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Theorem ruclem6 16124
Description: Lemma for ruc 16132. Domain and codomain of the interval sequence. (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)
Assertion
Ref Expression
ruclem6 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
Distinct variable groups:   π‘₯,π‘š,𝑦,𝐹   π‘š,𝐺,π‘₯,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦,π‘š)   𝐢(π‘₯,𝑦,π‘š)   𝐷(π‘₯,𝑦,π‘š)

Proof of Theorem ruclem6
Dummy variables 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ruc.5 . . . . . . 7 𝐺 = seq0(𝐷, 𝐢)
21fveq1i 6848 . . . . . 6 (πΊβ€˜0) = (seq0(𝐷, 𝐢)β€˜0)
3 0z 12517 . . . . . . 7 0 ∈ β„€
4 seq1 13926 . . . . . . 7 (0 ∈ β„€ β†’ (seq0(𝐷, 𝐢)β€˜0) = (πΆβ€˜0))
53, 4ax-mp 5 . . . . . 6 (seq0(𝐷, 𝐢)β€˜0) = (πΆβ€˜0)
62, 5eqtri 2765 . . . . 5 (πΊβ€˜0) = (πΆβ€˜0)
7 ruc.1 . . . . . 6 (πœ‘ β†’ 𝐹:β„•βŸΆβ„)
8 ruc.2 . . . . . 6 (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‘₯)⟩)))
9 ruc.4 . . . . . 6 𝐢 = ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)
107, 8, 9, 1ruclem4 16123 . . . . 5 (πœ‘ β†’ (πΊβ€˜0) = ⟨0, 1⟩)
116, 10eqtr3id 2791 . . . 4 (πœ‘ β†’ (πΆβ€˜0) = ⟨0, 1⟩)
12 0re 11164 . . . . 5 0 ∈ ℝ
13 1re 11162 . . . . 5 1 ∈ ℝ
14 opelxpi 5675 . . . . 5 ((0 ∈ ℝ ∧ 1 ∈ ℝ) β†’ ⟨0, 1⟩ ∈ (ℝ Γ— ℝ))
1512, 13, 14mp2an 691 . . . 4 ⟨0, 1⟩ ∈ (ℝ Γ— ℝ)
1611, 15eqeltrdi 2846 . . 3 (πœ‘ β†’ (πΆβ€˜0) ∈ (ℝ Γ— ℝ))
17 1st2nd2 7965 . . . . . 6 (𝑧 ∈ (ℝ Γ— ℝ) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
1817ad2antrl 727 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ ℝ)) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
1918oveq1d 7377 . . . 4 ((πœ‘ ∧ (𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ ℝ)) β†’ (𝑧𝐷𝑀) = (⟨(1st β€˜π‘§), (2nd β€˜π‘§)βŸ©π·π‘€))
207adantr 482 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ ℝ)) β†’ 𝐹:β„•βŸΆβ„)
218adantr 482 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ ℝ)) 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‘₯)⟩)))
22 xp1st 7958 . . . . . . 7 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (1st β€˜π‘§) ∈ ℝ)
2322ad2antrl 727 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ ℝ)) β†’ (1st β€˜π‘§) ∈ ℝ)
24 xp2nd 7959 . . . . . . 7 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜π‘§) ∈ ℝ)
2524ad2antrl 727 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ ℝ)) β†’ (2nd β€˜π‘§) ∈ ℝ)
26 simprr 772 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ ℝ)) β†’ 𝑀 ∈ ℝ)
27 eqid 2737 . . . . . 6 (1st β€˜(⟨(1st β€˜π‘§), (2nd β€˜π‘§)βŸ©π·π‘€)) = (1st β€˜(⟨(1st β€˜π‘§), (2nd β€˜π‘§)βŸ©π·π‘€))
28 eqid 2737 . . . . . 6 (2nd β€˜(⟨(1st β€˜π‘§), (2nd β€˜π‘§)βŸ©π·π‘€)) = (2nd β€˜(⟨(1st β€˜π‘§), (2nd β€˜π‘§)βŸ©π·π‘€))
2920, 21, 23, 25, 26, 27, 28ruclem1 16120 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ ℝ)) β†’ ((⟨(1st β€˜π‘§), (2nd β€˜π‘§)βŸ©π·π‘€) ∈ (ℝ Γ— ℝ) ∧ (1st β€˜(⟨(1st β€˜π‘§), (2nd β€˜π‘§)βŸ©π·π‘€)) = if((((1st β€˜π‘§) + (2nd β€˜π‘§)) / 2) < 𝑀, (1st β€˜π‘§), (((((1st β€˜π‘§) + (2nd β€˜π‘§)) / 2) + (2nd β€˜π‘§)) / 2)) ∧ (2nd β€˜(⟨(1st β€˜π‘§), (2nd β€˜π‘§)βŸ©π·π‘€)) = if((((1st β€˜π‘§) + (2nd β€˜π‘§)) / 2) < 𝑀, (((1st β€˜π‘§) + (2nd β€˜π‘§)) / 2), (2nd β€˜π‘§))))
3029simp1d 1143 . . . 4 ((πœ‘ ∧ (𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ ℝ)) β†’ (⟨(1st β€˜π‘§), (2nd β€˜π‘§)βŸ©π·π‘€) ∈ (ℝ Γ— ℝ))
3119, 30eqeltrd 2838 . . 3 ((πœ‘ ∧ (𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ ℝ)) β†’ (𝑧𝐷𝑀) ∈ (ℝ Γ— ℝ))
32 nn0uz 12812 . . 3 β„•0 = (β„€β‰₯β€˜0)
33 0zd 12518 . . 3 (πœ‘ β†’ 0 ∈ β„€)
34 0p1e1 12282 . . . . . . 7 (0 + 1) = 1
3534fveq2i 6850 . . . . . 6 (β„€β‰₯β€˜(0 + 1)) = (β„€β‰₯β€˜1)
36 nnuz 12813 . . . . . 6 β„• = (β„€β‰₯β€˜1)
3735, 36eqtr4i 2768 . . . . 5 (β„€β‰₯β€˜(0 + 1)) = β„•
3837eleq2i 2830 . . . 4 (𝑧 ∈ (β„€β‰₯β€˜(0 + 1)) ↔ 𝑧 ∈ β„•)
399equncomi 4120 . . . . . . . 8 𝐢 = (𝐹 βˆͺ {⟨0, ⟨0, 1⟩⟩})
4039fveq1i 6848 . . . . . . 7 (πΆβ€˜π‘§) = ((𝐹 βˆͺ {⟨0, ⟨0, 1⟩⟩})β€˜π‘§)
41 nnne0 12194 . . . . . . . . 9 (𝑧 ∈ β„• β†’ 𝑧 β‰  0)
4241necomd 3000 . . . . . . . 8 (𝑧 ∈ β„• β†’ 0 β‰  𝑧)
43 fvunsn 7130 . . . . . . . 8 (0 β‰  𝑧 β†’ ((𝐹 βˆͺ {⟨0, ⟨0, 1⟩⟩})β€˜π‘§) = (πΉβ€˜π‘§))
4442, 43syl 17 . . . . . . 7 (𝑧 ∈ β„• β†’ ((𝐹 βˆͺ {⟨0, ⟨0, 1⟩⟩})β€˜π‘§) = (πΉβ€˜π‘§))
4540, 44eqtrid 2789 . . . . . 6 (𝑧 ∈ β„• β†’ (πΆβ€˜π‘§) = (πΉβ€˜π‘§))
4645adantl 483 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ β„•) β†’ (πΆβ€˜π‘§) = (πΉβ€˜π‘§))
477ffvelcdmda 7040 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ β„•) β†’ (πΉβ€˜π‘§) ∈ ℝ)
4846, 47eqeltrd 2838 . . . 4 ((πœ‘ ∧ 𝑧 ∈ β„•) β†’ (πΆβ€˜π‘§) ∈ ℝ)
4938, 48sylan2b 595 . . 3 ((πœ‘ ∧ 𝑧 ∈ (β„€β‰₯β€˜(0 + 1))) β†’ (πΆβ€˜π‘§) ∈ ℝ)
5016, 31, 32, 33, 49seqf2 13934 . 2 (πœ‘ β†’ seq0(𝐷, 𝐢):β„•0⟢(ℝ Γ— ℝ))
511feq1i 6664 . 2 (𝐺:β„•0⟢(ℝ Γ— ℝ) ↔ seq0(𝐷, 𝐢):β„•0⟢(ℝ Γ— ℝ))
5250, 51sylibr 233 1 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  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   / 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:  ruclem8  16126  ruclem9  16127  ruclem10  16128  ruclem11  16129  ruclem12  16130
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