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| Mirrors > Home > MPE Home > Th. List > rpnnen2lem3 | Structured version Visualization version GIF version | ||
| Description: Lemma for rpnnen2 16262. (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 11261 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
| 2 | 3nn 12345 | . . . . . . 7 ⊢ 3 ∈ ℕ | |
| 3 | nndivre 12307 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℕ) → (1 / 3) ∈ ℝ) | |
| 4 | 1, 2, 3 | mp2an 692 | . . . . . 6 ⊢ (1 / 3) ∈ ℝ |
| 5 | 4 | recni 11275 | . . . . 5 ⊢ (1 / 3) ∈ ℂ |
| 6 | 5 | a1i 11 | . . . 4 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
| 7 | 0re 11263 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
| 8 | 3re 12346 | . . . . . . . . 9 ⊢ 3 ∈ ℝ | |
| 9 | 3pos 12371 | . . . . . . . . 9 ⊢ 0 < 3 | |
| 10 | 8, 9 | recgt0ii 12174 | . . . . . . . 8 ⊢ 0 < (1 / 3) |
| 11 | 7, 4, 10 | ltleii 11384 | . . . . . . 7 ⊢ 0 ≤ (1 / 3) |
| 12 | absid 15335 | . . . . . . 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 12439 | . . . . . . 7 ⊢ 1 < 3 | |
| 15 | recgt1 12164 | . . . . . . . 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 5164 | . . . . 5 ⊢ (abs‘(1 / 3)) < 1 |
| 19 | 18 | a1i 11 | . . . 4 ⊢ (⊤ → (abs‘(1 / 3)) < 1) |
| 20 | 1nn0 12542 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
| 21 | 20 | a1i 11 | . . . 4 ⊢ (⊤ → 1 ∈ ℕ0) |
| 22 | ssid 4006 | . . . . . 6 ⊢ ℕ ⊆ ℕ | |
| 23 | simpr 484 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ (ℤ≥‘1)) → 𝑘 ∈ (ℤ≥‘1)) | |
| 24 | nnuz 12921 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
| 25 | 23, 24 | eleqtrrdi 2852 | . . . . . 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 16250 | . . . . . 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 4533 | . . . . 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 15907 | . . 3 ⊢ (⊤ → seq1( + , (𝐹‘ℕ)) ⇝ (((1 / 3)↑1) / (1 − (1 / 3)))) |
| 32 | 31 | mptru 1547 | . 2 ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (((1 / 3)↑1) / (1 − (1 / 3))) |
| 33 | exp1 14108 | . . . . 5 ⊢ ((1 / 3) ∈ ℂ → ((1 / 3)↑1) = (1 / 3)) | |
| 34 | 5, 33 | ax-mp 5 | . . . 4 ⊢ ((1 / 3)↑1) = (1 / 3) |
| 35 | 3cn 12347 | . . . . . 6 ⊢ 3 ∈ ℂ | |
| 36 | ax-1cn 11213 | . . . . . 6 ⊢ 1 ∈ ℂ | |
| 37 | 3ne0 12372 | . . . . . . 7 ⊢ 3 ≠ 0 | |
| 38 | 35, 37 | pm3.2i 470 | . . . . . 6 ⊢ (3 ∈ ℂ ∧ 3 ≠ 0) |
| 39 | divsubdir 11961 | . . . . . 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 12394 | . . . . . 6 ⊢ (3 − 1) = 2 | |
| 42 | 41 | oveq1i 7441 | . . . . 5 ⊢ ((3 − 1) / 3) = (2 / 3) |
| 43 | 35, 37 | dividi 12000 | . . . . . 6 ⊢ (3 / 3) = 1 |
| 44 | 43 | oveq1i 7441 | . . . . 5 ⊢ ((3 / 3) − (1 / 3)) = (1 − (1 / 3)) |
| 45 | 40, 42, 44 | 3eqtr3ri 2774 | . . . 4 ⊢ (1 − (1 / 3)) = (2 / 3) |
| 46 | 34, 45 | oveq12i 7443 | . . 3 ⊢ (((1 / 3)↑1) / (1 − (1 / 3))) = ((1 / 3) / (2 / 3)) |
| 47 | 2cnne0 12476 | . . . 4 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
| 48 | divcan7 11976 | . . . 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 2765 | . 2 ⊢ (((1 / 3)↑1) / (1 − (1 / 3))) = (1 / 2) |
| 51 | 32, 50 | breqtri 5168 | 1 ⊢ seq1( + , (𝐹‘ℕ)) ⇝ (1 / 2) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ifcif 4525 𝒫 cpw 4600 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 3c3 12322 ℕ0cn0 12526 ℤ≥cuz 12878 seqcseq 14042 ↑cexp 14102 abscabs 15273 ⇝ cli 15520 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 |
| This theorem is referenced by: rpnnen2lem5 16254 rpnnen2lem12 16261 |
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