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Mirrors > Home > MPE Home > Th. List > rpnnen1lem6 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen1 12028. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 15-Aug-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
rpnnen1lem.1 | ⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} |
rpnnen1lem.2 | ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))) |
rpnnen1lem.n | ⊢ ℕ ∈ V |
rpnnen1lem.q | ⊢ ℚ ∈ V |
Ref | Expression |
---|---|
rpnnen1lem6 | ⊢ ℝ ≼ (ℚ ↑𝑚 ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 6827 | . 2 ⊢ (ℚ ↑𝑚 ℕ) ∈ V | |
2 | rpnnen1lem.1 | . . . 4 ⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} | |
3 | rpnnen1lem.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))) | |
4 | rpnnen1lem.n | . . . 4 ⊢ ℕ ∈ V | |
5 | rpnnen1lem.q | . . . 4 ⊢ ℚ ∈ V | |
6 | 2, 3, 4, 5 | rpnnen1lem1 12023 | . . 3 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) ∈ (ℚ ↑𝑚 ℕ)) |
7 | rneq 5488 | . . . . . 6 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → ran (𝐹‘𝑥) = ran (𝐹‘𝑦)) | |
8 | 7 | supeq1d 8512 | . . . . 5 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → sup(ran (𝐹‘𝑥), ℝ, < ) = sup(ran (𝐹‘𝑦), ℝ, < )) |
9 | 2, 3, 4, 5 | rpnnen1lem5 12026 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥) |
10 | fveq2 6333 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
11 | 10 | rneqd 5490 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ran (𝐹‘𝑥) = ran (𝐹‘𝑦)) |
12 | 11 | supeq1d 8512 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → sup(ran (𝐹‘𝑥), ℝ, < ) = sup(ran (𝐹‘𝑦), ℝ, < )) |
13 | id 22 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
14 | 12, 13 | eqeq12d 2786 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥 ↔ sup(ran (𝐹‘𝑦), ℝ, < ) = 𝑦)) |
15 | 14, 9 | vtoclga 3423 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → sup(ran (𝐹‘𝑦), ℝ, < ) = 𝑦) |
16 | 9, 15 | eqeqan12d 2787 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) = sup(ran (𝐹‘𝑦), ℝ, < ) ↔ 𝑥 = 𝑦)) |
17 | 8, 16 | syl5ib 234 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) |
18 | 17, 10 | impbid1 215 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦)) |
19 | 6, 18 | dom2 8156 | . 2 ⊢ ((ℚ ↑𝑚 ℕ) ∈ V → ℝ ≼ (ℚ ↑𝑚 ℕ)) |
20 | 1, 19 | ax-mp 5 | 1 ⊢ ℝ ≼ (ℚ ↑𝑚 ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 = wceq 1631 ∈ wcel 2145 {crab 3065 Vcvv 3351 class class class wbr 4787 ↦ cmpt 4864 ran crn 5251 ‘cfv 6030 (class class class)co 6796 ↑𝑚 cmap 8013 ≼ cdom 8111 supcsup 8506 ℝcr 10141 < clt 10280 / cdiv 10890 ℕcn 11226 ℤcz 11584 ℚcq 11996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-er 7900 df-map 8015 df-en 8114 df-dom 8115 df-sdom 8116 df-sup 8508 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-n0 11500 df-z 11585 df-q 11997 |
This theorem is referenced by: rpnnen1 12028 |
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