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Mirrors > Home > MPE Home > Th. List > fsuppmapnn0ub | Structured version Visualization version GIF version |
Description: If a function over the nonnegative integers is finitely supported, then there is an upper bound for the arguments resulting in nonzero values. (Contributed by AV, 6-Oct-2019.) |
Ref | Expression |
---|---|
fsuppmapnn0ub | ⊢ ((𝐹 ∈ (𝑅 ↑m ℕ0) ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝐹‘𝑥) = 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . 4 ⊢ (((𝐹 ∈ (𝑅 ↑m ℕ0) ∧ 𝑍 ∈ 𝑉) ∧ 𝐹 finSupp 𝑍) → 𝐹 finSupp 𝑍) | |
2 | 1 | fsuppimpd 9383 | . . 3 ⊢ (((𝐹 ∈ (𝑅 ↑m ℕ0) ∧ 𝑍 ∈ 𝑉) ∧ 𝐹 finSupp 𝑍) → (𝐹 supp 𝑍) ∈ Fin) |
3 | 2 | ex 412 | . 2 ⊢ ((𝐹 ∈ (𝑅 ↑m ℕ0) ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 → (𝐹 supp 𝑍) ∈ Fin)) |
4 | elmapfn 8873 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 ↑m ℕ0) → 𝐹 Fn ℕ0) | |
5 | 4 | adantr 480 | . . . . 5 ⊢ ((𝐹 ∈ (𝑅 ↑m ℕ0) ∧ 𝑍 ∈ 𝑉) → 𝐹 Fn ℕ0) |
6 | nn0ex 12494 | . . . . . 6 ⊢ ℕ0 ∈ V | |
7 | 6 | a1i 11 | . . . . 5 ⊢ ((𝐹 ∈ (𝑅 ↑m ℕ0) ∧ 𝑍 ∈ 𝑉) → ℕ0 ∈ V) |
8 | simpr 484 | . . . . 5 ⊢ ((𝐹 ∈ (𝑅 ↑m ℕ0) ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ 𝑉) | |
9 | suppvalfn 8165 | . . . . 5 ⊢ ((𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) = {𝑥 ∈ ℕ0 ∣ (𝐹‘𝑥) ≠ 𝑍}) | |
10 | 5, 7, 8, 9 | syl3anc 1369 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 ↑m ℕ0) ∧ 𝑍 ∈ 𝑉) → (𝐹 supp 𝑍) = {𝑥 ∈ ℕ0 ∣ (𝐹‘𝑥) ≠ 𝑍}) |
11 | 10 | eleq1d 2813 | . . 3 ⊢ ((𝐹 ∈ (𝑅 ↑m ℕ0) ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) ∈ Fin ↔ {𝑥 ∈ ℕ0 ∣ (𝐹‘𝑥) ≠ 𝑍} ∈ Fin)) |
12 | rabssnn0fi 13969 | . . . 4 ⊢ ({𝑥 ∈ ℕ0 ∣ (𝐹‘𝑥) ≠ 𝑍} ∈ Fin ↔ ∃𝑚 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → ¬ (𝐹‘𝑥) ≠ 𝑍)) | |
13 | nne 2939 | . . . . . . 7 ⊢ (¬ (𝐹‘𝑥) ≠ 𝑍 ↔ (𝐹‘𝑥) = 𝑍) | |
14 | 13 | imbi2i 336 | . . . . . 6 ⊢ ((𝑚 < 𝑥 → ¬ (𝐹‘𝑥) ≠ 𝑍) ↔ (𝑚 < 𝑥 → (𝐹‘𝑥) = 𝑍)) |
15 | 14 | ralbii 3088 | . . . . 5 ⊢ (∀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → ¬ (𝐹‘𝑥) ≠ 𝑍) ↔ ∀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝐹‘𝑥) = 𝑍)) |
16 | 15 | rexbii 3089 | . . . 4 ⊢ (∃𝑚 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → ¬ (𝐹‘𝑥) ≠ 𝑍) ↔ ∃𝑚 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝐹‘𝑥) = 𝑍)) |
17 | 12, 16 | sylbb 218 | . . 3 ⊢ ({𝑥 ∈ ℕ0 ∣ (𝐹‘𝑥) ≠ 𝑍} ∈ Fin → ∃𝑚 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝐹‘𝑥) = 𝑍)) |
18 | 11, 17 | biimtrdi 252 | . 2 ⊢ ((𝐹 ∈ (𝑅 ↑m ℕ0) ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) ∈ Fin → ∃𝑚 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝐹‘𝑥) = 𝑍))) |
19 | 3, 18 | syld 47 | 1 ⊢ ((𝐹 ∈ (𝑅 ↑m ℕ0) ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝐹‘𝑥) = 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ∃wrex 3065 {crab 3427 Vcvv 3469 class class class wbr 5142 Fn wfn 6537 ‘cfv 6542 (class class class)co 7414 supp csupp 8157 ↑m cmap 8834 Fincfn 8953 finSupp cfsupp 9375 < clt 11264 ℕ0cn0 12488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 |
This theorem is referenced by: fsuppmapnn0fz 13979 nn0gsumfz 19923 mptcoe1fsupp 22108 coe1ae0 22109 gsummoncoe1 22201 mptcoe1matfsupp 22678 mp2pm2mplem4 22685 pm2mp 22701 cayhamlem4 22764 |
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