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Mirrors > Home > MPE Home > Th. List > rpnnen2lem3 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen2 15579. (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 10641 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
2 | 3nn 11717 | . . . . . . 7 ⊢ 3 ∈ ℕ | |
3 | nndivre 11679 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℕ) → (1 / 3) ∈ ℝ) | |
4 | 1, 2, 3 | mp2an 690 | . . . . . 6 ⊢ (1 / 3) ∈ ℝ |
5 | 4 | recni 10655 | . . . . 5 ⊢ (1 / 3) ∈ ℂ |
6 | 5 | a1i 11 | . . . 4 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
7 | 0re 10643 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
8 | 3re 11718 | . . . . . . . . 9 ⊢ 3 ∈ ℝ | |
9 | 3pos 11743 | . . . . . . . . 9 ⊢ 0 < 3 | |
10 | 8, 9 | recgt0ii 11546 | . . . . . . . 8 ⊢ 0 < (1 / 3) |
11 | 7, 4, 10 | ltleii 10763 | . . . . . . 7 ⊢ 0 ≤ (1 / 3) |
12 | absid 14656 | . . . . . . 7 ⊢ (((1 / 3) ∈ ℝ ∧ 0 ≤ (1 / 3)) → (abs‘(1 / 3)) = (1 / 3)) | |
13 | 4, 11, 12 | mp2an 690 | . . . . . 6 ⊢ (abs‘(1 / 3)) = (1 / 3) |
14 | 1lt3 11811 | . . . . . . 7 ⊢ 1 < 3 | |
15 | recgt1 11536 | . . . . . . . 8 ⊢ ((3 ∈ ℝ ∧ 0 < 3) → (1 < 3 ↔ (1 / 3) < 1)) | |
16 | 8, 9, 15 | mp2an 690 | . . . . . . 7 ⊢ (1 < 3 ↔ (1 / 3) < 1) |
17 | 14, 16 | mpbi 232 | . . . . . 6 ⊢ (1 / 3) < 1 |
18 | 13, 17 | eqbrtri 5087 | . . . . 5 ⊢ (abs‘(1 / 3)) < 1 |
19 | 18 | a1i 11 | . . . 4 ⊢ (⊤ → (abs‘(1 / 3)) < 1) |
20 | 1nn0 11914 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
21 | 20 | a1i 11 | . . . 4 ⊢ (⊤ → 1 ∈ ℕ0) |
22 | ssid 3989 | . . . . . 6 ⊢ ℕ ⊆ ℕ | |
23 | simpr 487 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → 𝑘 ∈ (ℤ≥‘1)) | |
24 | nnuz 12282 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
25 | 23, 24 | eleqtrrdi 2924 | . . . . . 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 15567 | . . . . . 6 ⊢ ((ℕ ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘ℕ)‘𝑘) = if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0)) |
28 | 22, 25, 27 | sylancr 589 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0)) |
29 | 25 | iftrued 4475 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0) = ((1 / 3)↑𝑘)) |
30 | 28, 29 | eqtrd 2856 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = ((1 / 3)↑𝑘)) |
31 | 6, 19, 21, 30 | geolim2 15227 | . . 3 ⊢ (⊤ → seq1( + , (𝐹‘ℕ)) ⇝ (((1 / 3)↑1) / (1 − (1 / 3)))) |
32 | 31 | mptru 1544 | . 2 ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (((1 / 3)↑1) / (1 − (1 / 3))) |
33 | exp1 13436 | . . . . 5 ⊢ ((1 / 3) ∈ ℂ → ((1 / 3)↑1) = (1 / 3)) | |
34 | 5, 33 | ax-mp 5 | . . . 4 ⊢ ((1 / 3)↑1) = (1 / 3) |
35 | 3cn 11719 | . . . . . 6 ⊢ 3 ∈ ℂ | |
36 | ax-1cn 10595 | . . . . . 6 ⊢ 1 ∈ ℂ | |
37 | 3ne0 11744 | . . . . . . 7 ⊢ 3 ≠ 0 | |
38 | 35, 37 | pm3.2i 473 | . . . . . 6 ⊢ (3 ∈ ℂ ∧ 3 ≠ 0) |
39 | divsubdir 11334 | . . . . . 6 ⊢ ((3 ∈ ℂ ∧ 1 ∈ ℂ ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((3 − 1) / 3) = ((3 / 3) − (1 / 3))) | |
40 | 35, 36, 38, 39 | mp3an 1457 | . . . . 5 ⊢ ((3 − 1) / 3) = ((3 / 3) − (1 / 3)) |
41 | 3m1e2 11766 | . . . . . 6 ⊢ (3 − 1) = 2 | |
42 | 41 | oveq1i 7166 | . . . . 5 ⊢ ((3 − 1) / 3) = (2 / 3) |
43 | 35, 37 | dividi 11373 | . . . . . 6 ⊢ (3 / 3) = 1 |
44 | 43 | oveq1i 7166 | . . . . 5 ⊢ ((3 / 3) − (1 / 3)) = (1 − (1 / 3)) |
45 | 40, 42, 44 | 3eqtr3ri 2853 | . . . 4 ⊢ (1 − (1 / 3)) = (2 / 3) |
46 | 34, 45 | oveq12i 7168 | . . 3 ⊢ (((1 / 3)↑1) / (1 − (1 / 3))) = ((1 / 3) / (2 / 3)) |
47 | 2cnne0 11848 | . . . 4 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
48 | divcan7 11349 | . . . 4 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((1 / 3) / (2 / 3)) = (1 / 2)) | |
49 | 36, 47, 38, 48 | mp3an 1457 | . . 3 ⊢ ((1 / 3) / (2 / 3)) = (1 / 2) |
50 | 46, 49 | eqtri 2844 | . 2 ⊢ (((1 / 3)↑1) / (1 − (1 / 3))) = (1 / 2) |
51 | 32, 50 | breqtri 5091 | 1 ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (1 / 2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ≠ wne 3016 ⊆ wss 3936 ifcif 4467 𝒫 cpw 4539 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 − cmin 10870 / cdiv 11297 ℕcn 11638 2c2 11693 3c3 11694 ℕ0cn0 11898 ℤ≥cuz 12244 seqcseq 13370 ↑cexp 13430 abscabs 14593 ⇝ cli 14841 |
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-inf2 9104 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 ax-pre-sup 10615 |
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-int 4877 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-se 5515 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-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 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-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 df-sum 15043 |
This theorem is referenced by: rpnnen2lem5 15571 rpnnen2lem12 15578 |
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