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Mirrors > Home > MPE Home > Th. List > mptnn0fsupp | Structured version Visualization version GIF version |
Description: A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 23-Dec-2019.) |
Ref | Expression |
---|---|
mptnn0fsupp.0 | ⊢ (𝜑 → 0 ∈ 𝑉) |
mptnn0fsupp.c | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ 𝐵) |
mptnn0fsupp.s | ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
Ref | Expression |
---|---|
mptnn0fsupp | ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptnn0fsupp.c | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ 𝐵) | |
2 | 1 | ralrimiva 3182 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) |
3 | eqid 2821 | . . . . . 6 ⊢ (𝑘 ∈ ℕ0 ↦ 𝐶) = (𝑘 ∈ ℕ0 ↦ 𝐶) | |
4 | 3 | fnmpt 6488 | . . . . 5 ⊢ (∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵 → (𝑘 ∈ ℕ0 ↦ 𝐶) Fn ℕ0) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) Fn ℕ0) |
6 | nn0ex 11904 | . . . . 5 ⊢ ℕ0 ∈ V | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
8 | mptnn0fsupp.0 | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝑉) | |
9 | 8 | elexd 3514 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
10 | suppvalfn 7837 | . . . 4 ⊢ (((𝑘 ∈ ℕ0 ↦ 𝐶) Fn ℕ0 ∧ ℕ0 ∈ V ∧ 0 ∈ V) → ((𝑘 ∈ ℕ0 ↦ 𝐶) supp 0 ) = {𝑥 ∈ ℕ0 ∣ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 }) | |
11 | 5, 7, 9, 10 | syl3anc 1367 | . . 3 ⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ 𝐶) supp 0 ) = {𝑥 ∈ ℕ0 ∣ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 }) |
12 | mptnn0fsupp.s | . . . . 5 ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) | |
13 | nne 3020 | . . . . . . . . 9 ⊢ (¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 ↔ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 ) | |
14 | simpr 487 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0) | |
15 | 2 | ad2antrr 724 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) |
16 | rspcsbela 4387 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) → ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵) | |
17 | 14, 15, 16 | syl2anc 586 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵) |
18 | 3 | fvmpts 6771 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0 ∧ ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵) → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐶) |
19 | 14, 17, 18 | syl2anc 586 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐶) |
20 | 19 | eqeq1d 2823 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 ↔ ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
21 | 13, 20 | syl5bb 285 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 ↔ ⦋𝑥 / 𝑘⦌𝐶 = 0 )) |
22 | 21 | imbi2d 343 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑠 < 𝑥 → ¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 ) ↔ (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ))) |
23 | 22 | ralbidva 3196 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 ) ↔ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ))) |
24 | 23 | rexbidva 3296 | . . . . 5 ⊢ (𝜑 → (∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 ) ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ))) |
25 | 12, 24 | mpbird 259 | . . . 4 ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 )) |
26 | rabssnn0fi 13355 | . . . 4 ⊢ ({𝑥 ∈ ℕ0 ∣ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 } ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 )) | |
27 | 25, 26 | sylibr 236 | . . 3 ⊢ (𝜑 → {𝑥 ∈ ℕ0 ∣ ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) ≠ 0 } ∈ Fin) |
28 | 11, 27 | eqeltrd 2913 | . 2 ⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ 𝐶) supp 0 ) ∈ Fin) |
29 | funmpt 6393 | . . 3 ⊢ Fun (𝑘 ∈ ℕ0 ↦ 𝐶) | |
30 | 6 | mptex 6986 | . . 3 ⊢ (𝑘 ∈ ℕ0 ↦ 𝐶) ∈ V |
31 | funisfsupp 8838 | . . 3 ⊢ ((Fun (𝑘 ∈ ℕ0 ↦ 𝐶) ∧ (𝑘 ∈ ℕ0 ↦ 𝐶) ∈ V ∧ 0 ∈ V) → ((𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ↔ ((𝑘 ∈ ℕ0 ↦ 𝐶) supp 0 ) ∈ Fin)) | |
32 | 29, 30, 9, 31 | mp3an12i 1461 | . 2 ⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ↔ ((𝑘 ∈ ℕ0 ↦ 𝐶) supp 0 ) ∈ Fin)) |
33 | 28, 32 | mpbird 259 | 1 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ⦋csb 3883 class class class wbr 5066 ↦ cmpt 5146 Fun wfun 6349 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 supp csupp 7830 Fincfn 8509 finSupp cfsupp 8833 < clt 10675 ℕ0cn0 11898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 |
This theorem is referenced by: mptnn0fsuppd 13367 mptcoe1fsupp 20383 mptcoe1matfsupp 21410 pm2mp 21433 chfacffsupp 21464 chfacfscmulfsupp 21467 chfacfpmmulfsupp 21471 cayhamlem4 21496 ply1mulgsumlem3 44462 ply1mulgsumlem4 44463 |
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