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Mirrors > Home > MPE Home > Th. List > rpnnen1lem6 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen1 12376. (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 7182 | . 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 12371 | . . 3 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) ∈ (ℚ ↑m ℕ)) |
7 | rneq 5799 | . . . . . 6 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → ran (𝐹‘𝑥) = ran (𝐹‘𝑦)) | |
8 | 7 | supeq1d 8903 | . . . . 5 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → sup(ran (𝐹‘𝑥), ℝ, < ) = sup(ran (𝐹‘𝑦), ℝ, < )) |
9 | 2, 3, 4, 5 | rpnnen1lem5 12374 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥) |
10 | fveq2 6663 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
11 | 10 | rneqd 5801 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ran (𝐹‘𝑥) = ran (𝐹‘𝑦)) |
12 | 11 | supeq1d 8903 | . . . . . . . 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 2836 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥 ↔ sup(ran (𝐹‘𝑦), ℝ, < ) = 𝑦)) |
15 | 14, 9 | vtoclga 3571 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → sup(ran (𝐹‘𝑦), ℝ, < ) = 𝑦) |
16 | 9, 15 | eqeqan12d 2837 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) = sup(ran (𝐹‘𝑦), ℝ, < ) ↔ 𝑥 = 𝑦)) |
17 | 8, 16 | syl5ib 246 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) |
18 | 17, 10 | impbid1 227 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦)) |
19 | 6, 18 | dom2 8545 | . 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 398 = wceq 1536 ∈ wcel 2113 {crab 3141 Vcvv 3491 class class class wbr 5059 ↦ cmpt 5139 ran crn 5549 ‘cfv 6348 (class class class)co 7149 ↑m cmap 8399 ≼ cdom 8500 supcsup 8897 ℝcr 10529 < clt 10668 / cdiv 11290 ℕcn 11631 ℤcz 11975 ℚcq 12342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-sup 8899 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-n0 11892 df-z 11976 df-q 12343 |
This theorem is referenced by: rpnnen1 12376 |
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