Theorem ruclem10 16287

Description: Lemma for ruc 16291. 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 16283	. . . 4 ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ))
6		ruclem10.6	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
7	5, 6	ffvelcdmd 7119	. . 3 ⊢ (𝜑 → (𝐺‘𝑀) ∈ (ℝ × ℝ))
8		xp1st 8062	. . 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 4594	. . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0)
12	5, 11	ffvelcdmd 7119	. . 3 ⊢ (𝜑 → (𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ))
13		xp1st 8062	. . 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 7119	. . 3 ⊢ (𝜑 → (𝐺‘𝑁) ∈ (ℝ × ℝ))
16		xp2nd 8063	. . 3 ⊢ ((𝐺‘𝑁) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘𝑁)) ∈ ℝ)
17	15, 16	syl 17	. 2 ⊢ (𝜑 → (2nd ‘(𝐺‘𝑁)) ∈ ℝ)
18	6	nn0red 12614	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
19	10	nn0red 12614	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
20		max1 13247	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
21	18, 19, 20	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
22	6	nn0zd 12665	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
23	11	nn0zd 12665	. . . . . 6 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℤ)
24		eluz 12917	. . . . . 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 583	. . . . 5 ⊢ (𝜑 → (if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
26	21, 25	mpbird 257	. . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑀))
27	1, 2, 3, 4, 6, 26	ruclem9 16286	. . 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 494	. 2 ⊢ (𝜑 → (1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))))
29		xp2nd 8063	. . . 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 16285	. . . 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 686	. . 3 ⊢ (𝜑 → (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))))
33		max2 13249	. . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
34	18, 19, 33	syl2anc 583	. . . . . 6 ⊢ (𝜑 → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
35	10	nn0zd 12665	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
36		eluz 12917	. . . . . . 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 583	. . . . . 6 ⊢ (𝜑 → (if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑁) ↔ 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
38	34, 37	mpbird 257	. . . . 5 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ (ℤ≥‘𝑁))
39	1, 2, 3, 4, 10, 38	ruclem9 16286	. . . 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 495	. . 3 ⊢ (𝜑 → (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺‘𝑁)))
41	14, 30, 17, 32, 40	ltletrd 11450	. 2 ⊢ (𝜑 → (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘𝑁)))
42	9, 14, 17, 28, 41	lelttrd 11448	1 ⊢ (𝜑 → (1st ‘(𝐺‘𝑀)) < (2nd ‘(𝐺‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  ⦋csb 3921   ∪ cun 3974  ifcif 4548  {csn 4648  ⟨cop 4654   class class class wbr 5166   × cxp 5698  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1^st c1st 8028  2^nd c2nd 8029  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   < clt 11324   ≤ cle 11325   / cdiv 11947  ℕcn 12293  2c2 12348  ℕ₀cn0 12553  ℤcz 12639  ℤ_≥cuz 12903  seqcseq 14052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-seq 14053

This theorem is referenced by:  ruclem11  16288  ruclem12  16289