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Theorem rpnnen1lem6 11763
 Description: Lemma for rpnnen1 11764. (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 6632 . 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 11759 . . 3 (𝑥 ∈ ℝ → (𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ))
7 rneq 5311 . . . . . 6 ((𝐹𝑥) = (𝐹𝑦) → ran (𝐹𝑥) = ran (𝐹𝑦))
87supeq1d 8296 . . . . 5 ((𝐹𝑥) = (𝐹𝑦) → sup(ran (𝐹𝑥), ℝ, < ) = sup(ran (𝐹𝑦), ℝ, < ))
92, 3, 4, 5rpnnen1lem5 11762 . . . . . 6 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) = 𝑥)
10 fveq2 6148 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1110rneqd 5313 . . . . . . . . 9 (𝑥 = 𝑦 → ran (𝐹𝑥) = ran (𝐹𝑦))
1211supeq1d 8296 . . . . . . . 8 (𝑥 = 𝑦 → sup(ran (𝐹𝑥), ℝ, < ) = sup(ran (𝐹𝑦), ℝ, < ))
13 id 22 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
1412, 13eqeq12d 2636 . . . . . . 7 (𝑥 = 𝑦 → (sup(ran (𝐹𝑥), ℝ, < ) = 𝑥 ↔ sup(ran (𝐹𝑦), ℝ, < ) = 𝑦))
1514, 9vtoclga 3258 . . . . . 6 (𝑦 ∈ ℝ → sup(ran (𝐹𝑦), ℝ, < ) = 𝑦)
169, 15eqeqan12d 2637 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) = sup(ran (𝐹𝑦), ℝ, < ) ↔ 𝑥 = 𝑦))
178, 16syl5ib 234 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
1817, 10impbid1 215 . . 3 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝑥 = 𝑦))
196, 18dom2 7942 . 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 384   = wceq 1480   ∈ wcel 1987  {crab 2911  Vcvv 3186   class class class wbr 4613   ↦ cmpt 4673  ran crn 5075  ‘cfv 5847  (class class class)co 6604   ↑𝑚 cmap 7802   ≼ cdom 7897  supcsup 8290  ℝcr 9879   < clt 10018   / cdiv 10628  ℕcn 10964  ℤcz 11321  ℚcq 11732 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-n0 11237  df-z 11322  df-q 11733 This theorem is referenced by:  rpnnen1  11764
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