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Mirrors > Home > MPE Home > Th. List > rpnnen1lem4 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen1 12383. (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 12378 | . . . 4 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) ∈ (ℚ ↑m ℕ)) |
6 | 4, 3 | elmap 8435 | . . . 4 ⊢ ((𝐹‘𝑥) ∈ (ℚ ↑m ℕ) ↔ (𝐹‘𝑥):ℕ⟶ℚ) |
7 | 5, 6 | sylib 220 | . . 3 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥):ℕ⟶ℚ) |
8 | frn 6520 | . . . 4 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ⊆ ℚ) | |
9 | qssre 12359 | . . . 4 ⊢ ℚ ⊆ ℝ | |
10 | 8, 9 | sstrdi 3979 | . . 3 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ⊆ ℝ) |
11 | 7, 10 | syl 17 | . 2 ⊢ (𝑥 ∈ ℝ → ran (𝐹‘𝑥) ⊆ ℝ) |
12 | 1nn 11649 | . . . . . 6 ⊢ 1 ∈ ℕ | |
13 | 12 | ne0ii 4303 | . . . . 5 ⊢ ℕ ≠ ∅ |
14 | fdm 6522 | . . . . . 6 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → dom (𝐹‘𝑥) = ℕ) | |
15 | 14 | neeq1d 3075 | . . . . 5 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → (dom (𝐹‘𝑥) ≠ ∅ ↔ ℕ ≠ ∅)) |
16 | 13, 15 | mpbiri 260 | . . . 4 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → dom (𝐹‘𝑥) ≠ ∅) |
17 | dm0rn0 5795 | . . . . 5 ⊢ (dom (𝐹‘𝑥) = ∅ ↔ ran (𝐹‘𝑥) = ∅) | |
18 | 17 | necon3bii 3068 | . . . 4 ⊢ (dom (𝐹‘𝑥) ≠ ∅ ↔ ran (𝐹‘𝑥) ≠ ∅) |
19 | 16, 18 | sylib 220 | . . 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 12379 | . . 3 ⊢ (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥) |
22 | breq2 5070 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑛 ≤ 𝑦 ↔ 𝑛 ≤ 𝑥)) | |
23 | 22 | ralbidv 3197 | . . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦 ↔ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥)) |
24 | 23 | rspcev 3623 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦) |
25 | 21, 24 | mpdan 685 | . 2 ⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦) |
26 | suprcl 11601 | . 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 1367 | 1 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 supcsup 8904 ℝcr 10536 1c1 10538 < clt 10675 ≤ cle 10676 / cdiv 11297 ℕcn 11638 ℤcz 11982 ℚcq 12349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-n0 11899 df-z 11983 df-q 12350 |
This theorem is referenced by: rpnnen1lem5 12381 |
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