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Mirrors > Home > MPE Home > Th. List > rpnnen2lem3 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen2 16115. (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 11162 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
2 | 3nn 12239 | . . . . . . 7 ⊢ 3 ∈ ℕ | |
3 | nndivre 12201 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℕ) → (1 / 3) ∈ ℝ) | |
4 | 1, 2, 3 | mp2an 691 | . . . . . 6 ⊢ (1 / 3) ∈ ℝ |
5 | 4 | recni 11176 | . . . . 5 ⊢ (1 / 3) ∈ ℂ |
6 | 5 | a1i 11 | . . . 4 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
7 | 0re 11164 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
8 | 3re 12240 | . . . . . . . . 9 ⊢ 3 ∈ ℝ | |
9 | 3pos 12265 | . . . . . . . . 9 ⊢ 0 < 3 | |
10 | 8, 9 | recgt0ii 12068 | . . . . . . . 8 ⊢ 0 < (1 / 3) |
11 | 7, 4, 10 | ltleii 11285 | . . . . . . 7 ⊢ 0 ≤ (1 / 3) |
12 | absid 15188 | . . . . . . 7 ⊢ (((1 / 3) ∈ ℝ ∧ 0 ≤ (1 / 3)) → (abs‘(1 / 3)) = (1 / 3)) | |
13 | 4, 11, 12 | mp2an 691 | . . . . . 6 ⊢ (abs‘(1 / 3)) = (1 / 3) |
14 | 1lt3 12333 | . . . . . . 7 ⊢ 1 < 3 | |
15 | recgt1 12058 | . . . . . . . 8 ⊢ ((3 ∈ ℝ ∧ 0 < 3) → (1 < 3 ↔ (1 / 3) < 1)) | |
16 | 8, 9, 15 | mp2an 691 | . . . . . . 7 ⊢ (1 < 3 ↔ (1 / 3) < 1) |
17 | 14, 16 | mpbi 229 | . . . . . 6 ⊢ (1 / 3) < 1 |
18 | 13, 17 | eqbrtri 5131 | . . . . 5 ⊢ (abs‘(1 / 3)) < 1 |
19 | 18 | a1i 11 | . . . 4 ⊢ (⊤ → (abs‘(1 / 3)) < 1) |
20 | 1nn0 12436 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
21 | 20 | a1i 11 | . . . 4 ⊢ (⊤ → 1 ∈ ℕ0) |
22 | ssid 3971 | . . . . . 6 ⊢ ℕ ⊆ ℕ | |
23 | simpr 486 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → 𝑘 ∈ (ℤ≥‘1)) | |
24 | nnuz 12813 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
25 | 23, 24 | eleqtrrdi 2849 | . . . . . 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 16103 | . . . . . 6 ⊢ ((ℕ ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘ℕ)‘𝑘) = if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0)) |
28 | 22, 25, 27 | sylancr 588 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0)) |
29 | 25 | iftrued 4499 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0) = ((1 / 3)↑𝑘)) |
30 | 28, 29 | eqtrd 2777 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = ((1 / 3)↑𝑘)) |
31 | 6, 19, 21, 30 | geolim2 15763 | . . 3 ⊢ (⊤ → seq1( + , (𝐹‘ℕ)) ⇝ (((1 / 3)↑1) / (1 − (1 / 3)))) |
32 | 31 | mptru 1549 | . 2 ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (((1 / 3)↑1) / (1 − (1 / 3))) |
33 | exp1 13980 | . . . . 5 ⊢ ((1 / 3) ∈ ℂ → ((1 / 3)↑1) = (1 / 3)) | |
34 | 5, 33 | ax-mp 5 | . . . 4 ⊢ ((1 / 3)↑1) = (1 / 3) |
35 | 3cn 12241 | . . . . . 6 ⊢ 3 ∈ ℂ | |
36 | ax-1cn 11116 | . . . . . 6 ⊢ 1 ∈ ℂ | |
37 | 3ne0 12266 | . . . . . . 7 ⊢ 3 ≠ 0 | |
38 | 35, 37 | pm3.2i 472 | . . . . . 6 ⊢ (3 ∈ ℂ ∧ 3 ≠ 0) |
39 | divsubdir 11856 | . . . . . 6 ⊢ ((3 ∈ ℂ ∧ 1 ∈ ℂ ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((3 − 1) / 3) = ((3 / 3) − (1 / 3))) | |
40 | 35, 36, 38, 39 | mp3an 1462 | . . . . 5 ⊢ ((3 − 1) / 3) = ((3 / 3) − (1 / 3)) |
41 | 3m1e2 12288 | . . . . . 6 ⊢ (3 − 1) = 2 | |
42 | 41 | oveq1i 7372 | . . . . 5 ⊢ ((3 − 1) / 3) = (2 / 3) |
43 | 35, 37 | dividi 11895 | . . . . . 6 ⊢ (3 / 3) = 1 |
44 | 43 | oveq1i 7372 | . . . . 5 ⊢ ((3 / 3) − (1 / 3)) = (1 − (1 / 3)) |
45 | 40, 42, 44 | 3eqtr3ri 2774 | . . . 4 ⊢ (1 − (1 / 3)) = (2 / 3) |
46 | 34, 45 | oveq12i 7374 | . . 3 ⊢ (((1 / 3)↑1) / (1 − (1 / 3))) = ((1 / 3) / (2 / 3)) |
47 | 2cnne0 12370 | . . . 4 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
48 | divcan7 11871 | . . . 4 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((1 / 3) / (2 / 3)) = (1 / 2)) | |
49 | 36, 47, 38, 48 | mp3an 1462 | . . 3 ⊢ ((1 / 3) / (2 / 3)) = (1 / 2) |
50 | 46, 49 | eqtri 2765 | . 2 ⊢ (((1 / 3)↑1) / (1 − (1 / 3))) = (1 / 2) |
51 | 32, 50 | breqtri 5135 | 1 ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (1 / 2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ≠ wne 2944 ⊆ wss 3915 ifcif 4491 𝒫 cpw 4565 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6501 (class class class)co 7362 ℂcc 11056 ℝcr 11057 0cc0 11058 1c1 11059 + caddc 11061 < clt 11196 ≤ cle 11197 − cmin 11392 / cdiv 11819 ℕcn 12160 2c2 12215 3c3 12216 ℕ0cn0 12420 ℤ≥cuz 12770 seqcseq 13913 ↑cexp 13974 abscabs 15126 ⇝ cli 15373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-rlim 15378 df-sum 15578 |
This theorem is referenced by: rpnnen2lem5 16107 rpnnen2lem12 16114 |
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