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Mirrors > Home > MPE Home > Th. List > ruclem11 | Structured version Visualization version GIF version |
Description: Lemma for ruc 15434. 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 15426 | . . . 4 ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ)) |
6 | 1stcof 7580 | . . . 4 ⊢ (𝐺:ℕ0⟶(ℝ × ℝ) → (1st ∘ 𝐺):ℕ0⟶ℝ) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (1st ∘ 𝐺):ℕ0⟶ℝ) |
8 | 7 | frnd 6394 | . 2 ⊢ (𝜑 → ran (1st ∘ 𝐺) ⊆ ℝ) |
9 | 7 | fdmd 6396 | . . . 4 ⊢ (𝜑 → dom (1st ∘ 𝐺) = ℕ0) |
10 | 0nn0 11765 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
11 | ne0i 4224 | . . . . 5 ⊢ (0 ∈ ℕ0 → ℕ0 ≠ ∅) | |
12 | 10, 11 | mp1i 13 | . . . 4 ⊢ (𝜑 → ℕ0 ≠ ∅) |
13 | 9, 12 | eqnetrd 3051 | . . 3 ⊢ (𝜑 → dom (1st ∘ 𝐺) ≠ ∅) |
14 | dm0rn0 5684 | . . . 4 ⊢ (dom (1st ∘ 𝐺) = ∅ ↔ ran (1st ∘ 𝐺) = ∅) | |
15 | 14 | necon3bii 3036 | . . 3 ⊢ (dom (1st ∘ 𝐺) ≠ ∅ ↔ ran (1st ∘ 𝐺) ≠ ∅) |
16 | 13, 15 | sylib 219 | . 2 ⊢ (𝜑 → ran (1st ∘ 𝐺) ≠ ∅) |
17 | fvco3 6632 | . . . . . 6 ⊢ ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → ((1st ∘ 𝐺)‘𝑛) = (1st ‘(𝐺‘𝑛))) | |
18 | 5, 17 | sylan 580 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((1st ∘ 𝐺)‘𝑛) = (1st ‘(𝐺‘𝑛))) |
19 | 1 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐹:ℕ⟶ℝ) |
20 | 2 | adantr 481 | . . . . . . . 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 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0) | |
22 | 10 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 0 ∈ ℕ0) |
23 | 19, 20, 3, 4, 21, 22 | ruclem10 15430 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) < (2nd ‘(𝐺‘0))) |
24 | 1, 2, 3, 4 | ruclem4 15425 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘0) = 〈0, 1〉) |
25 | 24 | fveq2d 6547 | . . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(𝐺‘0)) = (2nd ‘〈0, 1〉)) |
26 | c0ex 10486 | . . . . . . . . . 10 ⊢ 0 ∈ V | |
27 | 1ex 10488 | . . . . . . . . . 10 ⊢ 1 ∈ V | |
28 | 26, 27 | op2nd 7559 | . . . . . . . . 9 ⊢ (2nd ‘〈0, 1〉) = 1 |
29 | 25, 28 | syl6eq 2847 | . . . . . . . 8 ⊢ (𝜑 → (2nd ‘(𝐺‘0)) = 1) |
30 | 29 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (2nd ‘(𝐺‘0)) = 1) |
31 | 23, 30 | breqtrd 4992 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) < 1) |
32 | 5 | ffvelrnda 6721 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) ∈ (ℝ × ℝ)) |
33 | xp1st 7582 | . . . . . . . 8 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘𝑛)) ∈ ℝ) | |
34 | 32, 33 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (1st ‘(𝐺‘𝑛)) ∈ ℝ) |
35 | 1re 10492 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
36 | ltle 10581 | . . . . . . 7 ⊢ (((1st ‘(𝐺‘𝑛)) ∈ ℝ ∧ 1 ∈ ℝ) → ((1st ‘(𝐺‘𝑛)) < 1 → (1st ‘(𝐺‘𝑛)) ≤ 1)) | |
37 | 34, 35, 36 | sylancl 586 | . . . . . 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 4988 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((1st ∘ 𝐺)‘𝑛) ≤ 1) |
40 | 39 | ralrimiva 3149 | . . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 ((1st ∘ 𝐺)‘𝑛) ≤ 1) |
41 | 7 | ffnd 6388 | . . . 4 ⊢ (𝜑 → (1st ∘ 𝐺) Fn ℕ0) |
42 | breq1 4969 | . . . . 5 ⊢ (𝑧 = ((1st ∘ 𝐺)‘𝑛) → (𝑧 ≤ 1 ↔ ((1st ∘ 𝐺)‘𝑛) ≤ 1)) | |
43 | 42 | ralrn 6724 | . . . 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 258 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ ran (1st ∘ 𝐺)𝑧 ≤ 1) |
46 | 8, 16, 45 | 3jca 1121 | 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 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 ⦋csb 3815 ∪ cun 3861 ⊆ wss 3863 ∅c0 4215 ifcif 4385 {csn 4476 〈cop 4482 class class class wbr 4966 × cxp 5446 dom cdm 5448 ran crn 5449 ∘ ccom 5452 Fn wfn 6225 ⟶wf 6226 ‘cfv 6230 (class class class)co 7021 ∈ cmpo 7023 1st c1st 7548 2nd c2nd 7549 ℝcr 10387 0cc0 10388 1c1 10389 + caddc 10391 < clt 10526 ≤ cle 10527 / cdiv 11150 ℕcn 11491 2c2 11545 ℕ0cn0 11750 seqcseq 13224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-div 11151 df-nn 11492 df-2 11553 df-n0 11751 df-z 11835 df-uz 12099 df-fz 12748 df-seq 13225 |
This theorem is referenced by: ruclem12 15432 |
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