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Theorem rpnnen1lem6 12369
Description: Lemma for rpnnen1 12370. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 15-Aug-2021.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
rpnnen1lem.1 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
rpnnen1lem.2 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
rpnnen1lem.n ℕ ∈ V
rpnnen1lem.q ℚ ∈ V
Assertion
Ref Expression
rpnnen1lem6 ℝ ≼ (ℚ ↑m ℕ)
Distinct variable groups:   𝑘,𝐹,𝑛,𝑥   𝑇,𝑛
Allowed substitution hints:   𝑇(𝑥,𝑘)

Proof of Theorem rpnnen1lem6
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ovex 7178 . 2 (ℚ ↑m ℕ) ∈ V
2 rpnnen1lem.1 . . . 4 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
3 rpnnen1lem.2 . . . 4 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
4 rpnnen1lem.n . . . 4 ℕ ∈ V
5 rpnnen1lem.q . . . 4 ℚ ∈ V
62, 3, 4, 5rpnnen1lem1 12365 . . 3 (𝑥 ∈ ℝ → (𝐹𝑥) ∈ (ℚ ↑m ℕ))
7 rneq 5799 . . . . . 6 ((𝐹𝑥) = (𝐹𝑦) → ran (𝐹𝑥) = ran (𝐹𝑦))
87supeq1d 8898 . . . . 5 ((𝐹𝑥) = (𝐹𝑦) → sup(ran (𝐹𝑥), ℝ, < ) = sup(ran (𝐹𝑦), ℝ, < ))
92, 3, 4, 5rpnnen1lem5 12368 . . . . . 6 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) = 𝑥)
10 fveq2 6663 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1110rneqd 5801 . . . . . . . . 9 (𝑥 = 𝑦 → ran (𝐹𝑥) = ran (𝐹𝑦))
1211supeq1d 8898 . . . . . . . 8 (𝑥 = 𝑦 → sup(ran (𝐹𝑥), ℝ, < ) = sup(ran (𝐹𝑦), ℝ, < ))
13 id 22 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
1412, 13eqeq12d 2834 . . . . . . 7 (𝑥 = 𝑦 → (sup(ran (𝐹𝑥), ℝ, < ) = 𝑥 ↔ sup(ran (𝐹𝑦), ℝ, < ) = 𝑦))
1514, 9vtoclga 3571 . . . . . 6 (𝑦 ∈ ℝ → sup(ran (𝐹𝑦), ℝ, < ) = 𝑦)
169, 15eqeqan12d 2835 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) = sup(ran (𝐹𝑦), ℝ, < ) ↔ 𝑥 = 𝑦))
178, 16syl5ib 245 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
1817, 10impbid1 226 . . 3 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝑥 = 𝑦))
196, 18dom2 8540 . 2 ((ℚ ↑m ℕ) ∈ V → ℝ ≼ (ℚ ↑m ℕ))
201, 19ax-mp 5 1 ℝ ≼ (ℚ ↑m ℕ)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492   class class class wbr 5057  cmpt 5137  ran crn 5549  cfv 6348  (class class class)co 7145  m cmap 8395  cdom 8495  supcsup 8892  cr 10524   < clt 10663   / cdiv 11285  cn 11626  cz 11969  cq 12336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-n0 11886  df-z 11970  df-q 12337
This theorem is referenced by:  rpnnen1  12370
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