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Mirrors > Home > MPE Home > Th. List > suppssfifsupp | Structured version Visualization version GIF version |
Description: If the support of a function is a subset of a finite set, the function is finitely supported. (Contributed by AV, 15-Jul-2019.) |
Ref | Expression |
---|---|
suppssfifsupp | ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ (𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹)) → 𝐺 finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssfi 9173 | . . 3 ⊢ ((𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹) → (𝐺 supp 𝑍) ∈ Fin) | |
2 | 1 | adantl 483 | . 2 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ (𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹)) → (𝐺 supp 𝑍) ∈ Fin) |
3 | 3ancoma 1099 | . . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ↔ (Fun 𝐺 ∧ 𝐺 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) | |
4 | 3 | biimpi 215 | . . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) → (Fun 𝐺 ∧ 𝐺 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) |
5 | 4 | adantr 482 | . . 3 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ (𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹)) → (Fun 𝐺 ∧ 𝐺 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) |
6 | funisfsupp 9367 | . . 3 ⊢ ((Fun 𝐺 ∧ 𝐺 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ (𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹)) → (𝐺 finSupp 𝑍 ↔ (𝐺 supp 𝑍) ∈ Fin)) |
8 | 2, 7 | mpbird 257 | 1 ⊢ (((𝐺 ∈ 𝑉 ∧ Fun 𝐺 ∧ 𝑍 ∈ 𝑊) ∧ (𝐹 ∈ Fin ∧ (𝐺 supp 𝑍) ⊆ 𝐹)) → 𝐺 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 ⊆ wss 3949 class class class wbr 5149 Fun wfun 6538 (class class class)co 7409 supp csupp 8146 Fincfn 8939 finSupp cfsupp 9361 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-om 7856 df-1o 8466 df-en 8940 df-fin 8943 df-fsupp 9362 |
This theorem is referenced by: fsuppsssupp 9379 fsfnn0gsumfsffz 19851 mptscmfsupp0 20537 uvcff 21346 uvcresum 21348 frlmup1 21353 psrass1lemOLD 21493 psrass1lem 21496 psrlidm 21523 psrridm 21524 psrass1 21525 psrass23l 21528 psrcom 21529 psrass23 21530 mvrcl 21551 mplsubrglem 21563 mplsubrg 21564 mplmon 21590 mplmonmul 21591 mplcoe1 21592 mplcoe5 21595 mplbas2 21597 psrbagev1 21638 psrbagev1OLD 21639 evlslem2 21642 evlslem3 21643 evlslem6 21644 psropprmul 21760 coe1mul2 21791 plypf1 25726 tayl0 25874 fsuppcurry1 31950 fsuppcurry2 31951 gsummptres2 32205 evls1fpws 32646 ply1degltdimlem 32707 fedgmullem1 32714 fedgmullem2 32715 fsuppsssuppgd 41064 lincresunit2 47159 |
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