| 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 6796 | . . 3 ⊢ (𝑍 ∈ 𝑊 → (𝐵 × {𝑍}) Fn 𝐵) | |
| 3 | fnfun 6668 | . . 3 ⊢ ((𝐵 × {𝑍}) Fn 𝐵 → Fun (𝐵 × {𝑍})) | |
| 4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → Fun (𝐵 × {𝑍})) |
| 5 | fczsupp0 8218 | . . . 4 ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅ | |
| 6 | 0fi 9082 | . . . 4 ⊢ ∅ ∈ Fin | |
| 7 | 5, 6 | eqeltri 2837 | . . 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 5436 | . . . 4 ⊢ {𝑍} ∈ V | |
| 11 | xpexg 7770 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑍} ∈ V) → (𝐵 × {𝑍}) ∈ V) | |
| 12 | 9, 10, 11 | sylancl 586 | . . 3 ⊢ (𝜑 → (𝐵 × {𝑍}) ∈ V) |
| 13 | isfsupp 9405 | . . 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 2108 Vcvv 3480 ∅c0 4333 {csn 4626 class class class wbr 5143 × cxp 5683 Fun wfun 6555 Fn wfn 6556 (class class class)co 7431 supp csupp 8185 Fincfn 8985 finSupp cfsupp 9401 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8186 df-en 8986 df-fin 8989 df-fsupp 9402 |
| This theorem is referenced by: cantnf0 9715 cantnf 9733 dprdsubg 20044 tsms0 24150 tgptsmscls 24158 dchrptlem3 27310 elrgspnlem1 33246 elrspunidl 33456 cantnfresb 43337 naddcnffo 43377 naddcnfid1 43380 naddcnfid2 43381 |
| Copyright terms: Public domain | W3C validator |