Theorem ruclem10 16176

Description: Lemma for ruc 16180. 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 16172	. . . 4 ⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ × ℝ))
6		ruclem10.6	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
7	5, 6	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → (𝐺‘𝑀) ∈ (ℝ × ℝ))
8		xp1st 7975	. . 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 4528	. . . 4 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℕ0)
12	5, 11	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → (𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ))
13		xp1st 7975	. . 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 7039	. . 3 ⊢ (𝜑 → (𝐺‘𝑁) ∈ (ℝ × ℝ))
16		xp2nd 7976	. . 3 ⊢ ((𝐺‘𝑁) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘𝑁)) ∈ ℝ)
17	15, 16	syl 17	. 2 ⊢ (𝜑 → (2nd ‘(𝐺‘𝑁)) ∈ ℝ)
18	6	nn0red 12475	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
19	10	nn0red 12475	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
20		max1 13112	. . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
21	18, 19, 20	syl2anc 585	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
22	6	nn0zd 12525	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
23	11	nn0zd 12525	. . . . . 6 ⊢ (𝜑 → if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ∈ ℤ)
24		eluz 12777	. . . . . 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 585	. . . . 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 16175	. . 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 7976	. . . 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 16174	. . . 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 688	. . 3 ⊢ (𝜑 → (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))))
33		max2 13114	. . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
34	18, 19, 33	syl2anc 585	. . . . . 6 ⊢ (𝜑 → 𝑁 ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
35	10	nn0zd 12525	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
36		eluz 12777	. . . . . . 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 585	. . . . . 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 16175	. . . 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 11305	. 2 ⊢ (𝜑 → (1st ‘(𝐺‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘𝑁)))
42	9, 14, 17, 28, 41	lelttrd 11303	1 ⊢ (𝜑 → (1st ‘(𝐺‘𝑀)) < (2nd ‘(𝐺‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ⦋csb 3851   ∪ cun 3901  ifcif 4481  {csn 4582  ⟨cop 4588   class class class wbr 5100   × cxp 5630  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1^st c1st 7941  2^nd c2nd 7942  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   < clt 11178   ≤ cle 11179   / cdiv 11806  ℕcn 12157  2c2 12212  ℕ₀cn0 12413  ℤcz 12500  ℤ_≥cuz 12763  seqcseq 13936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-n0 12414  df-z 12501  df-uz 12764  df-fz 13436  df-seq 13937

This theorem is referenced by:  ruclem11  16177  ruclem12  16178