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Theorem rpnnen1OLD 11701
 Description: One half of rpnnen 14795, 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 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. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 16-Jun-2013.) Obsolete version of rpnnen1 11696 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
rpnnen1.1OLD 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
rpnnen1.2OLD 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
Assertion
Ref Expression
rpnnen1OLD ℝ ≼ (ℚ ↑𝑚 ℕ)
Distinct variable groups:   𝑘,𝐹,𝑛,𝑥   𝑇,𝑛
Allowed substitution hints:   𝑇(𝑥,𝑘)

Proof of Theorem rpnnen1OLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ovex 6577 . 2 (ℚ ↑𝑚 ℕ) ∈ V
2 rpnnen1.1OLD . . . 4 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
3 rpnnen1.2OLD . . . 4 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
42, 3rpnnen1lem1OLD 11697 . . 3 (𝑥 ∈ ℝ → (𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ))
5 rneq 5272 . . . . . 6 ((𝐹𝑥) = (𝐹𝑦) → ran (𝐹𝑥) = ran (𝐹𝑦))
65supeq1d 8235 . . . . 5 ((𝐹𝑥) = (𝐹𝑦) → sup(ran (𝐹𝑥), ℝ, < ) = sup(ran (𝐹𝑦), ℝ, < ))
72, 3rpnnen1lem5OLD 11700 . . . . . 6 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) = 𝑥)
8 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
98rneqd 5274 . . . . . . . . 9 (𝑥 = 𝑦 → ran (𝐹𝑥) = ran (𝐹𝑦))
109supeq1d 8235 . . . . . . . 8 (𝑥 = 𝑦 → sup(ran (𝐹𝑥), ℝ, < ) = sup(ran (𝐹𝑦), ℝ, < ))
11 id 22 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
1210, 11eqeq12d 2625 . . . . . . 7 (𝑥 = 𝑦 → (sup(ran (𝐹𝑥), ℝ, < ) = 𝑥 ↔ sup(ran (𝐹𝑦), ℝ, < ) = 𝑦))
1312, 7vtoclga 3245 . . . . . 6 (𝑦 ∈ ℝ → sup(ran (𝐹𝑦), ℝ, < ) = 𝑦)
147, 13eqeqan12d 2626 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) = sup(ran (𝐹𝑦), ℝ, < ) ↔ 𝑥 = 𝑦))
156, 14syl5ib 233 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
1615, 8impbid1 214 . . 3 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝑥 = 𝑦))
174, 16dom2 7884 . 2 ((ℚ ↑𝑚 ℕ) ∈ V → ℝ ≼ (ℚ ↑𝑚 ℕ))
181, 17ax-mp 5 1 ℝ ≼ (ℚ ↑𝑚 ℕ)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744   ≼ cdom 7839  supcsup 8229  ℝcr 9814   < clt 9953   / cdiv 10563  ℕcn 10897  ℤcz 11254  ℚcq 11664 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-n0 11170  df-z 11255  df-q 11665 This theorem is referenced by: (None)
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