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Mirrors > Home > MPE Home > Th. List > rpnnen1 | Structured version Visualization version GIF version |
Description: One half of rpnnen 16224, where we show an injection from the real numbers to sequences of rational numbers. Specifically, we map a real number 𝑥 to the sequence (𝐹‘𝑥):ℕ⟶ℚ (see rpnnen1lem6 13013) such that ((𝐹‘𝑥)‘𝑘) is the largest rational number with denominator 𝑘 that is strictly less than 𝑥. In this manner, we get a monotonically increasing sequence that converges to 𝑥, and since each sequence converges to a unique real number, this mapping from reals to sequences of rational numbers is injective. Note: The ℕ and ℚ existence hypotheses provide for use with either nnex 12265 and qex 12992, or nnexALT 12261 and qexALT 12995. The proof should not be modified to use any of those 4 theorems. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 16-Jun-2013.) (Revised by NM, 15-Aug-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
rpnnen1.n | ⊢ ℕ ∈ V |
rpnnen1.q | ⊢ ℚ ∈ V |
Ref | Expression |
---|---|
rpnnen1 | ⊢ ℝ ≼ (ℚ ↑m ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7430 | . . . 4 ⊢ (𝑚 = 𝑛 → (𝑚 / 𝑘) = (𝑛 / 𝑘)) | |
2 | 1 | breq1d 5162 | . . 3 ⊢ (𝑚 = 𝑛 → ((𝑚 / 𝑘) < 𝑥 ↔ (𝑛 / 𝑘) < 𝑥)) |
3 | 2 | cbvrabv 3429 | . 2 ⊢ {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥} = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} |
4 | oveq2 7431 | . . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝑚 / 𝑗) = (𝑚 / 𝑘)) | |
5 | 4 | breq1d 5162 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝑚 / 𝑗) < 𝑦 ↔ (𝑚 / 𝑘) < 𝑦)) |
6 | 5 | rabbidv 3426 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → {𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦} = {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}) |
7 | 6 | supeq1d 9485 | . . . . . 6 ⊢ (𝑗 = 𝑘 → sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) = sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < )) |
8 | id 22 | . . . . . 6 ⊢ (𝑗 = 𝑘 → 𝑗 = 𝑘) | |
9 | 7, 8 | oveq12d 7441 | . . . . 5 ⊢ (𝑗 = 𝑘 → (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗) = (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘)) |
10 | 9 | cbvmptv 5265 | . . . 4 ⊢ (𝑗 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗)) = (𝑘 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘)) |
11 | breq2 5156 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑚 / 𝑘) < 𝑦 ↔ (𝑚 / 𝑘) < 𝑥)) | |
12 | 11 | rabbidv 3426 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦} = {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}) |
13 | 12 | supeq1d 9485 | . . . . . 6 ⊢ (𝑦 = 𝑥 → sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) = sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < )) |
14 | 13 | oveq1d 7438 | . . . . 5 ⊢ (𝑦 = 𝑥 → (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘) = (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ) / 𝑘)) |
15 | 14 | mpteq2dv 5254 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑘 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘)) = (𝑘 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ) / 𝑘))) |
16 | 10, 15 | eqtrid 2777 | . . 3 ⊢ (𝑦 = 𝑥 → (𝑗 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗)) = (𝑘 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ) / 𝑘))) |
17 | 16 | cbvmptv 5265 | . 2 ⊢ (𝑦 ∈ ℝ ↦ (𝑗 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗))) = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ) / 𝑘))) |
18 | rpnnen1.n | . 2 ⊢ ℕ ∈ V | |
19 | rpnnen1.q | . 2 ⊢ ℚ ∈ V | |
20 | 3, 17, 18, 19 | rpnnen1lem6 13013 | 1 ⊢ ℝ ≼ (ℚ ↑m ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 {crab 3418 Vcvv 3461 class class class wbr 5152 ↦ cmpt 5235 (class class class)co 7423 ↑m cmap 8854 ≼ cdom 8971 supcsup 9479 ℝcr 11153 < clt 11294 / cdiv 11917 ℕcn 12259 ℤcz 12605 ℚcq 12979 |
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 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 |
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 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-map 8856 df-en 8974 df-dom 8975 df-sdom 8976 df-sup 9481 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-n0 12520 df-z 12606 df-q 12980 |
This theorem is referenced by: reexALT 13015 rpnnen 16224 |
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