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Mirrors > Home > MPE Home > Th. List > ruclem10 | Structured version Visualization version GIF version |
Description: Lemma for ruc 16227. Every first component of the 𝐺 sequence is less than every second component. That is, the sequences form a chain a1 < a2 <... < b2 < b1, where ai are the first components and bi are the second components. (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) |
ruclem10.6 | ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
ruclem10.7 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
ruclem10 | ⊢ (𝜑 → (1st ‘(𝐺‘𝑀)) < (2nd ‘(𝐺‘𝑁))) |
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 16219 | . . . 4 ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ)) |
6 | ruclem10.6 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
7 | 5, 6 | ffvelcdmd 7100 | . . 3 ⊢ (𝜑 → (𝐺‘𝑀) ∈ (ℝ × ℝ)) |
8 | xp1st 8031 | . . 3 ⊢ ((𝐺‘𝑀) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘𝑀)) ∈ ℝ) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → (1st ‘(𝐺‘𝑀)) ∈ ℝ) |
10 | ruclem10.7 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
11 | 10, 6 | ifcld 4578 | . . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0) |
12 | 5, 11 | ffvelcdmd 7100 | . . 3 ⊢ (𝜑 → (𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ)) |
13 | xp1st 8031 | . . 3 ⊢ ((𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) ∈ ℝ) | |
14 | 12, 13 | syl 17 | . 2 ⊢ (𝜑 → (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) ∈ ℝ) |
15 | 5, 10 | ffvelcdmd 7100 | . . 3 ⊢ (𝜑 → (𝐺‘𝑁) ∈ (ℝ × ℝ)) |
16 | xp2nd 8032 | . . 3 ⊢ ((𝐺‘𝑁) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘𝑁)) ∈ ℝ) | |
17 | 15, 16 | syl 17 | . 2 ⊢ (𝜑 → (2nd ‘(𝐺‘𝑁)) ∈ ℝ) |
18 | 6 | nn0red 12571 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
19 | 10 | nn0red 12571 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
20 | max1 13204 | . . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) | |
21 | 18, 19, 20 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) |
22 | 6 | nn0zd 12622 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
23 | 11 | nn0zd 12622 | . . . . . 6 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℤ) |
24 | eluz 12874 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℤ) → (if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) | |
25 | 22, 23, 24 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → (if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) |
26 | 21, 25 | mpbird 256 | . . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑀)) |
27 | 1, 2, 3, 4, 6, 26 | ruclem9 16222 | . . 3 ⊢ (𝜑 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) ∧ (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺‘𝑀)))) |
28 | 27 | simpld 493 | . 2 ⊢ (𝜑 → (1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))) |
29 | xp2nd 8032 | . . . 4 ⊢ ((𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) ∈ ℝ) | |
30 | 12, 29 | syl 17 | . . 3 ⊢ (𝜑 → (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) ∈ ℝ) |
31 | 1, 2, 3, 4 | ruclem8 16221 | . . . 4 ⊢ ((𝜑 ∧ if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0) → (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))) |
32 | 11, 31 | mpdan 685 | . . 3 ⊢ (𝜑 → (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))) |
33 | max2 13206 | . . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) | |
34 | 18, 19, 33 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) |
35 | 10 | nn0zd 12622 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
36 | eluz 12874 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℤ) → (if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑁) ↔ 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) | |
37 | 35, 23, 36 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → (if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑁) ↔ 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) |
38 | 34, 37 | mpbird 256 | . . . . 5 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑁)) |
39 | 1, 2, 3, 4, 10, 38 | ruclem9 16222 | . . . 4 ⊢ (𝜑 → ((1st ‘(𝐺‘𝑁)) ≤ (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) ∧ (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺‘𝑁)))) |
40 | 39 | simprd 494 | . . 3 ⊢ (𝜑 → (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺‘𝑁))) |
41 | 14, 30, 17, 32, 40 | ltletrd 11412 | . 2 ⊢ (𝜑 → (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘𝑁))) |
42 | 9, 14, 17, 28, 41 | lelttrd 11410 | 1 ⊢ (𝜑 → (1st ‘(𝐺‘𝑀)) < (2nd ‘(𝐺‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ⦋csb 3894 ∪ cun 3947 ifcif 4532 {csn 4632 〈cop 4638 class class class wbr 5152 × cxp 5680 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 1st c1st 7997 2nd c2nd 7998 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 < clt 11286 ≤ cle 11287 / cdiv 11909 ℕcn 12250 2c2 12305 ℕ0cn0 12510 ℤcz 12596 ℤ≥cuz 12860 seqcseq 14006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-seq 14007 |
This theorem is referenced by: ruclem11 16224 ruclem12 16225 |
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