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Mirrors > Home > MPE Home > Th. List > rpnnen1lem4 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen1 12370. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-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 |
---|---|
rpnnen1lem4 | ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpnnen1lem.1 | . . . . 5 ⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} | |
2 | rpnnen1lem.2 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))) | |
3 | rpnnen1lem.n | . . . . 5 ⊢ ℕ ∈ V | |
4 | rpnnen1lem.q | . . . . 5 ⊢ ℚ ∈ V | |
5 | 1, 2, 3, 4 | rpnnen1lem1 12365 | . . . 4 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) ∈ (ℚ ↑m ℕ)) |
6 | 4, 3 | elmap 8418 | . . . 4 ⊢ ((𝐹‘𝑥) ∈ (ℚ ↑m ℕ) ↔ (𝐹‘𝑥):ℕ⟶ℚ) |
7 | 5, 6 | sylib 221 | . . 3 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥):ℕ⟶ℚ) |
8 | frn 6493 | . . . 4 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ⊆ ℚ) | |
9 | qssre 12346 | . . . 4 ⊢ ℚ ⊆ ℝ | |
10 | 8, 9 | sstrdi 3927 | . . 3 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ⊆ ℝ) |
11 | 7, 10 | syl 17 | . 2 ⊢ (𝑥 ∈ ℝ → ran (𝐹‘𝑥) ⊆ ℝ) |
12 | 1nn 11636 | . . . . . 6 ⊢ 1 ∈ ℕ | |
13 | 12 | ne0ii 4253 | . . . . 5 ⊢ ℕ ≠ ∅ |
14 | fdm 6495 | . . . . . 6 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → dom (𝐹‘𝑥) = ℕ) | |
15 | 14 | neeq1d 3046 | . . . . 5 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → (dom (𝐹‘𝑥) ≠ ∅ ↔ ℕ ≠ ∅)) |
16 | 13, 15 | mpbiri 261 | . . . 4 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → dom (𝐹‘𝑥) ≠ ∅) |
17 | dm0rn0 5759 | . . . . 5 ⊢ (dom (𝐹‘𝑥) = ∅ ↔ ran (𝐹‘𝑥) = ∅) | |
18 | 17 | necon3bii 3039 | . . . 4 ⊢ (dom (𝐹‘𝑥) ≠ ∅ ↔ ran (𝐹‘𝑥) ≠ ∅) |
19 | 16, 18 | sylib 221 | . . 3 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ≠ ∅) |
20 | 7, 19 | syl 17 | . 2 ⊢ (𝑥 ∈ ℝ → ran (𝐹‘𝑥) ≠ ∅) |
21 | 1, 2, 3, 4 | rpnnen1lem3 12366 | . . 3 ⊢ (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥) |
22 | breq2 5034 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑛 ≤ 𝑦 ↔ 𝑛 ≤ 𝑥)) | |
23 | 22 | ralbidv 3162 | . . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦 ↔ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥)) |
24 | 23 | rspcev 3571 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦) |
25 | 21, 24 | mpdan 686 | . 2 ⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦) |
26 | suprcl 11588 | . 2 ⊢ ((ran (𝐹‘𝑥) ⊆ ℝ ∧ ran (𝐹‘𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦) → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ) | |
27 | 11, 20, 25, 26 | syl3anc 1368 | 1 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 {crab 3110 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 supcsup 8888 ℝcr 10525 1c1 10527 < clt 10664 ≤ cle 10665 / cdiv 11286 ℕcn 11625 ℤcz 11969 ℚcq 12336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-q 12337 |
This theorem is referenced by: rpnnen1lem5 12368 |
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