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Theorem ruclem10 16225
Description: Lemma for ruc 16229. 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 16221 . . . 4 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
6 ruclem10.6 . . . 4 (πœ‘ β†’ 𝑀 ∈ β„•0)
75, 6ffvelcdmd 7100 . . 3 (πœ‘ β†’ (πΊβ€˜π‘€) ∈ (ℝ Γ— ℝ))
8 xp1st 8033 . . 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 4578 . . . 4 (πœ‘ β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ β„•0)
125, 11ffvelcdmd 7100 . . 3 (πœ‘ β†’ (πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀)) ∈ (ℝ Γ— ℝ))
13 xp1st 8033 . . 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 7100 . . 3 (πœ‘ β†’ (πΊβ€˜π‘) ∈ (ℝ Γ— ℝ))
16 xp2nd 8034 . . 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 12573 . . . . . 6 (πœ‘ β†’ 𝑀 ∈ ℝ)
1910nn0red 12573 . . . . . 6 (πœ‘ β†’ 𝑁 ∈ ℝ)
20 max1 13206 . . . . . 6 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) β†’ 𝑀 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
2118, 19, 20syl2anc 582 . . . . 5 (πœ‘ β†’ 𝑀 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
226nn0zd 12624 . . . . . 6 (πœ‘ β†’ 𝑀 ∈ β„€)
2311nn0zd 12624 . . . . . 6 (πœ‘ β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ β„€)
24 eluz 12876 . . . . . 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 582 . . . . 5 (πœ‘ β†’ (if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ (β„€β‰₯β€˜π‘€) ↔ 𝑀 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀)))
2621, 25mpbird 256 . . . 4 (πœ‘ β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ (β„€β‰₯β€˜π‘€))
271, 2, 3, 4, 6, 26ruclem9 16224 . . 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 493 . 2 (πœ‘ β†’ (1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))))
29 xp2nd 8034 . . . 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 16223 . . . 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 685 . . 3 (πœ‘ β†’ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) < (2nd β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))))
33 max2 13208 . . . . . . 7 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) β†’ 𝑁 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
3418, 19, 33syl2anc 582 . . . . . 6 (πœ‘ β†’ 𝑁 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
3510nn0zd 12624 . . . . . . 7 (πœ‘ β†’ 𝑁 ∈ β„€)
36 eluz 12876 . . . . . . 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 582 . . . . . 6 (πœ‘ β†’ (if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ (β„€β‰₯β€˜π‘) ↔ 𝑁 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀)))
3834, 37mpbird 256 . . . . 5 (πœ‘ β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ (β„€β‰₯β€˜π‘))
391, 2, 3, 4, 10, 38ruclem9 16224 . . . 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 494 . . 3 (πœ‘ β†’ (2nd β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) ≀ (2nd β€˜(πΊβ€˜π‘)))
4114, 30, 17, 32, 40ltletrd 11414 . 2 (πœ‘ β†’ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) < (2nd β€˜(πΊβ€˜π‘)))
429, 14, 17, 28, 41lelttrd 11412 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 1533   ∈ wcel 2098  β¦‹csb 3894   βˆͺ cun 3947  ifcif 4532  {csn 4632  βŸ¨cop 4638   class class class wbr 5152   Γ— cxp 5680  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426   ∈ cmpo 7428  1st c1st 7999  2nd c2nd 8000  β„cr 11147  0cc0 11148  1c1 11149   + caddc 11151   < clt 11288   ≀ cle 11289   / cdiv 11911  β„•cn 12252  2c2 12307  β„•0cn0 12512  β„€cz 12598  β„€β‰₯cuz 12862  seqcseq 14008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-er 8733  df-en 8973  df-dom 8974  df-sdom 8975  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-div 11912  df-nn 12253  df-2 12315  df-n0 12513  df-z 12599  df-uz 12863  df-fz 13527  df-seq 14009
This theorem is referenced by:  ruclem11  16226  ruclem12  16227
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