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Mirrors > Home > MPE Home > Th. List > rpnnen2lem3 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen2 15768. (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 |
---|---|
rpnnen2lem3 | ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (1 / 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10816 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
2 | 3nn 11892 | . . . . . . 7 ⊢ 3 ∈ ℕ | |
3 | nndivre 11854 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℕ) → (1 / 3) ∈ ℝ) | |
4 | 1, 2, 3 | mp2an 692 | . . . . . 6 ⊢ (1 / 3) ∈ ℝ |
5 | 4 | recni 10830 | . . . . 5 ⊢ (1 / 3) ∈ ℂ |
6 | 5 | a1i 11 | . . . 4 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
7 | 0re 10818 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
8 | 3re 11893 | . . . . . . . . 9 ⊢ 3 ∈ ℝ | |
9 | 3pos 11918 | . . . . . . . . 9 ⊢ 0 < 3 | |
10 | 8, 9 | recgt0ii 11721 | . . . . . . . 8 ⊢ 0 < (1 / 3) |
11 | 7, 4, 10 | ltleii 10938 | . . . . . . 7 ⊢ 0 ≤ (1 / 3) |
12 | absid 14843 | . . . . . . 7 ⊢ (((1 / 3) ∈ ℝ ∧ 0 ≤ (1 / 3)) → (abs‘(1 / 3)) = (1 / 3)) | |
13 | 4, 11, 12 | mp2an 692 | . . . . . 6 ⊢ (abs‘(1 / 3)) = (1 / 3) |
14 | 1lt3 11986 | . . . . . . 7 ⊢ 1 < 3 | |
15 | recgt1 11711 | . . . . . . . 8 ⊢ ((3 ∈ ℝ ∧ 0 < 3) → (1 < 3 ↔ (1 / 3) < 1)) | |
16 | 8, 9, 15 | mp2an 692 | . . . . . . 7 ⊢ (1 < 3 ↔ (1 / 3) < 1) |
17 | 14, 16 | mpbi 233 | . . . . . 6 ⊢ (1 / 3) < 1 |
18 | 13, 17 | eqbrtri 5064 | . . . . 5 ⊢ (abs‘(1 / 3)) < 1 |
19 | 18 | a1i 11 | . . . 4 ⊢ (⊤ → (abs‘(1 / 3)) < 1) |
20 | 1nn0 12089 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
21 | 20 | a1i 11 | . . . 4 ⊢ (⊤ → 1 ∈ ℕ0) |
22 | ssid 3913 | . . . . . 6 ⊢ ℕ ⊆ ℕ | |
23 | simpr 488 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → 𝑘 ∈ (ℤ≥‘1)) | |
24 | nnuz 12460 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
25 | 23, 24 | eleqtrrdi 2845 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → 𝑘 ∈ ℕ) |
26 | rpnnen2.1 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝒫 ℕ ↦ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ 𝑥, ((1 / 3)↑𝑛), 0))) | |
27 | 26 | rpnnen2lem1 15756 | . . . . . 6 ⊢ ((ℕ ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘ℕ)‘𝑘) = if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0)) |
28 | 22, 25, 27 | sylancr 590 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0)) |
29 | 25 | iftrued 4437 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0) = ((1 / 3)↑𝑘)) |
30 | 28, 29 | eqtrd 2774 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = ((1 / 3)↑𝑘)) |
31 | 6, 19, 21, 30 | geolim2 15416 | . . 3 ⊢ (⊤ → seq1( + , (𝐹‘ℕ)) ⇝ (((1 / 3)↑1) / (1 − (1 / 3)))) |
32 | 31 | mptru 1550 | . 2 ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (((1 / 3)↑1) / (1 − (1 / 3))) |
33 | exp1 13624 | . . . . 5 ⊢ ((1 / 3) ∈ ℂ → ((1 / 3)↑1) = (1 / 3)) | |
34 | 5, 33 | ax-mp 5 | . . . 4 ⊢ ((1 / 3)↑1) = (1 / 3) |
35 | 3cn 11894 | . . . . . 6 ⊢ 3 ∈ ℂ | |
36 | ax-1cn 10770 | . . . . . 6 ⊢ 1 ∈ ℂ | |
37 | 3ne0 11919 | . . . . . . 7 ⊢ 3 ≠ 0 | |
38 | 35, 37 | pm3.2i 474 | . . . . . 6 ⊢ (3 ∈ ℂ ∧ 3 ≠ 0) |
39 | divsubdir 11509 | . . . . . 6 ⊢ ((3 ∈ ℂ ∧ 1 ∈ ℂ ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((3 − 1) / 3) = ((3 / 3) − (1 / 3))) | |
40 | 35, 36, 38, 39 | mp3an 1463 | . . . . 5 ⊢ ((3 − 1) / 3) = ((3 / 3) − (1 / 3)) |
41 | 3m1e2 11941 | . . . . . 6 ⊢ (3 − 1) = 2 | |
42 | 41 | oveq1i 7212 | . . . . 5 ⊢ ((3 − 1) / 3) = (2 / 3) |
43 | 35, 37 | dividi 11548 | . . . . . 6 ⊢ (3 / 3) = 1 |
44 | 43 | oveq1i 7212 | . . . . 5 ⊢ ((3 / 3) − (1 / 3)) = (1 − (1 / 3)) |
45 | 40, 42, 44 | 3eqtr3ri 2771 | . . . 4 ⊢ (1 − (1 / 3)) = (2 / 3) |
46 | 34, 45 | oveq12i 7214 | . . 3 ⊢ (((1 / 3)↑1) / (1 − (1 / 3))) = ((1 / 3) / (2 / 3)) |
47 | 2cnne0 12023 | . . . 4 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
48 | divcan7 11524 | . . . 4 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((1 / 3) / (2 / 3)) = (1 / 2)) | |
49 | 36, 47, 38, 48 | mp3an 1463 | . . 3 ⊢ ((1 / 3) / (2 / 3)) = (1 / 2) |
50 | 46, 49 | eqtri 2762 | . 2 ⊢ (((1 / 3)↑1) / (1 − (1 / 3))) = (1 / 2) |
51 | 32, 50 | breqtri 5068 | 1 ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (1 / 2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 ≠ wne 2935 ⊆ wss 3857 ifcif 4429 𝒫 cpw 4503 class class class wbr 5043 ↦ cmpt 5124 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 ℝcr 10711 0cc0 10712 1c1 10713 + caddc 10715 < clt 10850 ≤ cle 10851 − cmin 11045 / cdiv 11472 ℕcn 11813 2c2 11868 3c3 11869 ℕ0cn0 12073 ℤ≥cuz 12421 seqcseq 13557 ↑cexp 13618 abscabs 14780 ⇝ cli 15028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-pm 8500 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-fz 13079 df-fzo 13222 df-fl 13350 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-rlim 15033 df-sum 15233 |
This theorem is referenced by: rpnnen2lem5 15760 rpnnen2lem12 15767 |
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