Theorem rpnnen1 12967

Description: One half of rpnnen 16170, 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 (see rpnnen1lem6 12966) 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. Note: The ℕ and ℚ existence hypotheses provide for use with either nnex 12218 and qex 12945, or nnexALT 12214 and qexALT 12948. 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	⊢ ℕ ∈ V
rpnnen1.q	⊢ ℚ ∈ V

Assertion

Ref	Expression
rpnnen1	⊢ ℝ ≼ (ℚ ↑m ℕ)

Proof of Theorem rpnnen1

Dummy variables 𝑗 𝑘 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7416	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑚 / 𝑘) = (𝑛 / 𝑘))
2	1	breq1d 5159	. . 3 ⊢ (𝑚 = 𝑛 → ((𝑚 / 𝑘) < 𝑥 ↔ (𝑛 / 𝑘) < 𝑥))
3	2	cbvrabv 3443	. 2 ⊢ {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥} = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
4		oveq2 7417	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝑚 / 𝑗) = (𝑚 / 𝑘))
5	4	breq1d 5159	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝑚 / 𝑗) < 𝑦 ↔ (𝑚 / 𝑘) < 𝑦))
6	5	rabbidv 3441	. . . . . . 7 ⊢ (𝑗 = 𝑘 → {𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦} = {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦})
7	6	supeq1d 9441	. . . . . 6 ⊢ (𝑗 = 𝑘 → sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) = sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ))
8		id 22	. . . . . 6 ⊢ (𝑗 = 𝑘 → 𝑗 = 𝑘)
9	7, 8	oveq12d 7427	. . . . 5 ⊢ (𝑗 = 𝑘 → (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗) = (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘))
10	9	cbvmptv 5262	. . . 4 ⊢ (𝑗 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗)) = (𝑘 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘))
11		breq2 5153	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑚 / 𝑘) < 𝑦 ↔ (𝑚 / 𝑘) < 𝑥))
12	11	rabbidv 3441	. . . . . . 7 ⊢ (𝑦 = 𝑥 → {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦} = {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥})
13	12	supeq1d 9441	. . . . . 6 ⊢ (𝑦 = 𝑥 → sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) = sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ))
14	13	oveq1d 7424	. . . . 5 ⊢ (𝑦 = 𝑥 → (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘) = (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ) / 𝑘))
15	14	mpteq2dv 5251	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑘 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘)) = (𝑘 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ) / 𝑘)))
16	10, 15	eqtrid 2785	. . 3 ⊢ (𝑦 = 𝑥 → (𝑗 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗)) = (𝑘 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ) / 𝑘)))
17	16	cbvmptv 5262	. 2 ⊢ (𝑦 ∈ ℝ ↦ (𝑗 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗))) = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ) / 𝑘)))
18		rpnnen1.n	. 2 ⊢ ℕ ∈ V
19		rpnnen1.q	. 2 ⊢ ℚ ∈ V
20	3, 17, 18, 19	rpnnen1lem6 12966	1 ⊢ ℝ ≼ (ℚ ↑m ℕ)

Syntax hints:   ∈ wcel 2107  {crab 3433  Vcvv 3475   class class class wbr 5149   ↦ cmpt 5232  (class class class)co 7409   ↑_m cmap 8820   ≼ cdom 8937  supcsup 9435  ℝcr 11109   < clt 11248   / cdiv 11871  ℕcn 12212  ℤcz 12558  ℚcq 12932

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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188

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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-n0 12473  df-z 12559  df-q 12933

This theorem is referenced by:  reexALT  12968  rpnnen  16170