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Theorem rpnnen1lem6 12236
Description: Lemma for rpnnen1 12237. (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 7053 . 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 12232 . . 3 (𝑥 ∈ ℝ → (𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ))
7 rneq 5693 . . . . . 6 ((𝐹𝑥) = (𝐹𝑦) → ran (𝐹𝑥) = ran (𝐹𝑦))
87supeq1d 8761 . . . . 5 ((𝐹𝑥) = (𝐹𝑦) → sup(ran (𝐹𝑥), ℝ, < ) = sup(ran (𝐹𝑦), ℝ, < ))
92, 3, 4, 5rpnnen1lem5 12235 . . . . . 6 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) = 𝑥)
10 fveq2 6543 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1110rneqd 5695 . . . . . . . . 9 (𝑥 = 𝑦 → ran (𝐹𝑥) = ran (𝐹𝑦))
1211supeq1d 8761 . . . . . . . 8 (𝑥 = 𝑦 → sup(ran (𝐹𝑥), ℝ, < ) = sup(ran (𝐹𝑦), ℝ, < ))
13 id 22 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
1412, 13eqeq12d 2810 . . . . . . 7 (𝑥 = 𝑦 → (sup(ran (𝐹𝑥), ℝ, < ) = 𝑥 ↔ sup(ran (𝐹𝑦), ℝ, < ) = 𝑦))
1514, 9vtoclga 3517 . . . . . 6 (𝑦 ∈ ℝ → sup(ran (𝐹𝑦), ℝ, < ) = 𝑦)
169, 15eqeqan12d 2811 . . . . 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 8405 . 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 1522  wcel 2081  {crab 3109  Vcvv 3437   class class class wbr 4966  cmpt 5045  ran crn 5449  cfv 6230  (class class class)co 7021  𝑚 cmap 8261  cdom 8360  supcsup 8755  cr 10387   < clt 10526   / cdiv 11150  cn 11491  cz 11834  cq 12202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465  ax-pre-sup 10466
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-om 7442  df-1st 7550  df-2nd 7551  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-er 8144  df-map 8263  df-en 8363  df-dom 8364  df-sdom 8365  df-sup 8757  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-div 11151  df-nn 11492  df-n0 11751  df-z 11835  df-q 12203
This theorem is referenced by:  rpnnen1  12237
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