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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 8556 | . . 3 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
3 | 2 | simprd 491 | . 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 2164 class class class wbr 4875 Fun wfun 6121 (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-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 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-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-iota 6090 df-fun 6129 df-fv 6135 df-ov 6913 df-fsupp 8551 |
This theorem is referenced by: fsuppsssupp 8566 fsuppxpfi 8567 fsuppun 8569 resfsupp 8577 fsuppmptif 8580 fsuppco 8582 fsuppco2 8583 fsuppcor 8584 cantnfcl 8848 cantnfp1lem1 8859 fsuppmapnn0fiublem 13091 fsuppmapnn0fiub 13092 fsuppmapnn0ub 13096 gsumzcl 18672 gsumcl 18676 gsumzadd 18682 gsumzmhm 18697 gsumzoppg 18704 gsum2dlem1 18729 gsum2dlem2 18730 gsum2d 18731 gsumdixp 18970 lcomfsupp 19266 mptscmfsupp0 19291 mplcoe1 19833 mplbas2 19838 psrbagev1 19877 evlslem2 19879 evlslem6 19880 regsumsupp 20336 frlmphllem 20493 uvcresum 20506 frlmsslsp 20509 frlmup1 20511 tsmsgsum 22319 rrxcph 23567 rrxfsupp 23577 mdegldg 24232 mdegcl 24235 plypf1 24374 rmfsupp 43016 mndpfsupp 43018 scmfsupp 43020 lincresunit2 43128 |
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