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

Theorem ruclem12 16184
Description: Lemma for ruc 16186. 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 16183 . . . . 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 11214 . . . . 5 1 ∈ ℝ
106simp3d 1145 . . . . 5 (πœ‘ β†’ βˆ€π‘§ ∈ ran (1st ∘ 𝐺)𝑧 ≀ 1)
11 brralrspcev 5209 . . . . 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 12177 . . 3 (πœ‘ β†’ sup(ran (1st ∘ 𝐺), ℝ, < ) ∈ ℝ)
141, 13eqeltrid 2838 . 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 16178 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺:β„•0⟢(ℝ Γ— ℝ))
18 nnm1nn0 12513 . . . . . . . . . . 11 (𝑛 ∈ β„• β†’ (𝑛 βˆ’ 1) ∈ β„•0)
19 ffvelcdm 7084 . . . . . . . . . . 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 8007 . . . . . . . . . 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 8008 . . . . . . . . . 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 7087 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) ∈ ℝ)
26 eqid 2733 . . . . . . . . 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 2733 . . . . . . . . 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 16180 . . . . . . . . . 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 16176 . . . . . . . 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 16179 . . . . . . . . . . . . 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 12220 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„• β†’ 𝑛 ∈ β„‚)
3433adantl 483 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„‚)
35 ax-1cn 11168 . . . . . . . . . . . . . 14 1 ∈ β„‚
36 npcan 11469 . . . . . . . . . . . . . 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 6896 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜((𝑛 βˆ’ 1) + 1)) = (πΊβ€˜π‘›))
39 1st2nd2 8014 . . . . . . . . . . . . . 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 6896 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜((𝑛 βˆ’ 1) + 1)) = (πΉβ€˜π‘›))
4240, 41oveq12d 7427 . . . . . . . . . . . 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 2781 . . . . . . . . . . 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 6896 . . . . . . . . . 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 5161 . . . . . . . . 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 6896 . . . . . . . . . 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 5159 . . . . . . . . 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 12479 . . . . . . . . . . . . 13 (𝑛 ∈ β„• β†’ 𝑛 ∈ β„•0)
54 fvco3 6991 . . . . . . . . . . . . 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 8005 . . . . . . . . . . . . . 14 (𝐺:β„•0⟢(ℝ Γ— ℝ) β†’ (1st ∘ 𝐺):β„•0βŸΆβ„)
58 ffn 6718 . . . . . . . . . . . . . 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 7083 . . . . . . . . . . . . 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 2835 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜π‘›)) ∈ ran (1st ∘ 𝐺))
6450, 51, 52, 63suprubd 12176 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜π‘›)) ≀ sup(ran (1st ∘ 𝐺), ℝ, < ))
6564, 1breqtrrdi 5191 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (1st β€˜(πΊβ€˜π‘›)) ≀ 𝑆)
66 ffvelcdm 7084 . . . . . . . . . . . 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 8007 . . . . . . . . . . 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 11306 . . . . . . . . . 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 6991 . . . . . . . . . . . . . . 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 7087 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ (πΊβ€˜π‘˜) ∈ (ℝ Γ— ℝ))
77 xp1st 8007 . . . . . . . . . . . . . . . 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 8008 . . . . . . . . . . . . . . . . 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 16182 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘˜)) < (2nd β€˜(πΊβ€˜π‘›)))
8778, 81, 86ltled 11362 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ (1st β€˜(πΊβ€˜π‘˜)) ≀ (2nd β€˜(πΊβ€˜π‘›)))
8875, 87eqbrtrd 5171 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑛 ∈ β„•) ∧ π‘˜ ∈ β„•0) β†’ ((1st ∘ 𝐺)β€˜π‘˜) ≀ (2nd β€˜(πΊβ€˜π‘›)))
8988ralrimiva 3147 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ βˆ€π‘˜ ∈ β„•0 ((1st ∘ 𝐺)β€˜π‘˜) ≀ (2nd β€˜(πΊβ€˜π‘›)))
90 breq1 5152 . . . . . . . . . . . . . 14 (𝑧 = ((1st ∘ 𝐺)β€˜π‘˜) β†’ (𝑧 ≀ (2nd β€˜(πΊβ€˜π‘›)) ↔ ((1st ∘ 𝐺)β€˜π‘˜) ≀ (2nd β€˜(πΊβ€˜π‘›))))
9190ralrn 7090 . . . . . . . . . . . . 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 12180 . . . . . . . . . . . 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 5184 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑆 ≀ (2nd β€˜(πΊβ€˜π‘›)))
98 lelttr 11304 . . . . . . . . . 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 11353 . . . . . 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 2946 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ Β¬ (πΉβ€˜π‘›) = 𝑆)
106105nrexdv 3150 . . 3 (πœ‘ β†’ Β¬ βˆƒπ‘› ∈ β„• (πΉβ€˜π‘›) = 𝑆)
107 risset 3231 . . . 4 (𝑆 ∈ ran 𝐹 ↔ βˆƒπ‘§ ∈ ran 𝐹 𝑧 = 𝑆)
108 ffn 6718 . . . . 5 (𝐹:β„•βŸΆβ„ β†’ 𝐹 Fn β„•)
109 eqeq1 2737 . . . . . 6 (𝑧 = (πΉβ€˜π‘›) β†’ (𝑧 = 𝑆 ↔ (πΉβ€˜π‘›) = 𝑆))
110109rexrn 7089 . . . . 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 3960 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 2941  βˆ€wral 3062  βˆƒwrex 3071  β¦‹csb 3894   βˆ– cdif 3946   βˆͺ cun 3947   βŠ† wss 3949  βˆ…c0 4323  ifcif 4529  {csn 4629  βŸ¨cop 4635   class class class wbr 5149   Γ— cxp 5675  ran crn 5678   ∘ ccom 5681   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1st c1st 7973  2nd c2nd 7974  supcsup 9435  β„‚cc 11108  β„cr 11109  0cc0 11110  1c1 11111   + caddc 11113   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444   / cdiv 11871  β„•cn 12212  2c2 12267  β„•0cn0 12472  seqcseq 13966
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 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-seq 13967
This theorem is referenced by:  ruclem13  16185
  Copyright terms: Public domain W3C validator