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Mirrors > Home > MPE Home > Th. List > ruclem11 | Structured version Visualization version GIF version |
Description: Lemma for ruc 15588. 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 15580 | . . . 4 ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ)) |
6 | 1stcof 7711 | . . . 4 ⊢ (𝐺:ℕ0⟶(ℝ × ℝ) → (1st ∘ 𝐺):ℕ0⟶ℝ) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (1st ∘ 𝐺):ℕ0⟶ℝ) |
8 | 7 | frnd 6514 | . 2 ⊢ (𝜑 → ran (1st ∘ 𝐺) ⊆ ℝ) |
9 | 7 | fdmd 6516 | . . . 4 ⊢ (𝜑 → dom (1st ∘ 𝐺) = ℕ0) |
10 | 0nn0 11904 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
11 | ne0i 4298 | . . . . 5 ⊢ (0 ∈ ℕ0 → ℕ0 ≠ ∅) | |
12 | 10, 11 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℕ0 ≠ ∅) |
13 | 9, 12 | eqnetrd 3081 | . . 3 ⊢ (𝜑 → dom (1st ∘ 𝐺) ≠ ∅) |
14 | dm0rn0 5788 | . . . 4 ⊢ (dom (1st ∘ 𝐺) = ∅ ↔ ran (1st ∘ 𝐺) = ∅) | |
15 | 14 | necon3bii 3066 | . . 3 ⊢ (dom (1st ∘ 𝐺) ≠ ∅ ↔ ran (1st ∘ 𝐺) ≠ ∅) |
16 | 13, 15 | sylib 220 | . 2 ⊢ (𝜑 → ran (1st ∘ 𝐺) ≠ ∅) |
17 | fvco3 6753 | . . . . . 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 15584 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) < (2nd ‘(𝐺‘0))) |
24 | 1, 2, 3, 4 | ruclem4 15579 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘0) = 〈0, 1〉) |
25 | 24 | fveq2d 6667 | . . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘〈0, 1〉)) |
26 | c0ex 10627 | . . . . . . . . . 10 ⊢ 0 ∈ V | |
27 | 1ex 10629 | . . . . . . . . . 10 ⊢ 1 ∈ V | |
28 | 26, 27 | op2nd 7690 | . . . . . . . . 9 ⊢ (2nd ‘〈0, 1〉) = 1 |
29 | 25, 28 | syl6eq 2870 | . . . . . . . 8 ⊢ (𝜑 → (2nd ‘(𝐺‘0)) = 1) |
30 | 29 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (2nd ‘(𝐺‘0)) = 1) |
31 | 23, 30 | breqtrd 5083 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) < 1) |
32 | 5 | ffvelrnda 6844 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) ∈ (ℝ × ℝ)) |
33 | xp1st 7713 | . . . . . . . 8 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘𝑛)) ∈ ℝ) | |
34 | 32, 33 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) ∈ ℝ) |
35 | 1re 10633 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
36 | ltle 10721 | . . . . . . 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 5079 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((1st ∘ 𝐺)‘𝑛) ≤ 1) |
40 | 39 | ralrimiva 3180 | . . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 ((1st ∘ 𝐺)‘𝑛) ≤ 1) |
41 | 7 | ffnd 6508 | . . . 4 ⊢ (𝜑 → (1st ∘ 𝐺) Fn ℕ0) |
42 | breq1 5060 | . . . . 5 ⊢ (𝑧 = ((1st ∘ 𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1st ∘ 𝐺)‘𝑛) ≤ 1)) | |
43 | 42 | ralrn 6847 | . . . 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 1123 | 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 1082 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∀wral 3136 ⦋csb 3881 ∪ cun 3932 ⊆ wss 3934 ∅c0 4289 ifcif 4465 {csn 4559 〈cop 4565 class class class wbr 5057 × cxp 5546 dom cdm 5548 ran crn 5549 ∘ ccom 5552 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ∈ cmpo 7150 1st c1st 7679 2nd c2nd 7680 ℝcr 10528 0cc0 10529 1c1 10530 + caddc 10532 < clt 10667 ≤ cle 10668 / cdiv 11289 ℕcn 11630 2c2 11684 ℕ0cn0 11889 seqcseq 13361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 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 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-seq 13362 |
This theorem is referenced by: ruclem12 15586 |
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