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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 6093 | . . 3 ⊢ (𝑍 ∈ 𝑊 → (𝐵 × {𝑍}) Fn 𝐵) | |
3 | fnfun 5988 | . . 3 ⊢ ((𝐵 × {𝑍}) Fn 𝐵 → Fun (𝐵 × {𝑍})) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → Fun (𝐵 × {𝑍})) |
5 | fczsupp0 7324 | . . . 4 ⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅ | |
6 | 0fin 8188 | . . . 4 ⊢ ∅ ∈ Fin | |
7 | 5, 6 | eqeltri 2697 | . . 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 4908 | . . . 4 ⊢ {𝑍} ∈ V | |
11 | xpexg 6960 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ {𝑍} ∈ V) → (𝐵 × {𝑍}) ∈ V) | |
12 | 9, 10, 11 | sylancl 694 | . . 3 ⊢ (𝜑 → (𝐵 × {𝑍}) ∈ V) |
13 | isfsupp 8279 | . . 3 ⊢ (((𝐵 × {𝑍}) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐵 × {𝑍}) finSupp 𝑍 ↔ (Fun (𝐵 × {𝑍}) ∧ ((𝐵 × {𝑍}) supp 𝑍) ∈ Fin))) | |
14 | 12, 1, 13 | syl2anc 693 | . 2 ⊢ (𝜑 → ((𝐵 × {𝑍}) finSupp 𝑍 ↔ (Fun (𝐵 × {𝑍}) ∧ ((𝐵 × {𝑍}) supp 𝑍) ∈ Fin))) |
15 | 4, 8, 14 | mpbir2and 957 | 1 ⊢ (𝜑 → (𝐵 × {𝑍}) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 {csn 4177 class class class wbr 4653 × cxp 5112 Fun wfun 5882 Fn wfn 5883 (class class class)co 6650 supp csupp 7295 Fincfn 7955 finSupp cfsupp 8275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-supp 7296 df-en 7956 df-fin 7959 df-fsupp 8276 |
This theorem is referenced by: cantnf0 8572 cantnf 8590 dprdsubg 18423 tsms0 21945 tgptsmscls 21953 dchrptlem3 24991 |
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