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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 8322 | . . 3 ⊢ (𝐹 finSupp 𝑍 → (Fun 𝐹 ∧ (𝐹 supp 𝑍) ∈ Fin)) | |
3 | 2 | simprd 478 | . 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 2030 class class class wbr 4685 Fun wfun 5920 (class class class)co 6690 supp csupp 7340 Fincfn 7997 finSupp cfsupp 8316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-iota 5889 df-fun 5928 df-fv 5934 df-ov 6693 df-fsupp 8317 |
This theorem is referenced by: fsuppsssupp 8332 fsuppxpfi 8333 fsuppun 8335 resfsupp 8343 fsuppmptif 8346 fsuppco 8348 fsuppco2 8349 fsuppcor 8350 cantnfcl 8602 cantnfp1lem1 8613 fsuppmapnn0fiublem 12829 fsuppmapnn0fiub 12830 fsuppmapnn0fiubOLD 12831 fsuppmapnn0ub 12835 gsumzcl 18358 gsumcl 18362 gsumzadd 18368 gsumzmhm 18383 gsumzoppg 18390 gsum2dlem1 18415 gsum2dlem2 18416 gsum2d 18417 gsumdixp 18655 lcomfsupp 18951 mptscmfsupp0 18976 mplcoe1 19513 mplbas2 19518 psrbagev1 19558 evlslem2 19560 evlslem6 19561 regsumsupp 20016 frlmphllem 20167 uvcresum 20180 frlmsslsp 20183 frlmup1 20185 tsmsgsum 21989 rrxcph 23226 rrxfsupp 23231 mdegldg 23871 mdegcl 23874 plypf1 24013 rmfsupp 42480 mndpfsupp 42482 scmfsupp 42484 lincresunit2 42592 |
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