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Mirrors > Home > MPE Home > Th. List > rpnnen2lem1 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen2 14794. (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 10903 | . . . . 5 ⊢ ℕ ∈ V | |
2 | 1 | elpw2 4755 | . . . 4 ⊢ (𝐴 ∈ 𝒫 ℕ ↔ 𝐴 ⊆ ℕ) |
3 | eleq2 2677 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑛 ∈ 𝑥 ↔ 𝑛 ∈ 𝐴)) | |
4 | 3 | ifbid 4058 | . . . . . 6 ⊢ (𝑥 = 𝐴 → if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0) = if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0)) |
5 | 4 | mpteq2dv 4673 | . . . . 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 6390 | . . . . 5 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0)) ∈ V |
8 | 5, 6, 7 | fvmpt 6191 | . . . 4 ⊢ (𝐴 ∈ 𝒫 ℕ → (𝐹‘𝐴) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0))) |
9 | 2, 8 | sylbir 224 | . . 3 ⊢ (𝐴 ⊆ ℕ → (𝐹‘𝐴) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0))) |
10 | 9 | fveq1d 6105 | . 2 ⊢ (𝐴 ⊆ ℕ → ((𝐹‘𝐴)‘𝑁) = ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0))‘𝑁)) |
11 | eleq1 2676 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴)) | |
12 | oveq2 6557 | . . . 4 ⊢ (𝑛 = 𝑁 → ((1 / 3)↑𝑛) = ((1 / 3)↑𝑁)) | |
13 | 11, 12 | ifbieq1d 4059 | . . 3 ⊢ (𝑛 = 𝑁 → if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0) = if(𝑁 ∈ 𝐴, ((1 / 3)↑𝑁), 0)) |
14 | eqid 2610 | . . 3 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0)) | |
15 | ovex 6577 | . . . 4 ⊢ ((1 / 3)↑𝑁) ∈ V | |
16 | c0ex 9913 | . . . 4 ⊢ 0 ∈ V | |
17 | 15, 16 | ifex 4106 | . . 3 ⊢ if(𝑁 ∈ 𝐴, ((1 / 3)↑𝑁), 0) ∈ V |
18 | 13, 14, 17 | fvmpt 6191 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝐴, ((1 / 3)↑𝑛), 0))‘𝑁) = if(𝑁 ∈ 𝐴, ((1 / 3)↑𝑁), 0)) |
19 | 10, 18 | sylan9eq 2664 | 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 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ifcif 4036 𝒫 cpw 4108 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 / cdiv 10563 ℕcn 10897 3c3 10948 ↑cexp 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 |
This theorem is referenced by: rpnnen2lem3 14784 rpnnen2lem4 14785 rpnnen2lem9 14790 rpnnen2lem10 14791 rpnnen2lem11 14792 |
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