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Mirrors > Home > MPE Home > Th. List > rpnnen1lem6 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen1 12237. (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 7053 | . 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 12232 | . . 3 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) ∈ (ℚ ↑𝑚 ℕ)) |
7 | rneq 5693 | . . . . . 6 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → ran (𝐹‘𝑥) = ran (𝐹‘𝑦)) | |
8 | 7 | supeq1d 8761 | . . . . 5 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → sup(ran (𝐹‘𝑥), ℝ, < ) = sup(ran (𝐹‘𝑦), ℝ, < )) |
9 | 2, 3, 4, 5 | rpnnen1lem5 12235 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥) |
10 | fveq2 6543 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
11 | 10 | rneqd 5695 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ran (𝐹‘𝑥) = ran (𝐹‘𝑦)) |
12 | 11 | supeq1d 8761 | . . . . . . . 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 2810 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥 ↔ sup(ran (𝐹‘𝑦), ℝ, < ) = 𝑦)) |
15 | 14, 9 | vtoclga 3517 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → sup(ran (𝐹‘𝑦), ℝ, < ) = 𝑦) |
16 | 9, 15 | eqeqan12d 2811 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) = sup(ran (𝐹‘𝑦), ℝ, < ) ↔ 𝑥 = 𝑦)) |
17 | 8, 16 | syl5ib 245 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) |
18 | 17, 10 | impbid1 226 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦)) |
19 | 6, 18 | dom2 8405 | . 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 396 = wceq 1522 ∈ wcel 2081 {crab 3109 Vcvv 3437 class class class wbr 4966 ↦ cmpt 5045 ran crn 5449 ‘cfv 6230 (class class class)co 7021 ↑𝑚 cmap 8261 ≼ cdom 8360 supcsup 8755 ℝcr 10387 < clt 10526 / cdiv 11150 ℕcn 11491 ℤcz 11834 ℚcq 12202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 ax-pre-sup 10466 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-er 8144 df-map 8263 df-en 8363 df-dom 8364 df-sdom 8365 df-sup 8757 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-div 11151 df-nn 11492 df-n0 11751 df-z 11835 df-q 12203 |
This theorem is referenced by: rpnnen1 12237 |
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