Theorem ruclem10 15959

Description: Lemma for ruc 15963. 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 15955	. . . 4 ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ))
6		ruclem10.6	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
7	5, 6	ffvelrnd 6959	. . 3 ⊢ (𝜑 → (𝐺‘𝑀) ∈ (ℝ × ℝ))
8		xp1st 7857	. . 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 4511	. . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0)
12	5, 11	ffvelrnd 6959	. . 3 ⊢ (𝜑 → (𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ))
13		xp1st 7857	. . 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 6959	. . 3 ⊢ (𝜑 → (𝐺‘𝑁) ∈ (ℝ × ℝ))
16		xp2nd 7858	. . 3 ⊢ ((𝐺‘𝑁) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘𝑁)) ∈ ℝ)
17	15, 16	syl 17	. 2 ⊢ (𝜑 → (2nd ‘(𝐺‘𝑁)) ∈ ℝ)
18	6	nn0red 12305	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
19	10	nn0red 12305	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
20		max1 12930	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
21	18, 19, 20	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
22	6	nn0zd 12435	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
23	11	nn0zd 12435	. . . . . 6 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℤ)
24		eluz 12607	. . . . . 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 584	. . . . 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 15958	. . 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 495	. 2 ⊢ (𝜑 → (1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))))
29		xp2nd 7858	. . . 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 15957	. . . 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 684	. . 3 ⊢ (𝜑 → (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))))
33		max2 12932	. . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
34	18, 19, 33	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
35	10	nn0zd 12435	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
36		eluz 12607	. . . . . . 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 584	. . . . . 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 15958	. . . 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 496	. . 3 ⊢ (𝜑 → (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺‘𝑁)))
41	14, 30, 17, 32, 40	ltletrd 11146	. 2 ⊢ (𝜑 → (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘𝑁)))
42	9, 14, 17, 28, 41	lelttrd 11144	1 ⊢ (𝜑 → (1st ‘(𝐺‘𝑀)) < (2nd ‘(𝐺‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2110  ⦋csb 3837   ∪ cun 3890  ifcif 4465  {csn 4567  ⟨cop 4573   class class class wbr 5079   × cxp 5588  ⟶wf 6428  ‘cfv 6432  (class class class)co 7272   ∈ cmpo 7274  1^st c1st 7823  2^nd c2nd 7824  ℝcr 10881  0cc0 10882  1c1 10883   + caddc 10885   < clt 11020   ≤ cle 11021   / cdiv 11643  ℕcn 11984  2c2 12039  ℕ₀cn0 12244  ℤcz 12330  ℤ_≥cuz 12593  seqcseq 13732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7583  ax-cnex 10938  ax-resscn 10939  ax-1cn 10940  ax-icn 10941  ax-addcl 10942  ax-addrcl 10943  ax-mulcl 10944  ax-mulrcl 10945  ax-mulcom 10946  ax-addass 10947  ax-mulass 10948  ax-distr 10949  ax-i2m1 10950  ax-1ne0 10951  ax-1rid 10952  ax-rnegex 10953  ax-rrecex 10954  ax-cnre 10955  ax-pre-lttri 10956  ax-pre-lttrn 10957  ax-pre-ltadd 10958  ax-pre-mulgt0 10959

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7229  df-ov 7275  df-oprab 7276  df-mpo 7277  df-om 7708  df-1st 7825  df-2nd 7826  df-frecs 8089  df-wrecs 8120  df-recs 8194  df-rdg 8233  df-er 8490  df-en 8726  df-dom 8727  df-sdom 8728  df-pnf 11022  df-mnf 11023  df-xr 11024  df-ltxr 11025  df-le 11026  df-sub 11218  df-neg 11219  df-div 11644  df-nn 11985  df-2 12047  df-n0 12245  df-z 12331  df-uz 12594  df-fz 13251  df-seq 13733

This theorem is referenced by:  ruclem11  15960  ruclem12  15961