Theorem rpnnen1 13111

Description: One half of rpnnen 16395, 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 13110) 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 12341 and qex 13088, or nnexALT 12337 and qexALT 13091. 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 7427	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑚 / 𝑘) = (𝑛 / 𝑘))
2	1	breq1d 5112	. . 3 ⊢ (𝑚 = 𝑛 → ((𝑚 / 𝑘) < 𝑥 ↔ (𝑛 / 𝑘) < 𝑥))
3	2	cbvrabv 3422	. 2 ⊢ {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥} = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
4		oveq2 7428	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝑚 / 𝑗) = (𝑚 / 𝑘))
5	4	breq1d 5112	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝑚 / 𝑗) < 𝑦 ↔ (𝑚 / 𝑘) < 𝑦))
6	5	rabbidv 3419	. . . . . . 7 ⊢ (𝑗 = 𝑘 → {𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦} = {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦})
7	6	supeq1d 9438	. . . . . 6 ⊢ (𝑗 = 𝑘 → sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) = sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ))
8		id 23	. . . . . 6 ⊢ (𝑗 = 𝑘 → 𝑗 = 𝑘)
9	7, 8	oveq12d 7438	. . . . 5 ⊢ (𝑗 = 𝑘 → (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗) = (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘))
10	9	cbvmptv 5208	. . . 4 ⊢ (𝑗 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑗) < 𝑦}, ℝ, < ) / 𝑗)) = (𝑘 ∈ ℕ ↦ (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘))
11		breq2 5106	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑚 / 𝑘) < 𝑦 ↔ (𝑚 / 𝑘) < 𝑥))
12	11	rabbidv 3419	. . . . . . 7 ⊢ (𝑦 = 𝑥 → {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦} = {𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥})
13	12	supeq1d 9438	. . . . . 6 ⊢ (𝑦 = 𝑥 → sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) = sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ))
14	13	oveq1d 7435	. . . . 5 ⊢ (𝑦 = 𝑥 → (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑦}, ℝ, < ) / 𝑘) = (sup({𝑚 ∈ ℤ ∣ (𝑚 / 𝑘) < 𝑥}, ℝ, < ) / 𝑘))
15	14	mpteq2dv 5198	. . . 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 2807	. . 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 5208	. 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 13110	1 ⊢ ℝ ≼ (ℚ ↑m ℕ)

This proof depends on syntax axioms:   ∈ wcel 2144  {crab 3412  Vcvv 3450   class class class wbr 5102   ↦ cmpt 5185  (class class class)co 7420   ↑_m cmap 8847   ≼ cdom 8971  supcsup 9432  ℝcr 11199   < clt 11343   / cdiv 11973  ℕcn 12335  ℤcz 12693  ℚcq 13075

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9434  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-n0 12607  df-z 12694  df-q 13076

This theorem is used by:  reexALT  13112  rpnnen  16395