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Mirrors > Home > MPE Home > Th. List > rpnnen2lem2 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen2 15579. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
rpnnen2.1 | ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0))) |
Ref | Expression |
---|---|
rpnnen2lem2 | ⊢ (𝐴 ⊆ ℕ → (𝐹‘𝐴):ℕ⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnex 11644 | . . . 4 ⊢ ℕ ∈ V | |
2 | 1 | elpw2 5248 | . . 3 ⊢ (𝐴 ∈ 𝒫 ℕ ↔ 𝐴 ⊆ ℕ) |
3 | eleq2 2901 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑛 ∈ 𝑥 ↔ 𝑛 ∈ 𝐴)) | |
4 | 3 | ifbid 4489 | . . . . 5 ⊢ (𝑥 = 𝐴 → if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0) = if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0)) |
5 | 4 | mpteq2dv 5162 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0))) |
6 | rpnnen2.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0))) | |
7 | 1 | mptex 6986 | . . . 4 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0)) ∈ V |
8 | 5, 6, 7 | fvmpt 6768 | . . 3 ⊢ (𝐴 ∈ 𝒫 ℕ → (𝐹‘𝐴) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0))) |
9 | 2, 8 | sylbir 237 | . 2 ⊢ (𝐴 ⊆ ℕ → (𝐹‘𝐴) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0))) |
10 | 1re 10641 | . . . . . 6 ⊢ 1 ∈ ℝ | |
11 | 3nn 11717 | . . . . . 6 ⊢ 3 ∈ ℕ | |
12 | nndivre 11679 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℕ) → (1 / 3) ∈ ℝ) | |
13 | 10, 11, 12 | mp2an 690 | . . . . 5 ⊢ (1 / 3) ∈ ℝ |
14 | nnnn0 11905 | . . . . 5 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
15 | reexpcl 13447 | . . . . 5 ⊢ (((1 / 3) ∈ ℝ ∧ 𝑛 ∈ ℕ0) → ((1 / 3)↑𝑛) ∈ ℝ) | |
16 | 13, 14, 15 | sylancr 589 | . . . 4 ⊢ (𝑛 ∈ ℕ → ((1 / 3)↑𝑛) ∈ ℝ) |
17 | 0re 10643 | . . . 4 ⊢ 0 ∈ ℝ | |
18 | ifcl 4511 | . . . 4 ⊢ ((((1 / 3)↑𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0) ∈ ℝ) | |
19 | 16, 17, 18 | sylancl 588 | . . 3 ⊢ (𝑛 ∈ ℕ → if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0) ∈ ℝ) |
20 | 19 | adantl 484 | . 2 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑛 ∈ ℕ) → if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0) ∈ ℝ) |
21 | 9, 20 | fmpt3d 6880 | 1 ⊢ (𝐴 ⊆ ℕ → (𝐹‘𝐴):ℕ⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ifcif 4467 𝒫 cpw 4539 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 / cdiv 11297 ℕcn 11638 3c3 11694 ℕ0cn0 11898 ↑cexp 13430 |
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-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-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 df-exp 13431 |
This theorem is referenced by: rpnnen2lem5 15571 rpnnen2lem6 15572 rpnnen2lem7 15573 rpnnen2lem8 15574 rpnnen2lem9 15575 rpnnen2lem10 15576 rpnnen2lem12 15578 |
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