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Mirrors > Home > MPE Home > Th. List > rpnnen1OLD | Structured version Visualization version GIF version |
Description: One half of rpnnen 14956, 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 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. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 16-Jun-2013.) Obsolete version of rpnnen1 11820 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
rpnnen1.1OLD | ⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} |
rpnnen1.2OLD | ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))) |
Ref | Expression |
---|---|
rpnnen1OLD | ⊢ ℝ ≼ (ℚ ↑𝑚 ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 6678 | . 2 ⊢ (ℚ ↑𝑚 ℕ) ∈ V | |
2 | rpnnen1.1OLD | . . . 4 ⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} | |
3 | rpnnen1.2OLD | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))) | |
4 | 2, 3 | rpnnen1lem1OLD 11821 | . . 3 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) ∈ (ℚ ↑𝑚 ℕ)) |
5 | rneq 5351 | . . . . . 6 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → ran (𝐹‘𝑥) = ran (𝐹‘𝑦)) | |
6 | 5 | supeq1d 8352 | . . . . 5 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → sup(ran (𝐹‘𝑥), ℝ, < ) = sup(ran (𝐹‘𝑦), ℝ, < )) |
7 | 2, 3 | rpnnen1lem5OLD 11824 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥) |
8 | fveq2 6191 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
9 | 8 | rneqd 5353 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ran (𝐹‘𝑥) = ran (𝐹‘𝑦)) |
10 | 9 | supeq1d 8352 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → sup(ran (𝐹‘𝑥), ℝ, < ) = sup(ran (𝐹‘𝑦), ℝ, < )) |
11 | id 22 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
12 | 10, 11 | eqeq12d 2637 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥 ↔ sup(ran (𝐹‘𝑦), ℝ, < ) = 𝑦)) |
13 | 12, 7 | vtoclga 3272 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → sup(ran (𝐹‘𝑦), ℝ, < ) = 𝑦) |
14 | 7, 13 | eqeqan12d 2638 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) = sup(ran (𝐹‘𝑦), ℝ, < ) ↔ 𝑥 = 𝑦)) |
15 | 6, 14 | syl5ib 234 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) |
16 | 15, 8 | impbid1 215 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦)) |
17 | 4, 16 | dom2 7998 | . 2 ⊢ ((ℚ ↑𝑚 ℕ) ∈ V → ℝ ≼ (ℚ ↑𝑚 ℕ)) |
18 | 1, 17 | ax-mp 5 | 1 ⊢ ℝ ≼ (ℚ ↑𝑚 ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 class class class wbr 4653 ↦ cmpt 4729 ran crn 5115 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 ≼ cdom 7953 supcsup 8346 ℝcr 9935 < clt 10074 / cdiv 10684 ℕcn 11020 ℤcz 11377 ℚcq 11788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-q 11789 |
This theorem is referenced by: (None) |
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