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Mirrors > Home > MPE Home > Th. List > fdmfifsupp | Structured version Visualization version GIF version |
Description: A function with a finite domain is always finitely supported. (Contributed by AV, 25-May-2019.) |
Ref | Expression |
---|---|
fdmfisuppfi.f | ⊢ (𝜑 → 𝐹:𝐷⟶𝑅) |
fdmfisuppfi.d | ⊢ (𝜑 → 𝐷 ∈ Fin) |
fdmfisuppfi.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
Ref | Expression |
---|---|
fdmfifsupp | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdmfisuppfi.f | . . 3 ⊢ (𝜑 → 𝐹:𝐷⟶𝑅) | |
2 | ffun 5961 | . . 3 ⊢ (𝐹:𝐷⟶𝑅 → Fun 𝐹) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → Fun 𝐹) |
4 | fdmfisuppfi.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ Fin) | |
5 | fdmfisuppfi.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
6 | 1, 4, 5 | fdmfisuppfi 8167 | . 2 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
7 | ffn 5958 | . . . . 5 ⊢ (𝐹:𝐷⟶𝑅 → 𝐹 Fn 𝐷) | |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐷) |
9 | fnex 6386 | . . . 4 ⊢ ((𝐹 Fn 𝐷 ∧ 𝐷 ∈ Fin) → 𝐹 ∈ V) | |
10 | 8, 4, 9 | syl2anc 691 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
11 | isfsupp 8162 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 ↔ (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin))) | |
12 | 10, 5, 11 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐹 finSupp 𝑍 ↔ (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin))) |
13 | 3, 6, 12 | mpbir2and 959 | 1 ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 (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: fsuppmptdm 8169 fndmfifsupp 8171 gsummptfif1o 18190 psrmulcllem 19208 frlmfibas 19924 elfilspd 19961 tmdgsum 21709 tsmslem1 21742 tsmssubm 21756 tsmsres 21757 tsmsf1o 21758 tsmsmhm 21759 tsmsadd 21760 tsmsxplem1 21766 tsmsxplem2 21767 imasdsf1olem 21988 xrge0gsumle 22444 xrge0tsms 22445 ehlbase 23002 jensenlem2 24514 jensen 24515 amgmlem 24516 amgm 24517 wilthlem2 24595 wilthlem3 24596 gsumle 29110 xrge0tsmsd 29116 esumpfinvalf 29465 k0004ss2 37470 rrxbasefi 39179 sge0tsms 39273 fsuppmptdmf 41956 linccl 41997 lcosn0 42003 islinindfis 42032 snlindsntor 42054 ldepspr 42056 zlmodzxzldeplem2 42084 amgmwlem 42357 amgmlemALT 42358 |
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