supp csupp 8184 Fincfn 8984 finSupp cfsupp 9399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8185 df-en 8985 df-fin 8988 df-fsupp 9400 |
This theorem is referenced by: cantnf0 9713 cantnf 9731 dprdsubg 20059 tsms0 24166 tgptsmscls 24174 dchrptlem3 27325 elrgspnlem1 33232 elrspunidl 33436 cantnfresb 43314 naddcnffo 43354 naddcnfid1 43357 naddcnfid2 43358 |
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