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Mirrors > Home > MPE Home > Th. List > ruclem13 | Structured version Visualization version GIF version |
Description: Lemma for ruc 16117. There is no function that maps ℕ onto ℝ. (Use nex 1802 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 6756 | . . . 4 ⊢ (𝐹:ℕ–onto→ℝ → ran 𝐹 = ℝ) | |
2 | 1 | difeq2d 4080 | . . 3 ⊢ (𝐹:ℕ–onto→ℝ → (ℝ ∖ ran 𝐹) = (ℝ ∖ ℝ)) |
3 | difid 4328 | . . 3 ⊢ (ℝ ∖ ℝ) = ∅ | |
4 | 2, 3 | eqtrdi 2792 | . 2 ⊢ (𝐹:ℕ–onto→ℝ → (ℝ ∖ ran 𝐹) = ∅) |
5 | reex 11138 | . . . . . 6 ⊢ ℝ ∈ V | |
6 | 5, 5 | xpex 7683 | . . . . 5 ⊢ (ℝ × ℝ) ∈ V |
7 | 6, 5 | mpoex 8008 | . . . 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 3458 | . . 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 6753 | . . . . . . . 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 2736 | . . . . . . 7 ⊢ ({〈0, 〈0, 1〉〉} ∪ 𝐹) = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
13 | eqid 2736 | . . . . . . 7 ⊢ seq0(𝑑, ({〈0, 〈0, 1〉〉} ∪ 𝐹)) = seq0(𝑑, ({〈0, 〈0, 1〉〉} ∪ 𝐹)) | |
14 | eqid 2736 | . . . . . . 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 16115 | . . . . . 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 4291 | . . . . . 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 1936 | . . 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 193 | 1 ⊢ ¬ 𝐹:ℕ–onto→ℝ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ⦋csb 3853 ∖ cdif 3905 ∪ cun 3906 ∅c0 4280 ifcif 4484 {csn 4584 〈cop 4590 class class class wbr 5103 × cxp 5629 ran crn 5632 ∘ ccom 5635 ⟶wf 6489 –onto→wfo 6491 ‘cfv 6493 (class class class)co 7353 ∈ cmpo 7355 1st c1st 7915 2nd c2nd 7916 supcsup 9372 ℝcr 11046 0cc0 11047 1c1 11048 + caddc 11050 < clt 11185 / cdiv 11808 ℕcn 12149 2c2 12204 seqcseq 13898 |
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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9374 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-seq 13899 |
This theorem is referenced by: ruc 16117 |
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