Theorem rpnnen1 13048

Description: One half of rpnnen 16275, 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 13047) 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 12299 and qex 13026, or nnexALT 12295 and qexALT 13029. 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 7455	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑚 / 𝑘) = (𝑛 / 𝑘))
2	1	breq1d 5176	. . 3 ⊢ (𝑚 = 𝑛 → ((𝑚 / 𝑘) < 𝑥 ↔ (𝑛 / 𝑘) < 𝑥))
3	2	cbvrabv 3454	. 2 ⊢ {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥} = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
4		oveq2 7456	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝑚 / 𝑗) = (𝑚 / 𝑘))
5	4	breq1d 5176	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝑚 / 𝑗) < 𝑦 ↔ (𝑚 / 𝑘) < 𝑦))
6	5	rabbidv 3451	. . . . . . 7 ⊢ (𝑗 = 𝑘 → {𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦} = {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦})
7	6	supeq1d 9515	. . . . . 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 7466	. . . . 5 ⊢ (𝑗 = 𝑘 → (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗) = (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘))
10	9	cbvmptv 5279	. . . 4 ⊢ (𝑗 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗)) = (𝑘 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘))
11		breq2 5170	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑚 / 𝑘) < 𝑦 ↔ (𝑚 / 𝑘) < 𝑥))
12	11	rabbidv 3451	. . . . . . 7 ⊢ (𝑦 = 𝑥 → {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦} = {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥})
13	12	supeq1d 9515	. . . . . 6 ⊢ (𝑦 = 𝑥 → sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) = sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ))
14	13	oveq1d 7463	. . . . 5 ⊢ (𝑦 = 𝑥 → (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘) = (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ) / 𝑘))
15	14	mpteq2dv 5268	. . . 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 2792	. . 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 5279	. 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 13047	1 ⊢ ℝ ≼ (ℚ ↑m ℕ)

Syntax hints:   ∈ wcel 2108  {crab 3443  Vcvv 3488   class class class wbr 5166   ↦ cmpt 5249  (class class class)co 7448   ↑_m cmap 8884   ≼ cdom 9001  supcsup 9509  ℝcr 11183   < clt 11324   / cdiv 11947  ℕcn 12293  ℤcz 12639  ℚcq 13013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-n0 12554  df-z 12640  df-q 13014

This theorem is referenced by:  reexALT  13049  rpnnen  16275