Theorem rpnnen1 13025

Description: One half of rpnnen 16263, 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 13024) 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 12272 and qex 13003, or nnexALT 12268 and qexALT 13006. 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 7438	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑚 / 𝑘) = (𝑛 / 𝑘))
2	1	breq1d 5153	. . 3 ⊢ (𝑚 = 𝑛 → ((𝑚 / 𝑘) < 𝑥 ↔ (𝑛 / 𝑘) < 𝑥))
3	2	cbvrabv 3447	. 2 ⊢ {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥} = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
4		oveq2 7439	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝑚 / 𝑗) = (𝑚 / 𝑘))
5	4	breq1d 5153	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝑚 / 𝑗) < 𝑦 ↔ (𝑚 / 𝑘) < 𝑦))
6	5	rabbidv 3444	. . . . . . 7 ⊢ (𝑗 = 𝑘 → {𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦} = {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦})
7	6	supeq1d 9486	. . . . . 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 7449	. . . . 5 ⊢ (𝑗 = 𝑘 → (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗) = (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘))
10	9	cbvmptv 5255	. . . 4 ⊢ (𝑗 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗)) = (𝑘 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘))
11		breq2 5147	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑚 / 𝑘) < 𝑦 ↔ (𝑚 / 𝑘) < 𝑥))
12	11	rabbidv 3444	. . . . . . 7 ⊢ (𝑦 = 𝑥 → {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦} = {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥})
13	12	supeq1d 9486	. . . . . 6 ⊢ (𝑦 = 𝑥 → sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) = sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ))
14	13	oveq1d 7446	. . . . 5 ⊢ (𝑦 = 𝑥 → (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘) = (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ) / 𝑘))
15	14	mpteq2dv 5244	. . . 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 2789	. . 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 5255	. 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 13024	1 ⊢ ℝ ≼ (ℚ ↑m ℕ)

Syntax hints:   ∈ wcel 2108  {crab 3436  Vcvv 3480   class class class wbr 5143   ↦ cmpt 5225  (class class class)co 7431   ↑_m cmap 8866   ≼ cdom 8983  supcsup 9480  ℝcr 11154   < clt 11295   / cdiv 11920  ℕcn 12266  ℤcz 12613  ℚcq 12990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-n0 12527  df-z 12614  df-q 12991

This theorem is referenced by:  reexALT  13026  rpnnen  16263