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Theorem ruclem10 14962
Description: Lemma for ruc 14966. 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.)
Hypotheses
Ref Expression
ruc.1 (𝜑𝐹:ℕ⟶ℝ)
ruc.2 (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
ruc.4 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
ruc.5 𝐺 = seq0(𝐷, 𝐶)
ruclem10.6 (𝜑𝑀 ∈ ℕ0)
ruclem10.7 (𝜑𝑁 ∈ ℕ0)
Assertion
Ref Expression
ruclem10 (𝜑 → (1st ‘(𝐺𝑀)) < (2nd ‘(𝐺𝑁)))
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
StepHypRef Expression
1 ruc.1 . . . . 5 (𝜑𝐹:ℕ⟶ℝ)
2 ruc.2 . . . . 5 (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
3 ruc.4 . . . . 5 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
4 ruc.5 . . . . 5 𝐺 = seq0(𝐷, 𝐶)
51, 2, 3, 4ruclem6 14958 . . . 4 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
6 ruclem10.6 . . . 4 (𝜑𝑀 ∈ ℕ0)
75, 6ffvelrnd 6358 . . 3 (𝜑 → (𝐺𝑀) ∈ (ℝ × ℝ))
8 xp1st 7195 . . 3 ((𝐺𝑀) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑀)) ∈ ℝ)
97, 8syl 17 . 2 (𝜑 → (1st ‘(𝐺𝑀)) ∈ ℝ)
10 ruclem10.7 . . . . 5 (𝜑𝑁 ∈ ℕ0)
1110, 6ifcld 4129 . . . 4 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
125, 11ffvelrnd 6358 . . 3 (𝜑 → (𝐺‘if(𝑀𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ))
13 xp1st 7195 . . 3 ((𝐺‘if(𝑀𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∈ ℝ)
1412, 13syl 17 . 2 (𝜑 → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∈ ℝ)
155, 10ffvelrnd 6358 . . 3 (𝜑 → (𝐺𝑁) ∈ (ℝ × ℝ))
16 xp2nd 7196 . . 3 ((𝐺𝑁) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑁)) ∈ ℝ)
1715, 16syl 17 . 2 (𝜑 → (2nd ‘(𝐺𝑁)) ∈ ℝ)
186nn0red 11349 . . . . . 6 (𝜑𝑀 ∈ ℝ)
1910nn0red 11349 . . . . . 6 (𝜑𝑁 ∈ ℝ)
20 max1 12013 . . . . . 6 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
2118, 19, 20syl2anc 693 . . . . 5 (𝜑𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
226nn0zd 11477 . . . . . 6 (𝜑𝑀 ∈ ℤ)
2311nn0zd 11477 . . . . . 6 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℤ)
24 eluz 11698 . . . . . 6 ((𝑀 ∈ ℤ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℤ) → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑀) ↔ 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
2522, 23, 24syl2anc 693 . . . . 5 (𝜑 → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑀) ↔ 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
2621, 25mpbird 247 . . . 4 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑀))
271, 2, 3, 4, 6, 26ruclem9 14961 . . 3 (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∧ (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺𝑀))))
2827simpld 475 . 2 (𝜑 → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))))
29 xp2nd 7196 . . . 4 ((𝐺‘if(𝑀𝑁, 𝑁, 𝑀)) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∈ ℝ)
3012, 29syl 17 . . 3 (𝜑 → (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∈ ℝ)
311, 2, 3, 4ruclem8 14960 . . . 4 ((𝜑 ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0) → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))))
3211, 31mpdan 702 . . 3 (𝜑 → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))))
33 max2 12015 . . . . . . 7 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
3418, 19, 33syl2anc 693 . . . . . 6 (𝜑𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
3510nn0zd 11477 . . . . . . 7 (𝜑𝑁 ∈ ℤ)
36 eluz 11698 . . . . . . 7 ((𝑁 ∈ ℤ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℤ) → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑁) ↔ 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
3735, 23, 36syl2anc 693 . . . . . 6 (𝜑 → (if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑁) ↔ 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
3834, 37mpbird 247 . . . . 5 (𝜑 → if(𝑀𝑁, 𝑁, 𝑀) ∈ (ℤ𝑁))
391, 2, 3, 4, 10, 38ruclem9 14961 . . . 4 (𝜑 → ((1st ‘(𝐺𝑁)) ≤ (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ∧ (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺𝑁))))
4039simprd 479 . . 3 (𝜑 → (2nd ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) ≤ (2nd ‘(𝐺𝑁)))
4114, 30, 17, 32, 40ltletrd 10194 . 2 (𝜑 → (1st ‘(𝐺‘if(𝑀𝑁, 𝑁, 𝑀))) < (2nd ‘(𝐺𝑁)))
429, 14, 17, 28, 41lelttrd 10192 1 (𝜑 → (1st ‘(𝐺𝑀)) < (2nd ‘(𝐺𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1482  wcel 1989  csb 3531  cun 3570  ifcif 4084  {csn 4175  cop 4181   class class class wbr 4651   × cxp 5110  wf 5882  cfv 5886  (class class class)co 6647  cmpt2 6649  1st c1st 7163  2nd c2nd 7164  cr 9932  0cc0 9933  1c1 9934   + caddc 9936   < clt 10071  cle 10072   / cdiv 10681  cn 11017  2c2 11067  0cn0 11289  cz 11374  cuz 11684  seqcseq 12796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-div 10682  df-nn 11018  df-2 11076  df-n0 11290  df-z 11375  df-uz 11685  df-fz 12324  df-seq 12797
This theorem is referenced by:  ruclem11  14963  ruclem12  14964
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