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Mirrors > Home > MPE Home > Th. List > rpnnen1lem4 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen1 12949. (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 12944 | . . . 4 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) ∈ (ℚ ↑m ℕ)) |
6 | 4, 3 | elmap 8848 | . . . 4 ⊢ ((𝐹‘𝑥) ∈ (ℚ ↑m ℕ) ↔ (𝐹‘𝑥):ℕ⟶ℚ) |
7 | 5, 6 | sylib 217 | . . 3 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥):ℕ⟶ℚ) |
8 | frn 6711 | . . . 4 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ⊆ ℚ) | |
9 | qssre 12925 | . . . 4 ⊢ ℚ ⊆ ℝ | |
10 | 8, 9 | sstrdi 3990 | . . 3 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ⊆ ℝ) |
11 | 7, 10 | syl 17 | . 2 ⊢ (𝑥 ∈ ℝ → ran (𝐹‘𝑥) ⊆ ℝ) |
12 | 1nn 12205 | . . . . . 6 ⊢ 1 ∈ ℕ | |
13 | 12 | ne0ii 4333 | . . . . 5 ⊢ ℕ ≠ ∅ |
14 | fdm 6713 | . . . . . 6 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → dom (𝐹‘𝑥) = ℕ) | |
15 | 14 | neeq1d 2999 | . . . . 5 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → (dom (𝐹‘𝑥) ≠ ∅ ↔ ℕ ≠ ∅)) |
16 | 13, 15 | mpbiri 257 | . . . 4 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → dom (𝐹‘𝑥) ≠ ∅) |
17 | dm0rn0 5916 | . . . . 5 ⊢ (dom (𝐹‘𝑥) = ∅ ↔ ran (𝐹‘𝑥) = ∅) | |
18 | 17 | necon3bii 2992 | . . . 4 ⊢ (dom (𝐹‘𝑥) ≠ ∅ ↔ ran (𝐹‘𝑥) ≠ ∅) |
19 | 16, 18 | sylib 217 | . . 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 12945 | . . 3 ⊢ (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥) |
22 | breq2 5145 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑛 ≤ 𝑦 ↔ 𝑛 ≤ 𝑥)) | |
23 | 22 | ralbidv 3176 | . . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦 ↔ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥)) |
24 | 23 | rspcev 3609 | . . 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 12156 | . 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 1371 | 1 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3431 Vcvv 3473 ⊆ wss 3944 ∅c0 4318 class class class wbr 5141 ↦ cmpt 5224 dom cdm 5669 ran crn 5670 ⟶wf 6528 ‘cfv 6532 (class class class)co 7393 ↑m cmap 8803 supcsup 9417 ℝcr 11091 1c1 11093 < clt 11230 ≤ cle 11231 / cdiv 11853 ℕcn 12194 ℤcz 12540 ℚcq 12914 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-sup 9419 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-n0 12455 df-z 12541 df-q 12915 |
This theorem is referenced by: rpnnen1lem5 12947 |
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