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Mirrors > Home > MPE Home > Th. List > fsuppmptif | Structured version Visualization version GIF version |
Description: A function mapping an argument to either a value of a finitely supported function or zero is finitely supported. (Contributed by AV, 6-Jun-2019.) |
Ref | Expression |
---|---|
fsuppmptif.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
fsuppmptif.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
fsuppmptif.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
fsuppmptif.s | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Ref | Expression |
---|---|
fsuppmptif | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6113 | . . . . 5 ⊢ (𝐹‘𝑘) ∈ V | |
2 | fsuppmptif.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
3 | 2 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑍 ∈ 𝑊) |
4 | ifexg 4107 | . . . . 5 ⊢ (((𝐹‘𝑘) ∈ V ∧ 𝑍 ∈ 𝑊) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) ∈ V) | |
5 | 1, 3, 4 | sylancr 694 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) ∈ V) |
6 | eqid 2610 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) | |
7 | 5, 6 | fmptd 6292 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)):𝐴⟶V) |
8 | ffun 5961 | . . 3 ⊢ ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)):𝐴⟶V → Fun (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍))) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → Fun (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍))) |
10 | fsuppmptif.s | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
11 | 10 | fsuppimpd 8165 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
12 | fsuppmptif.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
13 | ssid 3587 | . . . . . . . 8 ⊢ (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍) | |
14 | 13 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍)) |
15 | fsuppmptif.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
16 | 12, 14, 15, 2 | suppssr 7213 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑘) = 𝑍) |
17 | 16 | ifeq1d 4054 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) = if(𝑘 ∈ 𝐷, 𝑍, 𝑍)) |
18 | ifid 4075 | . . . . 5 ⊢ if(𝑘 ∈ 𝐷, 𝑍, 𝑍) = 𝑍 | |
19 | 17, 18 | syl6eq 2660 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) = 𝑍) |
20 | 19, 15 | suppss2 7216 | . . 3 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ⊆ (𝐹 supp 𝑍)) |
21 | ssfi 8065 | . . 3 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ∈ Fin) | |
22 | 11, 20, 21 | syl2anc 691 | . 2 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ∈ Fin) |
23 | mptexg 6389 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∈ V) | |
24 | 15, 23 | syl 17 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∈ V) |
25 | isfsupp 8162 | . . 3 ⊢ (((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍 ↔ (Fun (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∧ ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ∈ Fin))) | |
26 | 24, 2, 25 | syl2anc 691 | . 2 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍 ↔ (Fun (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∧ ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ∈ Fin))) |
27 | 9, 22, 26 | mpbir2and 959 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 supp csupp 7182 Fincfn 7841 finSupp cfsupp 8158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-supp 7183 df-er 7629 df-en 7842 df-fin 7845 df-fsupp 8159 |
This theorem is referenced by: cantnflem1d 8468 gsumzsplit 18150 |
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