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Mirrors > Home > MPE Home > Th. List > rpnnen2lem3 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen2 16259. (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 11259 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
2 | 3nn 12343 | . . . . . . 7 ⊢ 3 ∈ ℕ | |
3 | nndivre 12305 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℕ) → (1 / 3) ∈ ℝ) | |
4 | 1, 2, 3 | mp2an 692 | . . . . . 6 ⊢ (1 / 3) ∈ ℝ |
5 | 4 | recni 11273 | . . . . 5 ⊢ (1 / 3) ∈ ℂ |
6 | 5 | a1i 11 | . . . 4 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
7 | 0re 11261 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
8 | 3re 12344 | . . . . . . . . 9 ⊢ 3 ∈ ℝ | |
9 | 3pos 12369 | . . . . . . . . 9 ⊢ 0 < 3 | |
10 | 8, 9 | recgt0ii 12172 | . . . . . . . 8 ⊢ 0 < (1 / 3) |
11 | 7, 4, 10 | ltleii 11382 | . . . . . . 7 ⊢ 0 ≤ (1 / 3) |
12 | absid 15332 | . . . . . . 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 12437 | . . . . . . 7 ⊢ 1 < 3 | |
15 | recgt1 12162 | . . . . . . . 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 230 | . . . . . 6 ⊢ (1 / 3) < 1 |
18 | 13, 17 | eqbrtri 5169 | . . . . 5 ⊢ (abs‘(1 / 3)) < 1 |
19 | 18 | a1i 11 | . . . 4 ⊢ (⊤ → (abs‘(1 / 3)) < 1) |
20 | 1nn0 12540 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
21 | 20 | a1i 11 | . . . 4 ⊢ (⊤ → 1 ∈ ℕ0) |
22 | ssid 4018 | . . . . . 6 ⊢ ℕ ⊆ ℕ | |
23 | simpr 484 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → 𝑘 ∈ (ℤ≥‘1)) | |
24 | nnuz 12919 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
25 | 23, 24 | eleqtrrdi 2850 | . . . . . 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 16247 | . . . . . 6 ⊢ ((ℕ ⊆ ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘ℕ)‘𝑘) = if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0)) |
28 | 22, 25, 27 | sylancr 587 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0)) |
29 | 25 | iftrued 4539 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → if(𝑘 ∈ ℕ, ((1 / 3)↑𝑘), 0) = ((1 / 3)↑𝑘)) |
30 | 28, 29 | eqtrd 2775 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝐹‘ℕ)‘𝑘) = ((1 / 3)↑𝑘)) |
31 | 6, 19, 21, 30 | geolim2 15904 | . . 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 14105 | . . . . 5 ⊢ ((1 / 3) ∈ ℂ → ((1 / 3)↑1) = (1 / 3)) | |
34 | 5, 33 | ax-mp 5 | . . . 4 ⊢ ((1 / 3)↑1) = (1 / 3) |
35 | 3cn 12345 | . . . . . 6 ⊢ 3 ∈ ℂ | |
36 | ax-1cn 11211 | . . . . . 6 ⊢ 1 ∈ ℂ | |
37 | 3ne0 12370 | . . . . . . 7 ⊢ 3 ≠ 0 | |
38 | 35, 37 | pm3.2i 470 | . . . . . 6 ⊢ (3 ∈ ℂ ∧ 3 ≠ 0) |
39 | divsubdir 11959 | . . . . . 6 ⊢ ((3 ∈ ℂ ∧ 1 ∈ ℂ ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((3 − 1) / 3) = ((3 / 3) − (1 / 3))) | |
40 | 35, 36, 38, 39 | mp3an 1460 | . . . . 5 ⊢ ((3 − 1) / 3) = ((3 / 3) − (1 / 3)) |
41 | 3m1e2 12392 | . . . . . 6 ⊢ (3 − 1) = 2 | |
42 | 41 | oveq1i 7441 | . . . . 5 ⊢ ((3 − 1) / 3) = (2 / 3) |
43 | 35, 37 | dividi 11998 | . . . . . 6 ⊢ (3 / 3) = 1 |
44 | 43 | oveq1i 7441 | . . . . 5 ⊢ ((3 / 3) − (1 / 3)) = (1 − (1 / 3)) |
45 | 40, 42, 44 | 3eqtr3ri 2772 | . . . 4 ⊢ (1 − (1 / 3)) = (2 / 3) |
46 | 34, 45 | oveq12i 7443 | . . 3 ⊢ (((1 / 3)↑1) / (1 − (1 / 3))) = ((1 / 3) / (2 / 3)) |
47 | 2cnne0 12474 | . . . 4 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
48 | divcan7 11974 | . . . 4 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((1 / 3) / (2 / 3)) = (1 / 2)) | |
49 | 36, 47, 38, 48 | mp3an 1460 | . . 3 ⊢ ((1 / 3) / (2 / 3)) = (1 / 2) |
50 | 46, 49 | eqtri 2763 | . 2 ⊢ (((1 / 3)↑1) / (1 − (1 / 3))) = (1 / 2) |
51 | 32, 50 | breqtri 5173 | 1 ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (1 / 2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ifcif 4531 𝒫 cpw 4605 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℕcn 12264 2c2 12319 3c3 12320 ℕ0cn0 12524 ℤ≥cuz 12876 seqcseq 14039 ↑cexp 14099 abscabs 15270 ⇝ cli 15517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 |
This theorem is referenced by: rpnnen2lem5 16251 rpnnen2lem12 16258 |
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