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Theorem rpnnen1 12915
Description: One half of rpnnen 16116, where we show an injection from the real numbers to sequences of rational numbers. Specifically, we map a real number π‘₯ to the sequence (πΉβ€˜π‘₯):β„•βŸΆβ„š (see rpnnen1lem6 12914) such that ((πΉβ€˜π‘₯)β€˜π‘˜) is the largest rational number with denominator π‘˜ that is strictly less than π‘₯. In this manner, we get a monotonically increasing sequence that converges to π‘₯, and since each sequence converges to a unique real number, this mapping from reals to sequences of rational numbers is injective. Note: The β„• and β„š existence hypotheses provide for use with either nnex 12166 and qex 12893, or nnexALT 12162 and qexALT 12896. 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.)
Hypotheses
Ref Expression
rpnnen1.n β„• ∈ V
rpnnen1.q β„š ∈ V
Assertion
Ref Expression
rpnnen1 ℝ β‰Ό (β„š ↑m β„•)

Proof of Theorem rpnnen1
Dummy variables 𝑗 π‘˜ π‘š 𝑛 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7369 . . . 4 (π‘š = 𝑛 β†’ (π‘š / π‘˜) = (𝑛 / π‘˜))
21breq1d 5120 . . 3 (π‘š = 𝑛 β†’ ((π‘š / π‘˜) < π‘₯ ↔ (𝑛 / π‘˜) < π‘₯))
32cbvrabv 3420 . 2 {π‘š ∈ β„€ ∣ (π‘š / π‘˜) < π‘₯} = {𝑛 ∈ β„€ ∣ (𝑛 / π‘˜) < π‘₯}
4 oveq2 7370 . . . . . . . . 9 (𝑗 = π‘˜ β†’ (π‘š / 𝑗) = (π‘š / π‘˜))
54breq1d 5120 . . . . . . . 8 (𝑗 = π‘˜ β†’ ((π‘š / 𝑗) < 𝑦 ↔ (π‘š / π‘˜) < 𝑦))
65rabbidv 3418 . . . . . . 7 (𝑗 = π‘˜ β†’ {π‘š ∈ β„€ ∣ (π‘š / 𝑗) < 𝑦} = {π‘š ∈ β„€ ∣ (π‘š / π‘˜) < 𝑦})
76supeq1d 9389 . . . . . 6 (𝑗 = π‘˜ β†’ sup({π‘š ∈ β„€ ∣ (π‘š / 𝑗) < 𝑦}, ℝ, < ) = sup({π‘š ∈ β„€ ∣ (π‘š / π‘˜) < 𝑦}, ℝ, < ))
8 id 22 . . . . . 6 (𝑗 = π‘˜ β†’ 𝑗 = π‘˜)
97, 8oveq12d 7380 . . . . 5 (𝑗 = π‘˜ β†’ (sup({π‘š ∈ β„€ ∣ (π‘š / 𝑗) < 𝑦}, ℝ, < ) / 𝑗) = (sup({π‘š ∈ β„€ ∣ (π‘š / π‘˜) < 𝑦}, ℝ, < ) / π‘˜))
109cbvmptv 5223 . . . 4 (𝑗 ∈ β„• ↦ (sup({π‘š ∈ β„€ ∣ (π‘š / 𝑗) < 𝑦}, ℝ, < ) / 𝑗)) = (π‘˜ ∈ β„• ↦ (sup({π‘š ∈ β„€ ∣ (π‘š / π‘˜) < 𝑦}, ℝ, < ) / π‘˜))
11 breq2 5114 . . . . . . . 8 (𝑦 = π‘₯ β†’ ((π‘š / π‘˜) < 𝑦 ↔ (π‘š / π‘˜) < π‘₯))
1211rabbidv 3418 . . . . . . 7 (𝑦 = π‘₯ β†’ {π‘š ∈ β„€ ∣ (π‘š / π‘˜) < 𝑦} = {π‘š ∈ β„€ ∣ (π‘š / π‘˜) < π‘₯})
1312supeq1d 9389 . . . . . 6 (𝑦 = π‘₯ β†’ sup({π‘š ∈ β„€ ∣ (π‘š / π‘˜) < 𝑦}, ℝ, < ) = sup({π‘š ∈ β„€ ∣ (π‘š / π‘˜) < π‘₯}, ℝ, < ))
1413oveq1d 7377 . . . . 5 (𝑦 = π‘₯ β†’ (sup({π‘š ∈ β„€ ∣ (π‘š / π‘˜) < 𝑦}, ℝ, < ) / π‘˜) = (sup({π‘š ∈ β„€ ∣ (π‘š / π‘˜) < π‘₯}, ℝ, < ) / π‘˜))
1514mpteq2dv 5212 . . . 4 (𝑦 = π‘₯ β†’ (π‘˜ ∈ β„• ↦ (sup({π‘š ∈ β„€ ∣ (π‘š / π‘˜) < 𝑦}, ℝ, < ) / π‘˜)) = (π‘˜ ∈ β„• ↦ (sup({π‘š ∈ β„€ ∣ (π‘š / π‘˜) < π‘₯}, ℝ, < ) / π‘˜)))
1610, 15eqtrid 2789 . . 3 (𝑦 = π‘₯ β†’ (𝑗 ∈ β„• ↦ (sup({π‘š ∈ β„€ ∣ (π‘š / 𝑗) < 𝑦}, ℝ, < ) / 𝑗)) = (π‘˜ ∈ β„• ↦ (sup({π‘š ∈ β„€ ∣ (π‘š / π‘˜) < π‘₯}, ℝ, < ) / π‘˜)))
1716cbvmptv 5223 . 2 (𝑦 ∈ ℝ ↦ (𝑗 ∈ β„• ↦ (sup({π‘š ∈ β„€ ∣ (π‘š / 𝑗) < 𝑦}, ℝ, < ) / 𝑗))) = (π‘₯ ∈ ℝ ↦ (π‘˜ ∈ β„• ↦ (sup({π‘š ∈ β„€ ∣ (π‘š / π‘˜) < π‘₯}, ℝ, < ) / π‘˜)))
18 rpnnen1.n . 2 β„• ∈ V
19 rpnnen1.q . 2 β„š ∈ V
203, 17, 18, 19rpnnen1lem6 12914 1 ℝ β‰Ό (β„š ↑m β„•)
Colors of variables: wff setvar class
Syntax hints:   ∈ wcel 2107  {crab 3410  Vcvv 3448   class class class wbr 5110   ↦ cmpt 5193  (class class class)co 7362   ↑m cmap 8772   β‰Ό cdom 8888  supcsup 9383  β„cr 11057   < clt 11196   / cdiv 11819  β„•cn 12160  β„€cz 12506  β„šcq 12880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9385  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-n0 12421  df-z 12507  df-q 12881
This theorem is referenced by:  reexALT  12916  rpnnen  16116
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