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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 6731 | . . 3 ⊢ (𝑍 ∈ 𝑊 → (𝐵 × {𝑍}) Fn 𝐵) | |
3 | fnfun 6603 | . . 3 ⊢ ((𝐵 × {𝑍}) Fn 𝐵 → Fun (𝐵 × {𝑍})) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → Fun (𝐵 × {𝑍})) |
5 | fczsupp0 8125 | . . . 4 ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅ | |
6 | 0fin 9116 | . . . 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 5389 | . . . 4 ⊢ {𝑍} ∈ V | |
11 | xpexg 7685 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑍} ∈ V) → (𝐵 × {𝑍}) ∈ V) | |
12 | 9, 10, 11 | sylancl 587 | . . 3 ⊢ (𝜑 → (𝐵 × {𝑍}) ∈ V) |
13 | isfsupp 9310 | . . 3 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐵 × {𝑍}) finSupp 𝑍 ↔ (Fun (𝐵 × {𝑍}) ∧ ((𝐵 × {𝑍}) supp 𝑍) ∈ Fin))) | |
14 | 12, 1, 13 | syl2anc 585 | . 2 ⊢ (𝜑 → ((𝐵 × {𝑍}) finSupp 𝑍 ↔ (Fun (𝐵 × {𝑍}) ∧ ((𝐵 × {𝑍}) supp 𝑍) ∈ Fin))) |
15 | 4, 8, 14 | mpbir2and 712 | 1 ⊢ (𝜑 → (𝐵 × {𝑍}) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 Vcvv 3446 ∅c0 4283 {csn 4587 class class class wbr 5106 × cxp 5632 Fun wfun 6491 Fn wfn 6492 (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-pow 5321 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-en 8885 df-fin 8888 df-fsupp 9307 |
This theorem is referenced by: cantnf0 9612 cantnf 9630 dprdsubg 19804 tsms0 23496 tgptsmscls 23504 dchrptlem3 26617 elrspunidl 32206 cantnfresb 41661 naddcnffo 41681 naddcnfid1 41684 naddcnfid2 41685 |
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