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Mirrors > Home > MPE Home > Th. List > rpnnen2lem1 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen2 16206. (Contributed by Mario Carneiro, 13-May-2013.) |
Ref | Expression |
---|---|
rpnnen2.1 | ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0))) |
Ref | Expression |
---|---|
rpnnen2lem1 | ⊢ ((𝐴 ⊆ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐹‘𝐴)‘𝑁) = if(𝑁 ∈ 𝐴, ((1 / 3)↑𝑁), 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnex 12251 | . . . . 5 ⊢ ℕ ∈ V | |
2 | 1 | elpw2 5348 | . . . 4 ⊢ (𝐴 ∈ 𝒫 ℕ ↔ 𝐴 ⊆ ℕ) |
3 | eleq2 2814 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑛 ∈ 𝑥 ↔ 𝑛 ∈ 𝐴)) | |
4 | 3 | ifbid 4553 | . . . . . 6 ⊢ (𝑥 = 𝐴 → if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0) = if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0)) |
5 | 4 | mpteq2dv 5251 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0))) |
6 | rpnnen2.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0))) | |
7 | 1 | mptex 7235 | . . . . 5 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0)) ∈ V |
8 | 5, 6, 7 | fvmpt 7004 | . . . 4 ⊢ (𝐴 ∈ 𝒫 ℕ → (𝐹‘𝐴) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0))) |
9 | 2, 8 | sylbir 234 | . . 3 ⊢ (𝐴 ⊆ ℕ → (𝐹‘𝐴) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0))) |
10 | 9 | fveq1d 6898 | . 2 ⊢ (𝐴 ⊆ ℕ → ((𝐹‘𝐴)‘𝑁) = ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0))‘𝑁)) |
11 | eleq1 2813 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴)) | |
12 | oveq2 7427 | . . . 4 ⊢ (𝑛 = 𝑁 → ((1 / 3)↑𝑛) = ((1 / 3)↑𝑁)) | |
13 | 11, 12 | ifbieq1d 4554 | . . 3 ⊢ (𝑛 = 𝑁 → if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0) = if(𝑁 ∈ 𝐴, ((1 / 3)↑𝑁), 0)) |
14 | eqid 2725 | . . 3 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0)) | |
15 | ovex 7452 | . . . 4 ⊢ ((1 / 3)↑𝑁) ∈ V | |
16 | c0ex 11240 | . . . 4 ⊢ 0 ∈ V | |
17 | 15, 16 | ifex 4580 | . . 3 ⊢ if(𝑁 ∈ 𝐴, ((1 / 3)↑𝑁), 0) ∈ V |
18 | 13, 14, 17 | fvmpt 7004 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0))‘𝑁) = if(𝑁 ∈ 𝐴, ((1 / 3)↑𝑁), 0)) |
19 | 10, 18 | sylan9eq 2785 | 1 ⊢ ((𝐴 ⊆ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐹‘𝐴)‘𝑁) = if(𝑁 ∈ 𝐴, ((1 / 3)↑𝑁), 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ifcif 4530 𝒫 cpw 4604 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 0cc0 11140 1c1 11141 / cdiv 11903 ℕcn 12245 3c3 12301 ↑cexp 14062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-mulcl 11202 ax-i2m1 11208 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-nn 12246 |
This theorem is referenced by: rpnnen2lem3 16196 rpnnen2lem4 16197 rpnnen2lem9 16202 rpnnen2lem10 16203 rpnnen2lem11 16204 |
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