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Mirrors > Home > MPE Home > Th. List > ruclem11 | Structured version Visualization version GIF version |
Description: Lemma for ruc 15598. Closure lemmas for supremum. (Contributed by Mario Carneiro, 28-May-2014.) |
Ref | Expression |
---|---|
ruc.1 | ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) |
ruc.2 | ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) |
ruc.4 | ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) |
ruc.5 | ⊢ 𝐺 = seq0(𝐷, 𝐶) |
Ref | Expression |
---|---|
ruclem11 | ⊢ (𝜑 → (ran (1st ∘ 𝐺) ⊆ ℝ ∧ ran (1st ∘ 𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st ∘ 𝐺)𝑧 ≤ 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ruc.1 | . . . . 5 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ) | |
2 | ruc.2 | . . . . 5 ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) | |
3 | ruc.4 | . . . . 5 ⊢ 𝐶 = ({〈0, 〈0, 1〉〉} ∪ 𝐹) | |
4 | ruc.5 | . . . . 5 ⊢ 𝐺 = seq0(𝐷, 𝐶) | |
5 | 1, 2, 3, 4 | ruclem6 15590 | . . . 4 ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ)) |
6 | 1stcof 7721 | . . . 4 ⊢ (𝐺:ℕ0⟶(ℝ × ℝ) → (1st ∘ 𝐺):ℕ0⟶ℝ) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (1st ∘ 𝐺):ℕ0⟶ℝ) |
8 | 7 | frnd 6523 | . 2 ⊢ (𝜑 → ran (1st ∘ 𝐺) ⊆ ℝ) |
9 | 7 | fdmd 6525 | . . . 4 ⊢ (𝜑 → dom (1st ∘ 𝐺) = ℕ0) |
10 | 0nn0 11915 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
11 | ne0i 4302 | . . . . 5 ⊢ (0 ∈ ℕ0 → ℕ0 ≠ ∅) | |
12 | 10, 11 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℕ0 ≠ ∅) |
13 | 9, 12 | eqnetrd 3085 | . . 3 ⊢ (𝜑 → dom (1st ∘ 𝐺) ≠ ∅) |
14 | dm0rn0 5797 | . . . 4 ⊢ (dom (1st ∘ 𝐺) = ∅ ↔ ran (1st ∘ 𝐺) = ∅) | |
15 | 14 | necon3bii 3070 | . . 3 ⊢ (dom (1st ∘ 𝐺) ≠ ∅ ↔ ran (1st ∘ 𝐺) ≠ ∅) |
16 | 13, 15 | sylib 220 | . 2 ⊢ (𝜑 → ran (1st ∘ 𝐺) ≠ ∅) |
17 | fvco3 6762 | . . . . . 6 ⊢ ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → ((1st ∘ 𝐺)‘𝑛) = (1st ‘(𝐺‘𝑛))) | |
18 | 5, 17 | sylan 582 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((1st ∘ 𝐺)‘𝑛) = (1st ‘(𝐺‘𝑛))) |
19 | 1 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐹:ℕ⟶ℝ) |
20 | 2 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) |
21 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0) | |
22 | 10 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 0 ∈ ℕ0) |
23 | 19, 20, 3, 4, 21, 22 | ruclem10 15594 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) < (2nd ‘(𝐺‘0))) |
24 | 1, 2, 3, 4 | ruclem4 15589 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘0) = 〈0, 1〉) |
25 | 24 | fveq2d 6676 | . . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘〈0, 1〉)) |
26 | c0ex 10637 | . . . . . . . . . 10 ⊢ 0 ∈ V | |
27 | 1ex 10639 | . . . . . . . . . 10 ⊢ 1 ∈ V | |
28 | 26, 27 | op2nd 7700 | . . . . . . . . 9 ⊢ (2nd ‘〈0, 1〉) = 1 |
29 | 25, 28 | syl6eq 2874 | . . . . . . . 8 ⊢ (𝜑 → (2nd ‘(𝐺‘0)) = 1) |
30 | 29 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (2nd ‘(𝐺‘0)) = 1) |
31 | 23, 30 | breqtrd 5094 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) < 1) |
32 | 5 | ffvelrnda 6853 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) ∈ (ℝ × ℝ)) |
33 | xp1st 7723 | . . . . . . . 8 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘𝑛)) ∈ ℝ) | |
34 | 32, 33 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) ∈ ℝ) |
35 | 1re 10643 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
36 | ltle 10731 | . . . . . . 7 ⊢ (((1st ‘(𝐺‘𝑛)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1st ‘(𝐺‘𝑛)) < 1 → (1st ‘(𝐺‘𝑛)) ≤ 1)) | |
37 | 34, 35, 36 | sylancl 588 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((1st ‘(𝐺‘𝑛)) < 1 → (1st ‘(𝐺‘𝑛)) ≤ 1)) |
38 | 31, 37 | mpd 15 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) ≤ 1) |
39 | 18, 38 | eqbrtrd 5090 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((1st ∘ 𝐺)‘𝑛) ≤ 1) |
40 | 39 | ralrimiva 3184 | . . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 ((1st ∘ 𝐺)‘𝑛) ≤ 1) |
41 | 7 | ffnd 6517 | . . . 4 ⊢ (𝜑 → (1st ∘ 𝐺) Fn ℕ0) |
42 | breq1 5071 | . . . . 5 ⊢ (𝑧 = ((1st ∘ 𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1st ∘ 𝐺)‘𝑛) ≤ 1)) | |
43 | 42 | ralrn 6856 | . . . 4 ⊢ ((1st ∘ 𝐺) Fn ℕ0 → (∀𝑧 ∈ ran (1st ∘ 𝐺)𝑧 ≤ 1 ↔ ∀𝑛 ∈ ℕ0 ((1st ∘ 𝐺)‘𝑛) ≤ 1)) |
44 | 41, 43 | syl 17 | . . 3 ⊢ (𝜑 → (∀𝑧 ∈ ran (1st ∘ 𝐺)𝑧 ≤ 1 ↔ ∀𝑛 ∈ ℕ0 ((1st ∘ 𝐺)‘𝑛) ≤ 1)) |
45 | 40, 44 | mpbird 259 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ ran (1st ∘ 𝐺)𝑧 ≤ 1) |
46 | 8, 16, 45 | 3jca 1124 | 1 ⊢ (𝜑 → (ran (1st ∘ 𝐺) ⊆ ℝ ∧ ran (1st ∘ 𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st ∘ 𝐺)𝑧 ≤ 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ⦋csb 3885 ∪ cun 3936 ⊆ wss 3938 ∅c0 4293 ifcif 4469 {csn 4569 〈cop 4575 class class class wbr 5068 × cxp 5555 dom cdm 5557 ran crn 5558 ∘ ccom 5561 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 1st c1st 7689 2nd c2nd 7690 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 < clt 10677 ≤ cle 10678 / cdiv 11299 ℕcn 11640 2c2 11695 ℕ0cn0 11900 seqcseq 13372 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-seq 13373 |
This theorem is referenced by: ruclem12 15596 |
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