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Mirrors > Home > MPE Home > Th. List > ruclem13 | Structured version Visualization version GIF version |
Description: Lemma for ruc 15596. There is no function that maps ℕ onto ℝ. (Use nex 1801 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 6593 | . . . 4 ⊢ (𝐹:ℕ–onto→ℝ → ran 𝐹 = ℝ) | |
2 | 1 | difeq2d 4099 | . . 3 ⊢ (𝐹:ℕ–onto→ℝ → (ℝ ∖ ran 𝐹) = (ℝ ∖ ℝ)) |
3 | difid 4330 | . . 3 ⊢ (ℝ ∖ ℝ) = ∅ | |
4 | 2, 3 | syl6eq 2872 | . 2 ⊢ (𝐹:ℕ–onto→ℝ → (ℝ ∖ ran 𝐹) = ∅) |
5 | reex 10628 | . . . . . 6 ⊢ ℝ ∈ V | |
6 | 5, 5 | xpex 7476 | . . . . 5 ⊢ (ℝ × ℝ) ∈ V |
7 | 6, 5 | mpoex 7777 | . . . 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 3508 | . . 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 6590 | . . . . . . . 8 ⊢ (𝐹:ℕ–onto→ℝ → 𝐹:ℕ⟶ℝ) | |
10 | 9 | adantr 483 | . . . . . . 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 487 | . . . . . . 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 2821 | . . . . . . 7 ⊢ ({〈0, 〈0, 1〉〉} ∪ 𝐹) = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
13 | eqid 2821 | . . . . . . 7 ⊢ seq0(𝑑, ({〈0, 〈0, 1〉〉} ∪ 𝐹)) = seq0(𝑑, ({〈0, 〈0, 1〉〉} ∪ 𝐹)) | |
14 | eqid 2821 | . . . . . . 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 15594 | . . . . . 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 4299 | . . . . . 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 415 | . . . 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 1934 | . . 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 196 | 1 ⊢ ¬ 𝐹:ℕ–onto→ℝ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ⦋csb 3883 ∖ cdif 3933 ∪ cun 3934 ∅c0 4291 ifcif 4467 {csn 4567 〈cop 4573 class class class wbr 5066 × cxp 5553 ran crn 5556 ∘ ccom 5559 ⟶wf 6351 –onto→wfo 6353 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 1st c1st 7687 2nd c2nd 7688 supcsup 8904 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 / cdiv 11297 ℕcn 11638 2c2 11693 seqcseq 13370 |
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-cnex 10593 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-fal 1550 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-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-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-seq 13371 |
This theorem is referenced by: ruc 15596 |
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