![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6780 | . . 3 ⊢ (𝑍 ∈ 𝑊 → (𝐵 × {𝑍}) Fn 𝐵) | |
3 | fnfun 6649 | . . 3 ⊢ ((𝐵 × {𝑍}) Fn 𝐵 → Fun (𝐵 × {𝑍})) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → Fun (𝐵 × {𝑍})) |
5 | fczsupp0 8196 | . . . 4 ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅ | |
6 | 0fin 9194 | . . . 4 ⊢ ∅ ∈ Fin | |
7 | 5, 6 | eqeltri 2821 | . . 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 5427 | . . . 4 ⊢ {𝑍} ∈ V | |
11 | xpexg 7750 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑍} ∈ V) → (𝐵 × {𝑍}) ∈ V) | |
12 | 9, 10, 11 | sylancl 584 | . . 3 ⊢ (𝜑 → (𝐵 × {𝑍}) ∈ V) |
13 | isfsupp 9389 | . . 3 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐵 × {𝑍}) finSupp 𝑍 ↔ (Fun (𝐵 × {𝑍}) ∧ ((𝐵 × {𝑍}) supp 𝑍) ∈ Fin))) | |
14 | 12, 1, 13 | syl2anc 582 | . 2 ⊢ (𝜑 → ((𝐵 × {𝑍}) finSupp 𝑍 ↔ (Fun (𝐵 × {𝑍}) ∧ ((𝐵 × {𝑍}) supp 𝑍) ∈ Fin))) |
15 | 4, 8, 14 | mpbir2and 711 | 1 ⊢ (𝜑 → (𝐵 × {𝑍}) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 Vcvv 3463 ∅c0 4318 {csn 4624 class class class wbr 5143 × cxp 5670 Fun wfun 6537 Fn wfn 6538 (class class class)co 7416 supp csupp 8163 Fincfn 8962 finSupp cfsupp 9385 |
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-pow 5359 ax-pr 5423 ax-un 7738 |
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 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 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 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-supp 8164 df-en 8963 df-fin 8966 df-fsupp 9386 |
This theorem is referenced by: cantnf0 9698 cantnf 9716 dprdsubg 19985 tsms0 24064 tgptsmscls 24072 dchrptlem3 27217 elrspunidl 33193 cantnfresb 42818 naddcnffo 42858 naddcnfid1 42861 naddcnfid2 42862 |
Copyright terms: Public domain | W3C validator |