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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 6450 | . . . . 5 ⊢ (𝐹‘𝑘) ∈ V | |
2 | fsuppmptif.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
3 | 2 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑍 ∈ 𝑊) |
4 | ifexg 4355 | . . . . 5 ⊢ (((𝐹‘𝑘) ∈ V ∧ 𝑍 ∈ 𝑊) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) ∈ V) | |
5 | 1, 3, 4 | sylancr 581 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) ∈ V) |
6 | 5 | fmpttd 6639 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)):𝐴⟶V) |
7 | 6 | ffund 6286 | . 2 ⊢ (𝜑 → Fun (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍))) |
8 | fsuppmptif.s | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
9 | 8 | fsuppimpd 8557 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
10 | fsuppmptif.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
11 | ssidd 3849 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍)) | |
12 | fsuppmptif.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
13 | 10, 11, 12, 2 | suppssr 7596 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑘) = 𝑍) |
14 | 13 | ifeq1d 4326 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) = if(𝑘 ∈ 𝐷, 𝑍, 𝑍)) |
15 | ifid 4347 | . . . . 5 ⊢ if(𝑘 ∈ 𝐷, 𝑍, 𝑍) = 𝑍 | |
16 | 14, 15 | syl6eq 2877 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍) = 𝑍) |
17 | 16, 12 | suppss2 7599 | . . 3 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ⊆ (𝐹 supp 𝑍)) |
18 | ssfi 8455 | . . 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) | |
19 | 9, 17, 18 | syl2anc 579 | . 2 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) supp 𝑍) ∈ Fin) |
20 | mptexg 6745 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∈ V) | |
21 | 12, 20 | syl 17 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) ∈ V) |
22 | isfsupp 8554 | . . 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))) | |
23 | 21, 2, 22 | syl2anc 579 | . 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))) |
24 | 7, 19, 23 | mpbir2and 704 | 1 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 𝑍)) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2164 Vcvv 3414 ∖ cdif 3795 ⊆ wss 3798 ifcif 4308 class class class wbr 4875 ↦ cmpt 4954 Fun wfun 6121 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 supp csupp 7564 Fincfn 8228 finSupp cfsupp 8550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-supp 7565 df-er 8014 df-en 8229 df-fin 8232 df-fsupp 8551 |
This theorem is referenced by: cantnflem1d 8869 gsumzsplit 18687 |
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