MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ruclem12 Structured version   Visualization version   GIF version

Theorem ruclem12 16130
Description: Lemma for ruc 16132. The supremum of the increasing sequence 1st ∘ 𝐺 is a real number that is not in the range of 𝐹. (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)
ruc.6 𝑆 = sup(ran (1st ∘ 𝐺), ℝ, < )
Assertion
Ref Expression
ruclem12 (πœ‘ β†’ 𝑆 ∈ (ℝ βˆ– ran 𝐹))
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)   𝑆(π‘₯,𝑦,π‘š)

Proof of Theorem ruclem12
Dummy variables 𝑧 𝑛 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ruc.6 . . 3 𝑆 = sup(ran (1st ∘ 𝐺), ℝ, < )
2 ruc.1 . . . . . 6 (πœ‘ β†’ 𝐹:β„•βŸΆβ„)
3 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‘₯)⟩)))
4 ruc.4 . . . . . 6 𝐢 = ({⟨0, ⟨0, 1⟩⟩} βˆͺ 𝐹)
5 ruc.5 . . . . . 6 𝐺 = seq0(𝐷, 𝐢)
62, 3, 4, 5ruclem11 16129 . . . . 5 (πœ‘ β†’ (ran (1st ∘ 𝐺) βŠ† ℝ ∧ ran (1st ∘ 𝐺) β‰  βˆ… ∧ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 1))
76simp1d 1143 . . . 4 (πœ‘ β†’ ran (1st ∘ 𝐺) βŠ† ℝ)
86simp2d 1144 . . . 4 (πœ‘ β†’ ran (1st ∘ 𝐺) β‰  βˆ…)
9 1re 11162 . . . . 5 1 ∈ ℝ
106simp3d 1145 . . . . 5 (πœ‘ β†’ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 1)
11 brralrspcev 5170 . . . . 5 ((1 ∈ ℝ ∧ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 1) β†’ βˆƒπ‘› ∈ ℝ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 𝑛)
129, 10, 11sylancr 588 . . . 4 (πœ‘ β†’ βˆƒπ‘› ∈ ℝ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 𝑛)
137, 8, 12suprcld 12125 . . 3 (πœ‘ β†’ sup(ran (1st ∘ 𝐺), ℝ, < ) ∈ ℝ)
141, 13eqeltrid 2842 . 2 (πœ‘ β†’ 𝑆 ∈ ℝ)
152adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝐹:β„•βŸΆβ„)
163adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) 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‘₯)⟩)))
172, 3, 4, 5ruclem6 16124 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
18 nnm1nn0 12461 . . . . . . . . . . 11 (𝑛 ∈ β„• β†’ (𝑛 βˆ’ 1) ∈ β„•0)
19 ffvelcdm 7037 . . . . . . . . . . 11 ((𝐺:β„•0⟢(ℝ Γ— ℝ) ∧ (𝑛 βˆ’ 1) ∈ β„•0) β†’ (πΊβ€˜(𝑛 βˆ’ 1)) ∈ (ℝ Γ— ℝ))
2017, 18, 19syl2an 597 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜(𝑛 βˆ’ 1)) ∈ (ℝ Γ— ℝ))
21 xp1st 7958 . . . . . . . . . 10 ((πΊβ€˜(𝑛 βˆ’ 1)) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))) ∈ ℝ)
2220, 21syl 17 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))) ∈ ℝ)
23 xp2nd 7959 . . . . . . . . . 10 ((πΊβ€˜(𝑛 βˆ’ 1)) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1))) ∈ ℝ)
2420, 23syl 17 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1))) ∈ ℝ)
252ffvelcdmda 7040 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ ℝ)
26 eqid 2737 . . . . . . . . 9 (1st β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) = (1st β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›)))
27 eqid 2737 . . . . . . . . 9 (2nd β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) = (2nd β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›)))
282, 3, 4, 5ruclem8 16126 . . . . . . . . . 10 ((πœ‘ ∧ (𝑛 βˆ’ 1) ∈ β„•0) β†’ (1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))) < (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1))))
2918, 28sylan2 594 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))) < (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1))))
3015, 16, 22, 24, 25, 26, 27, 29ruclem3 16122 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›) < (1st β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) ∨ (2nd β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) < (πΉβ€˜π‘›)))
312, 3, 4, 5ruclem7 16125 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑛 βˆ’ 1) ∈ β„•0) β†’ (πΊβ€˜((𝑛 βˆ’ 1) + 1)) = ((πΊβ€˜(𝑛 βˆ’ 1))𝐷(πΉβ€˜((𝑛 βˆ’ 1) + 1))))
3218, 31sylan2 594 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜((𝑛 βˆ’ 1) + 1)) = ((πΊβ€˜(𝑛 βˆ’ 1))𝐷(πΉβ€˜((𝑛 βˆ’ 1) + 1))))
33 nncn 12168 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„• β†’ 𝑛 ∈ β„‚)
3433adantl 483 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„‚)
35 ax-1cn 11116 . . . . . . . . . . . . . 14 1 ∈ β„‚
36 npcan 11417 . . . . . . . . . . . . . 14 ((𝑛 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((𝑛 βˆ’ 1) + 1) = 𝑛)
3734, 35, 36sylancl 587 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((𝑛 βˆ’ 1) + 1) = 𝑛)
3837fveq2d 6851 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜((𝑛 βˆ’ 1) + 1)) = (πΊβ€˜π‘›))
39 1st2nd2 7965 . . . . . . . . . . . . . 14 ((πΊβ€˜(𝑛 βˆ’ 1)) ∈ (ℝ Γ— ℝ) β†’ (πΊβ€˜(𝑛 βˆ’ 1)) = ⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩)
4020, 39syl 17 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜(𝑛 βˆ’ 1)) = ⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩)
4137fveq2d 6851 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜((𝑛 βˆ’ 1) + 1)) = (πΉβ€˜π‘›))
4240, 41oveq12d 7380 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΊβ€˜(𝑛 βˆ’ 1))𝐷(πΉβ€˜((𝑛 βˆ’ 1) + 1))) = (⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›)))
4332, 38, 423eqtr3d 2785 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜π‘›) = (⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›)))
4443fveq2d 6851 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜π‘›)) = (1st β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))))
4544breq2d 5122 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) ↔ (πΉβ€˜π‘›) < (1st β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›)))))
4643fveq2d 6851 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (2nd β€˜(πΊβ€˜π‘›)) = (2nd β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))))
4746breq1d 5120 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›) ↔ (2nd β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) < (πΉβ€˜π‘›)))
4845, 47orbi12d 918 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) ∨ (2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›)) ↔ ((πΉβ€˜π‘›) < (1st β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) ∨ (2nd β€˜(⟨(1st β€˜(πΊβ€˜(𝑛 βˆ’ 1))), (2nd β€˜(πΊβ€˜(𝑛 βˆ’ 1)))⟩𝐷(πΉβ€˜π‘›))) < (πΉβ€˜π‘›))))
4930, 48mpbird 257 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) ∨ (2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›)))
507adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ran (1st ∘ 𝐺) βŠ† ℝ)
518adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ran (1st ∘ 𝐺) β‰  βˆ…)
5212adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ βˆƒπ‘› ∈ ℝ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 𝑛)
53 nnnn0 12427 . . . . . . . . . . . . 13 (𝑛 ∈ β„• β†’ 𝑛 ∈ β„•0)
54 fvco3 6945 . . . . . . . . . . . . 13 ((𝐺:β„•0⟢(ℝ Γ— ℝ) ∧ 𝑛 ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘›) = (1st β€˜(πΊβ€˜π‘›)))
5517, 53, 54syl2an 597 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((1st ∘ 𝐺)β€˜π‘›) = (1st β€˜(πΊβ€˜π‘›)))
5617adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
57 1stcof 7956 . . . . . . . . . . . . . 14 (𝐺:β„•0⟢(ℝ Γ— ℝ) β†’ (1st ∘ 𝐺):β„•0βŸΆβ„)
58 ffn 6673 . . . . . . . . . . . . . 14 ((1st ∘ 𝐺):β„•0βŸΆβ„ β†’ (1st ∘ 𝐺) Fn β„•0)
5956, 57, 583syl 18 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st ∘ 𝐺) Fn β„•0)
6053adantl 483 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„•0)
61 fnfvelrn 7036 . . . . . . . . . . . . 13 (((1st ∘ 𝐺) Fn β„•0 ∧ 𝑛 ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘›) ∈ ran (1st ∘ 𝐺))
6259, 60, 61syl2anc 585 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((1st ∘ 𝐺)β€˜π‘›) ∈ ran (1st ∘ 𝐺))
6355, 62eqeltrrd 2839 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜π‘›)) ∈ ran (1st ∘ 𝐺))
6450, 51, 52, 63suprubd 12124 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜π‘›)) ≀ sup(ran (1st ∘ 𝐺), ℝ, < ))
6564, 1breqtrrdi 5152 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜π‘›)) ≀ 𝑆)
66 ffvelcdm 7037 . . . . . . . . . . . 12 ((𝐺:β„•0⟢(ℝ Γ— ℝ) ∧ 𝑛 ∈ β„•0) β†’ (πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ))
6717, 53, 66syl2an 597 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ))
68 xp1st 7958 . . . . . . . . . . 11 ((πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΊβ€˜π‘›)) ∈ ℝ)
6967, 68syl 17 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜π‘›)) ∈ ℝ)
7014adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑆 ∈ ℝ)
71 ltletr 11254 . . . . . . . . . 10 (((πΉβ€˜π‘›) ∈ ℝ ∧ (1st β€˜(πΊβ€˜π‘›)) ∈ ℝ ∧ 𝑆 ∈ ℝ) β†’ (((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) ∧ (1st β€˜(πΊβ€˜π‘›)) ≀ 𝑆) β†’ (πΉβ€˜π‘›) < 𝑆))
7225, 69, 70, 71syl3anc 1372 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) ∧ (1st β€˜(πΊβ€˜π‘›)) ≀ 𝑆) β†’ (πΉβ€˜π‘›) < 𝑆))
7365, 72mpan2d 693 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) β†’ (πΉβ€˜π‘›) < 𝑆))
74 fvco3 6945 . . . . . . . . . . . . . . 15 ((𝐺:β„•0⟢(ℝ Γ— ℝ) ∧ π‘˜ ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘˜) = (1st β€˜(πΊβ€˜π‘˜)))
7556, 74sylan 581 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘˜) = (1st β€˜(πΊβ€˜π‘˜)))
7656ffvelcdmda 7040 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ (πΊβ€˜π‘˜) ∈ (ℝ Γ— ℝ))
77 xp1st 7958 . . . . . . . . . . . . . . . 16 ((πΊβ€˜π‘˜) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΊβ€˜π‘˜)) ∈ ℝ)
7876, 77syl 17 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘˜)) ∈ ℝ)
79 xp2nd 7959 . . . . . . . . . . . . . . . . 17 ((πΊβ€˜π‘›) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ)
8067, 79syl 17 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ)
8180adantr 482 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ)
8215adantr 482 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ 𝐹:β„•βŸΆβ„)
8316adantr 482 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) 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‘₯)⟩)))
84 simpr 486 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ π‘˜ ∈ β„•0)
8560adantr 482 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ 𝑛 ∈ β„•0)
8682, 83, 4, 5, 84, 85ruclem10 16128 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘˜)) < (2nd β€˜(πΊβ€˜π‘›)))
8778, 81, 86ltled 11310 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘›)))
8875, 87eqbrtrd 5132 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘˜) ≀ (2nd β€˜(πΊβ€˜π‘›)))
8988ralrimiva 3144 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ βˆ€π‘˜ ∈ β„•0 ((1st ∘ 𝐺)β€˜π‘˜) ≀ (2nd β€˜(πΊβ€˜π‘›)))
90 breq1 5113 . . . . . . . . . . . . . 14 (𝑧 = ((1st ∘ 𝐺)β€˜π‘˜) β†’ (𝑧 ≀ (2nd β€˜(πΊβ€˜π‘›)) ↔ ((1st ∘ 𝐺)β€˜π‘˜) ≀ (2nd β€˜(πΊβ€˜π‘›))))
9190ralrn 7043 . . . . . . . . . . . . 13 ((1st ∘ 𝐺) Fn β„•0 β†’ (βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ (2nd β€˜(πΊβ€˜π‘›)) ↔ βˆ€π‘˜ ∈ β„•0 ((1st ∘ 𝐺)β€˜π‘˜) ≀ (2nd β€˜(πΊβ€˜π‘›))))
9259, 91syl 17 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ (2nd β€˜(πΊβ€˜π‘›)) ↔ βˆ€π‘˜ ∈ β„•0 ((1st ∘ 𝐺)β€˜π‘˜) ≀ (2nd β€˜(πΊβ€˜π‘›))))
9389, 92mpbird 257 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ (2nd β€˜(πΊβ€˜π‘›)))
94 suprleub 12128 . . . . . . . . . . . 12 (((ran (1st ∘ 𝐺) βŠ† ℝ ∧ ran (1st ∘ 𝐺) β‰  βˆ… ∧ βˆƒπ‘› ∈ ℝ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 𝑛) ∧ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ) β†’ (sup(ran (1st ∘ 𝐺), ℝ, < ) ≀ (2nd β€˜(πΊβ€˜π‘›)) ↔ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ (2nd β€˜(πΊβ€˜π‘›))))
9550, 51, 52, 80, 94syl31anc 1374 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (sup(ran (1st ∘ 𝐺), ℝ, < ) ≀ (2nd β€˜(πΊβ€˜π‘›)) ↔ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ (2nd β€˜(πΊβ€˜π‘›))))
9693, 95mpbird 257 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ sup(ran (1st ∘ 𝐺), ℝ, < ) ≀ (2nd β€˜(πΊβ€˜π‘›)))
971, 96eqbrtrid 5145 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑆 ≀ (2nd β€˜(πΊβ€˜π‘›)))
98 lelttr 11252 . . . . . . . . . 10 ((𝑆 ∈ ℝ ∧ (2nd β€˜(πΊβ€˜π‘›)) ∈ ℝ ∧ (πΉβ€˜π‘›) ∈ ℝ) β†’ ((𝑆 ≀ (2nd β€˜(πΊβ€˜π‘›)) ∧ (2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›)) β†’ 𝑆 < (πΉβ€˜π‘›)))
9970, 80, 25, 98syl3anc 1372 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((𝑆 ≀ (2nd β€˜(πΊβ€˜π‘›)) ∧ (2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›)) β†’ 𝑆 < (πΉβ€˜π‘›)))
10097, 99mpand 694 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›) β†’ 𝑆 < (πΉβ€˜π‘›)))
10173, 100orim12d 964 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (((πΉβ€˜π‘›) < (1st β€˜(πΊβ€˜π‘›)) ∨ (2nd β€˜(πΊβ€˜π‘›)) < (πΉβ€˜π‘›)) β†’ ((πΉβ€˜π‘›) < 𝑆 ∨ 𝑆 < (πΉβ€˜π‘›))))
10249, 101mpd 15 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›) < 𝑆 ∨ 𝑆 < (πΉβ€˜π‘›)))
10325, 70lttri2d 11301 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜π‘›) β‰  𝑆 ↔ ((πΉβ€˜π‘›) < 𝑆 ∨ 𝑆 < (πΉβ€˜π‘›))))
104102, 103mpbird 257 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) β‰  𝑆)
105104neneqd 2949 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ Β¬ (πΉβ€˜π‘›) = 𝑆)
106105nrexdv 3147 . . 3 (πœ‘ β†’ Β¬ βˆƒπ‘› ∈ β„• (πΉβ€˜π‘›) = 𝑆)
107 risset 3224 . . . 4 (𝑆 ∈ ran 𝐹 ↔ βˆƒπ‘§ ∈ ran 𝐹 𝑧 = 𝑆)
108 ffn 6673 . . . . 5 (𝐹:β„•βŸΆβ„ β†’ 𝐹 Fn β„•)
109 eqeq1 2741 . . . . . 6 (𝑧 = (πΉβ€˜π‘›) β†’ (𝑧 = 𝑆 ↔ (πΉβ€˜π‘›) = 𝑆))
110109rexrn 7042 . . . . 5 (𝐹 Fn β„• β†’ (βˆƒπ‘§ ∈ ran 𝐹 𝑧 = 𝑆 ↔ βˆƒπ‘› ∈ β„• (πΉβ€˜π‘›) = 𝑆))
1112, 108, 1103syl 18 . . . 4 (πœ‘ β†’ (βˆƒπ‘§ ∈ ran 𝐹 𝑧 = 𝑆 ↔ βˆƒπ‘› ∈ β„• (πΉβ€˜π‘›) = 𝑆))
112107, 111bitrid 283 . . 3 (πœ‘ β†’ (𝑆 ∈ ran 𝐹 ↔ βˆƒπ‘› ∈ β„• (πΉβ€˜π‘›) = 𝑆))
113106, 112mtbird 325 . 2 (πœ‘ β†’ Β¬ 𝑆 ∈ ran 𝐹)
11414, 113eldifd 3926 1 (πœ‘ β†’ 𝑆 ∈ (ℝ βˆ– ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  β¦‹csb 3860   βˆ– cdif 3912   βˆͺ cun 3913   βŠ† wss 3915  βˆ…c0 4287  ifcif 4491  {csn 4591  βŸ¨cop 4597   class class class wbr 5110   Γ— cxp 5636  ran crn 5639   ∘ ccom 5642   Fn wfn 6496  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364  1st c1st 7924  2nd c2nd 7925  supcsup 9383  β„‚cc 11056  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   < clt 11196   ≀ cle 11197   βˆ’ cmin 11392   / cdiv 11819  β„•cn 12160  2c2 12215  β„•0cn0 12420  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  ax-pre-sup 11136
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-sup 9385  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:  ruclem13  16131
  Copyright terms: Public domain W3C validator