Theorem ruclem10 14968

Description: Lemma for ruc 14972. 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 a_i are the first components and b_i are the second components. (Contributed by Mario Carneiro, 28-May-2014.)

Hypotheses

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)

Assertion

Ref	Expression
ruclem10	⊢ (𝜑 → (1st ‘(𝐺‘𝑀)) < (2nd ‘(𝐺‘𝑁)))

Distinct variable groups:   𝑥,𝑚,𝑦,𝐹   𝑚,𝐺,𝑥,𝑦   𝑚,𝑀,𝑥,𝑦   𝑚,𝑁,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚)   𝐶(𝑥,𝑦,𝑚)   𝐷(𝑥,𝑦,𝑚)

Proof of Theorem ruclem10

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 14964	. . . 4 ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ))
6		ruclem10.6	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
7	5, 6	ffvelrnd 6360	. . 3 ⊢ (𝜑 → (𝐺‘𝑀) ∈ (ℝ × ℝ))
8		xp1st 7198	. . 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 4131	. . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0)
12	5, 11	ffvelrnd 6360	. . 3 ⊢ (𝜑 → (𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ))
13		xp1st 7198	. . 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	ffvelrnd 6360	. . 3 ⊢ (𝜑 → (𝐺‘𝑁) ∈ (ℝ × ℝ))
16		xp2nd 7199	. . 3 ⊢ ((𝐺‘𝑁) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘𝑁)) ∈ ℝ)
17	15, 16	syl 17	. 2 ⊢ (𝜑 → (2nd ‘(𝐺‘𝑁)) ∈ ℝ)
18	6	nn0red 11352	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
19	10	nn0red 11352	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
20		max1 12016	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
21	18, 19, 20	syl2anc 693	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
22	6	nn0zd 11480	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
23	11	nn0zd 11480	. . . . . 6 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℤ)
24		eluz 11701	. . . . . 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 693	. . . . 5 ⊢ (𝜑 → (if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
26	21, 25	mpbird 247	. . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑀))
27	1, 2, 3, 4, 6, 26	ruclem9 14967	. . 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 475	. 2 ⊢ (𝜑 → (1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))))
29		xp2nd 7199	. . . 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 14966	. . . 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 702	. . 3 ⊢ (𝜑 → (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))))
33		max2 12018	. . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
34	18, 19, 33	syl2anc 693	. . . . . 6 ⊢ (𝜑 → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
35	10	nn0zd 11480	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
36		eluz 11701	. . . . . . 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 693	. . . . . 6 ⊢ (𝜑 → (if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑁) ↔ 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
38	34, 37	mpbird 247	. . . . 5 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑁))
39	1, 2, 3, 4, 10, 38	ruclem9 14967	. . . 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 479	. . 3 ⊢ (𝜑 → (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺‘𝑁)))
41	14, 30, 17, 32, 40	ltletrd 10197	. 2 ⊢ (𝜑 → (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘𝑁)))
42	9, 14, 17, 28, 41	lelttrd 10195	1 ⊢ (𝜑 → (1st ‘(𝐺‘𝑀)) < (2nd ‘(𝐺‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  ⦋csb 3533   ∪ cun 3572  ifcif 4086  {csn 4177  ⟨cop 4183   class class class wbr 4653   × cxp 5112  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075   / cdiv 10684  ℕcn 11020  2c2 11070  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  seqcseq 12801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  ruclem11  14969  ruclem12  14970