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Mirrors > Home > MPE Home > Th. List > rpnnen2lem3 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen2 16197. (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 11239 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
2 | 3nn 12316 | . . . . . . 7 ⊢ 3 ∈ ℕ | |
3 | nndivre 12278 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℕ) → (1 / 3) ∈ ℝ) | |
4 | 1, 2, 3 | mp2an 690 | . . . . . 6 ⊢ (1 / 3) ∈ ℝ |
5 | 4 | recni 11253 | . . . . 5 ⊢ (1 / 3) ∈ ℂ |
6 | 5 | a1i 11 | . . . 4 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
7 | 0re 11241 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
8 | 3re 12317 | . . . . . . . . 9 ⊢ 3 ∈ ℝ | |
9 | 3pos 12342 | . . . . . . . . 9 ⊢ 0 < 3 | |
10 | 8, 9 | recgt0ii 12145 | . . . . . . . 8 ⊢ 0 < (1 / 3) |
11 | 7, 4, 10 | ltleii 11362 | . . . . . . 7 ⊢ 0 ≤ (1 / 3) |
12 | absid 15270 | . . . . . . 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 12410 | . . . . . . 7 ⊢ 1 < 3 | |
15 | recgt1 12135 | . . . . . . . 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 229 | . . . . . 6 ⊢ (1 / 3) < 1 |
18 | 13, 17 | eqbrtri 5165 | . . . . 5 ⊢ (abs‘(1 / 3)) < 1 |
19 | 18 | a1i 11 | . . . 4 ⊢ (⊤ → (abs‘(1 / 3)) < 1) |
20 | 1nn0 12513 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
21 | 20 | a1i 11 | . . . 4 ⊢ (⊤ → 1 ∈ ℕ0) |
22 | ssid 3996 | . . . . . 6 ⊢ ℕ ⊆ ℕ | |
23 | simpr 483 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → 𝑘 ∈ (ℤ≥‘1)) | |
24 | nnuz 12890 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
25 | 23, 24 | eleqtrrdi 2836 | . . . . . 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 16185 | . . . . . 6 ⊢ ((ℕ ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘ℕ)‘𝑘) = if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0)) |
28 | 22, 25, 27 | sylancr 585 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0)) |
29 | 25 | iftrued 4533 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0) = ((1 / 3)↑𝑘)) |
30 | 28, 29 | eqtrd 2765 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = ((1 / 3)↑𝑘)) |
31 | 6, 19, 21, 30 | geolim2 15844 | . . 3 ⊢ (⊤ → seq1( + , (𝐹‘ℕ)) ⇝ (((1 / 3)↑1) / (1 − (1 / 3)))) |
32 | 31 | mptru 1540 | . 2 ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (((1 / 3)↑1) / (1 − (1 / 3))) |
33 | exp1 14059 | . . . . 5 ⊢ ((1 / 3) ∈ ℂ → ((1 / 3)↑1) = (1 / 3)) | |
34 | 5, 33 | ax-mp 5 | . . . 4 ⊢ ((1 / 3)↑1) = (1 / 3) |
35 | 3cn 12318 | . . . . . 6 ⊢ 3 ∈ ℂ | |
36 | ax-1cn 11191 | . . . . . 6 ⊢ 1 ∈ ℂ | |
37 | 3ne0 12343 | . . . . . . 7 ⊢ 3 ≠ 0 | |
38 | 35, 37 | pm3.2i 469 | . . . . . 6 ⊢ (3 ∈ ℂ ∧ 3 ≠ 0) |
39 | divsubdir 11933 | . . . . . 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 12365 | . . . . . 6 ⊢ (3 − 1) = 2 | |
42 | 41 | oveq1i 7423 | . . . . 5 ⊢ ((3 − 1) / 3) = (2 / 3) |
43 | 35, 37 | dividi 11972 | . . . . . 6 ⊢ (3 / 3) = 1 |
44 | 43 | oveq1i 7423 | . . . . 5 ⊢ ((3 / 3) − (1 / 3)) = (1 − (1 / 3)) |
45 | 40, 42, 44 | 3eqtr3ri 2762 | . . . 4 ⊢ (1 − (1 / 3)) = (2 / 3) |
46 | 34, 45 | oveq12i 7425 | . . 3 ⊢ (((1 / 3)↑1) / (1 − (1 / 3))) = ((1 / 3) / (2 / 3)) |
47 | 2cnne0 12447 | . . . 4 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
48 | divcan7 11948 | . . . 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 2753 | . 2 ⊢ (((1 / 3)↑1) / (1 − (1 / 3))) = (1 / 2) |
51 | 32, 50 | breqtri 5169 | 1 ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (1 / 2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ≠ wne 2930 ⊆ wss 3941 ifcif 4525 𝒫 cpw 4599 class class class wbr 5144 ↦ cmpt 5227 ‘cfv 6543 (class class class)co 7413 ℂcc 11131 ℝcr 11132 0cc0 11133 1c1 11134 + caddc 11136 < clt 11273 ≤ cle 11274 − cmin 11469 / cdiv 11896 ℕcn 12237 2c2 12292 3c3 12293 ℕ0cn0 12497 ℤ≥cuz 12847 seqcseq 13993 ↑cexp 14053 abscabs 15208 ⇝ cli 15455 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-fl 13784 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-rlim 15460 df-sum 15660 |
This theorem is referenced by: rpnnen2lem5 16189 rpnnen2lem12 16196 |
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