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Theorem ruclem10 16189
Description: Lemma for ruc 16193. 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 16185 . . . 4 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
6 ruclem10.6 . . . 4 (πœ‘ β†’ 𝑀 ∈ β„•0)
75, 6ffvelcdmd 7081 . . 3 (πœ‘ β†’ (πΊβ€˜π‘€) ∈ (ℝ Γ— ℝ))
8 xp1st 8006 . . 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 4569 . . . 4 (πœ‘ β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ β„•0)
125, 11ffvelcdmd 7081 . . 3 (πœ‘ β†’ (πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀)) ∈ (ℝ Γ— ℝ))
13 xp1st 8006 . . 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 7081 . . 3 (πœ‘ β†’ (πΊβ€˜π‘) ∈ (ℝ Γ— ℝ))
16 xp2nd 8007 . . 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 12537 . . . . . 6 (πœ‘ β†’ 𝑀 ∈ ℝ)
1910nn0red 12537 . . . . . 6 (πœ‘ β†’ 𝑁 ∈ ℝ)
20 max1 13170 . . . . . 6 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) β†’ 𝑀 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
2118, 19, 20syl2anc 583 . . . . 5 (πœ‘ β†’ 𝑀 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
226nn0zd 12588 . . . . . 6 (πœ‘ β†’ 𝑀 ∈ β„€)
2311nn0zd 12588 . . . . . 6 (πœ‘ β†’ if(𝑀 ≀ 𝑁, 𝑁, 𝑀) ∈ β„€)
24 eluz 12840 . . . . . 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 583 . . . . 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 16188 . . 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 494 . 2 (πœ‘ β†’ (1st β€˜(πΊβ€˜π‘€)) ≀ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))))
29 xp2nd 8007 . . . 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 16187 . . . 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 684 . . 3 (πœ‘ β†’ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) < (2nd β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))))
33 max2 13172 . . . . . . 7 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) β†’ 𝑁 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
3418, 19, 33syl2anc 583 . . . . . 6 (πœ‘ β†’ 𝑁 ≀ if(𝑀 ≀ 𝑁, 𝑁, 𝑀))
3510nn0zd 12588 . . . . . . 7 (πœ‘ β†’ 𝑁 ∈ β„€)
36 eluz 12840 . . . . . . 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 583 . . . . . 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 16188 . . . 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 495 . . 3 (πœ‘ β†’ (2nd β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) ≀ (2nd β€˜(πΊβ€˜π‘)))
4114, 30, 17, 32, 40ltletrd 11378 . 2 (πœ‘ β†’ (1st β€˜(πΊβ€˜if(𝑀 ≀ 𝑁, 𝑁, 𝑀))) < (2nd β€˜(πΊβ€˜π‘)))
429, 14, 17, 28, 41lelttrd 11376 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 3888   βˆͺ cun 3941  ifcif 4523  {csn 4623  βŸ¨cop 4629   class class class wbr 5141   Γ— cxp 5667  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  1st c1st 7972  2nd c2nd 7973  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115   < clt 11252   ≀ cle 11253   / cdiv 11875  β„•cn 12216  2c2 12271  β„•0cn0 12476  β„€cz 12562  β„€β‰₯cuz 12826  seqcseq 13972
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-seq 13973
This theorem is referenced by:  ruclem11  16190  ruclem12  16191
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