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Mirrors > Home > MPE Home > Th. List > ruclem13 | Structured version Visualization version GIF version |
Description: Lemma for ruc 15584. There is no function that maps ℕ onto ℝ. (Use nex 1792 if you want this in the form ¬ ∃𝑓𝑓:ℕ–onto→ℝ.) (Contributed by NM, 14-Oct-2004.) (Proof shortened by Fan Zheng, 6-Jun-2016.) |
Ref | Expression |
---|---|
ruclem13 | ⊢ ¬ 𝐹:ℕ–onto→ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | forn 6586 | . . . 4 ⊢ (𝐹:ℕ–onto→ℝ → ran 𝐹 = ℝ) | |
2 | 1 | difeq2d 4096 | . . 3 ⊢ (𝐹:ℕ–onto→ℝ → (ℝ ∖ ran 𝐹) = (ℝ ∖ ℝ)) |
3 | difid 4327 | . . 3 ⊢ (ℝ ∖ ℝ) = ∅ | |
4 | 2, 3 | syl6eq 2869 | . 2 ⊢ (𝐹:ℕ–onto→ℝ → (ℝ ∖ ran 𝐹) = ∅) |
5 | reex 10616 | . . . . . 6 ⊢ ℝ ∈ V | |
6 | 5, 5 | xpex 7465 | . . . . 5 ⊢ (ℝ × ℝ) ∈ V |
7 | 6, 5 | mpoex 7766 | . . . 4 ⊢ (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉)) ∈ V |
8 | 7 | isseti 3506 | . . 3 ⊢ ∃𝑑 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉)) |
9 | fof 6583 | . . . . . . . 8 ⊢ (𝐹:ℕ–onto→ℝ → 𝐹:ℕ⟶ℝ) | |
10 | 9 | adantr 481 | . . . . . . 7 ⊢ ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) → 𝐹:ℕ⟶ℝ) |
11 | simpr 485 | . . . . . . 7 ⊢ ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) → 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) | |
12 | eqid 2818 | . . . . . . 7 ⊢ ({〈0, 〈0, 1〉〉} ∪ 𝐹) = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
13 | eqid 2818 | . . . . . . 7 ⊢ seq0(𝑑, ({〈0, 〈0, 1〉〉} ∪ 𝐹)) = seq0(𝑑, ({〈0, 〈0, 1〉〉} ∪ 𝐹)) | |
14 | eqid 2818 | . . . . . . 7 ⊢ sup(ran (1st ∘ seq0(𝑑, ({〈0, 〈0, 1〉〉} ∪ 𝐹))), ℝ, < ) = sup(ran (1st ∘ seq0(𝑑, ({〈0, 〈0, 1〉〉} ∪ 𝐹))), ℝ, < ) | |
15 | 10, 11, 12, 13, 14 | ruclem12 15582 | . . . . . 6 ⊢ ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) → sup(ran (1st ∘ seq0(𝑑, ({〈0, 〈0, 1〉〉} ∪ 𝐹))), ℝ, < ) ∈ (ℝ ∖ ran 𝐹)) |
16 | n0i 4296 | . . . . . 6 ⊢ (sup(ran (1st ∘ seq0(𝑑, ({〈0, 〈0, 1〉〉} ∪ 𝐹))), ℝ, < ) ∈ (ℝ ∖ ran 𝐹) → ¬ (ℝ ∖ ran 𝐹) = ∅) | |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ ((𝐹:ℕ–onto→ℝ ∧ 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) → ¬ (ℝ ∖ ran 𝐹) = ∅) |
18 | 17 | ex 413 | . . . 4 ⊢ (𝐹:ℕ–onto→ℝ → (𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉)) → ¬ (ℝ ∖ ran 𝐹) = ∅)) |
19 | 18 | exlimdv 1925 | . . 3 ⊢ (𝐹:ℕ–onto→ℝ → (∃𝑑 𝑑 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉)) → ¬ (ℝ ∖ ran 𝐹) = ∅)) |
20 | 8, 19 | mpi 20 | . 2 ⊢ (𝐹:ℕ–onto→ℝ → ¬ (ℝ ∖ ran 𝐹) = ∅) |
21 | 4, 20 | pm2.65i 195 | 1 ⊢ ¬ 𝐹:ℕ–onto→ℝ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ⦋csb 3880 ∖ cdif 3930 ∪ cun 3931 ∅c0 4288 ifcif 4463 {csn 4557 〈cop 4563 class class class wbr 5057 × cxp 5546 ran crn 5549 ∘ ccom 5552 ⟶wf 6344 –onto→wfo 6346 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 1st c1st 7676 2nd c2nd 7677 supcsup 8892 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 < clt 10663 / cdiv 11285 ℕcn 11626 2c2 11680 seqcseq 13357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-seq 13358 |
This theorem is referenced by: ruc 15584 |
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