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Theorem ruclem10 16128
Description: Lemma for ruc 16132. 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 (πœ‘ β†’ 𝐹:β„•βŸΆβ„)
ruc.2 (πœ‘ β†’ 𝐷 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)))
ruc.4 𝐢 = ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)
ruc.5 𝐺 = seq0(𝐷, 𝐢)
ruclem10.6 (πœ‘ β†’ 𝑀 ∈ β„•0)
ruclem10.7 (πœ‘ β†’ 𝑁 ∈ β„•0)
Assertion
Ref Expression
ruclem10 (πœ‘ β†’ (1st β€˜(πΊβ€˜π‘€)) < (2nd β€˜(πΊβ€˜π‘)))
Distinct variable groups:   π‘₯,π‘š,𝑦,𝐹   π‘š,𝐺,π‘₯,𝑦   π‘š,𝑀,π‘₯,𝑦   π‘š,𝑁,π‘₯,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦,π‘š)   𝐢(π‘₯,𝑦,π‘š)   𝐷(π‘₯,𝑦,π‘š)

Proof of Theorem ruclem10
StepHypRef Expression
1 ruc.1 . . . . 5 (πœ‘ β†’ 𝐹:β„•βŸΆβ„)
2 ruc.2 . . . . 5 (πœ‘ β†’ 𝐷 = (π‘₯ ∈ (ℝ Γ— ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1st β€˜π‘₯) + (2nd β€˜π‘₯)) / 2) / π‘šβ¦Œif(π‘š < 𝑦, ⟨(1st β€˜π‘₯), π‘šβŸ©, ⟨((π‘š + (2nd β€˜π‘₯)) / 2), (2nd β€˜π‘₯)⟩)))
3 ruc.4 . . . . 5 𝐢 = ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)
4 ruc.5 . . . . 5 𝐺 = seq0(𝐷, 𝐢)
51, 2, 3, 4ruclem6 16124 . . . 4 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
6 ruclem10.6 . . . 4 (πœ‘ β†’ 𝑀 ∈ β„•0)
75, 6ffvelcdmd 7041 . . 3 (πœ‘ β†’ (πΊβ€˜π‘€) ∈ (ℝ Γ— ℝ))
8 xp1st 7958 . . 3 ((πΊβ€˜π‘€) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΊβ€˜π‘€)) ∈ ℝ)
97, 8syl 17 . 2 (πœ‘ β†’ (1st β€˜(πΊβ€˜π‘€)) ∈ ℝ)
10 ruclem10.7 . . . . 5 (πœ‘ β†’ 𝑁 ∈ β„•0)
1110, 6ifcld 4537 . . . 4 (πœ‘ β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ β„•0)
125, 11ffvelcdmd 7041 . . 3 (πœ‘ β†’ (πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀)) ∈ (ℝ Γ— ℝ))
13 xp1st 7958 . . 3 ((πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀)) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) ∈ ℝ)
1412, 13syl 17 . 2 (πœ‘ β†’ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) ∈ ℝ)
155, 10ffvelcdmd 7041 . . 3 (πœ‘ β†’ (πΊβ€˜π‘) ∈ (ℝ Γ— ℝ))
16 xp2nd 7959 . . 3 ((πΊβ€˜π‘) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΊβ€˜π‘)) ∈ ℝ)
1715, 16syl 17 . 2 (πœ‘ β†’ (2nd β€˜(πΊβ€˜π‘)) ∈ ℝ)
186nn0red 12481 . . . . . 6 (πœ‘ β†’ 𝑀 ∈ ℝ)
1910nn0red 12481 . . . . . 6 (πœ‘ β†’ 𝑁 ∈ ℝ)
20 max1 13111 . . . . . 6 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) β†’ 𝑀 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
2118, 19, 20syl2anc 585 . . . . 5 (πœ‘ β†’ 𝑀 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
226nn0zd 12532 . . . . . 6 (πœ‘ β†’ 𝑀 ∈ β„€)
2311nn0zd 12532 . . . . . 6 (πœ‘ β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ β„€)
24 eluz 12784 . . . . . 6 ((𝑀 ∈ β„€ ∧ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ β„€) β†’ (if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ (β„€β‰₯β€˜π‘€) ↔ 𝑀 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀)))
2522, 23, 24syl2anc 585 . . . . 5 (πœ‘ β†’ (if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ (β„€β‰₯β€˜π‘€) ↔ 𝑀 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀)))
2621, 25mpbird 257 . . . 4 (πœ‘ β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ (β„€β‰₯β€˜π‘€))
271, 2, 3, 4, 6, 26ruclem9 16127 . . 3 (πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) ∧ (2nd β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) ≀ (2nd β€˜(πΊβ€˜π‘€))))
2827simpld 496 . 2 (πœ‘ β†’ (1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))))
29 xp2nd 7959 . . . 4 ((πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀)) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) ∈ ℝ)
3012, 29syl 17 . . 3 (πœ‘ β†’ (2nd β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) ∈ ℝ)
311, 2, 3, 4ruclem8 16126 . . . 4 ((πœ‘ ∧ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ β„•0) β†’ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) < (2nd β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))))
3211, 31mpdan 686 . . 3 (πœ‘ β†’ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) < (2nd β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))))
33 max2 13113 . . . . . . 7 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) β†’ 𝑁 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
3418, 19, 33syl2anc 585 . . . . . 6 (πœ‘ β†’ 𝑁 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
3510nn0zd 12532 . . . . . . 7 (πœ‘ β†’ 𝑁 ∈ β„€)
36 eluz 12784 . . . . . . 7 ((𝑁 ∈ β„€ ∧ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ β„€) β†’ (if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ (β„€β‰₯β€˜π‘) ↔ 𝑁 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀)))
3735, 23, 36syl2anc 585 . . . . . 6 (πœ‘ β†’ (if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ (β„€β‰₯β€˜π‘) ↔ 𝑁 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀)))
3834, 37mpbird 257 . . . . 5 (πœ‘ β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ (β„€β‰₯β€˜π‘))
391, 2, 3, 4, 10, 38ruclem9 16127 . . . 4 (πœ‘ β†’ ((1st β€˜(πΊβ€˜π‘)) ≀ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) ∧ (2nd β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) ≀ (2nd β€˜(πΊβ€˜π‘))))
4039simprd 497 . . 3 (πœ‘ β†’ (2nd β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) ≀ (2nd β€˜(πΊβ€˜π‘)))
4114, 30, 17, 32, 40ltletrd 11322 . 2 (πœ‘ β†’ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) < (2nd β€˜(πΊβ€˜π‘)))
429, 14, 17, 28, 41lelttrd 11320 1 (πœ‘ β†’ (1st β€˜(πΊβ€˜π‘€)) < (2nd β€˜(πΊβ€˜π‘)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  β¦‹csb 3860   βˆͺ cun 3913  ifcif 4491  {csn 4591  βŸ¨cop 4597   class class class wbr 5110   Γ— cxp 5636  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364  1st c1st 7924  2nd c2nd 7925  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   < clt 11196   ≀ cle 11197   / cdiv 11819  β„•cn 12160  2c2 12215  β„•0cn0 12420  β„€cz 12506  β„€β‰₯cuz 12770  seqcseq 13913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-seq 13914
This theorem is referenced by:  ruclem11  16129  ruclem12  16130
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