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| Mirrors > Home > MPE Home > Th. List > fsuppimpd | Structured version Visualization version GIF version | ||
| Description: A finitely supported function is a function with a finite support. (Contributed by AV, 6-Jun-2019.) |
| Ref | Expression |
|---|---|
| fsuppimpd.f | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
| Ref | Expression |
|---|---|
| fsuppimpd | ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fsuppimpd.f | . 2 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
| 2 | fsuppimp 9408 | . . 3 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
| 3 | 2 | simprd 495 | . 2 ⊢ (𝐹 finSupp 𝑍 → (𝐹 supp 𝑍) ∈ Fin) |
| 4 | 1, 3 | syl 17 | 1 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5143 Fun wfun 6555 (class class class)co 7431 supp csupp 8185 Fincfn 8985 finSupp cfsupp 9401 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-fsupp 9402 |
| This theorem is referenced by: fsuppsssupp 9421 fsuppsssuppgd 9422 fsuppxpfi 9425 fsuppun 9427 resfsupp 9436 fsuppmptif 9439 fsuppco 9442 fsuppco2 9443 fsuppcor 9444 cantnfcl 9707 cantnfp1lem1 9718 fsuppmapnn0fiublem 14031 fsuppmapnn0fiub 14032 fsuppmapnn0ub 14036 mndpfsupp 18780 gsumzcl 19929 gsumcl 19933 gsumzadd 19940 gsumzmhm 19955 gsumzoppg 19962 gsum2dlem1 19988 gsum2dlem2 19989 gsum2d 19990 gsumxp2 19998 gsumdixp 20316 lcomfsupp 20900 mptscmfsupp0 20925 regsumsupp 21640 frlmphllem 21800 uvcresum 21813 frlmsslsp 21816 frlmup1 21818 mplcoe1 22055 mplbas2 22060 psrbagev1 22101 evlslem2 22103 evlslem6 22105 psdmplcl 22166 evls1fpws 22373 tsmsgsum 24147 rrxcph 25426 rrxfsupp 25436 mdegldg 26105 mdegcl 26108 plypf1 26251 fsuppinisegfi 32696 fsupprnfi 32701 fsuppcurry1 32736 fsuppcurry2 32737 offinsupp1 32738 gsumfs2d 33058 gsumhashmul 33064 rmfsupp2 33242 elrgspnlem2 33247 elrgspnlem4 33249 elrgspnsubrunlem1 33251 elrgspnsubrunlem2 33252 elrspunidl 33456 elrspunsn 33457 rprmdvdsprod 33562 fedgmullem1 33680 fedgmullem2 33681 evls1fldgencl 33720 fldextrspunlsplem 33723 fldextrspunlsp 33724 zarcmplem 33880 fsuppind 42600 mnringmulrcld 44247 rmfsupp 48289 scmfsupp 48291 lincresunit2 48395 |
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