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Theorem ruclem12 15582
Description: Lemma for ruc 15584. The supremum of the increasing sequence 1st𝐺 is a real number that is not in the range of 𝐹. (Contributed by Mario Carneiro, 28-May-2014.)
Hypotheses
Ref Expression
ruc.1 (𝜑𝐹:ℕ⟶ℝ)
ruc.2 (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
ruc.4 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
ruc.5 𝐺 = seq0(𝐷, 𝐶)
ruc.6 𝑆 = sup(ran (1st𝐺), ℝ, < )
Assertion
Ref Expression
ruclem12 (𝜑𝑆 ∈ (ℝ ∖ ran 𝐹))
Distinct variable groups:   𝑥,𝑚,𝑦,𝐹   𝑚,𝐺,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚)   𝐶(𝑥,𝑦,𝑚)   𝐷(𝑥,𝑦,𝑚)   𝑆(𝑥,𝑦,𝑚)

Proof of Theorem ruclem12
Dummy variables 𝑧 𝑛 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ruc.6 . . 3 𝑆 = sup(ran (1st𝐺), ℝ, < )
2 ruc.1 . . . . . 6 (𝜑𝐹:ℕ⟶ℝ)
3 ruc.2 . . . . . 6 (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
4 ruc.4 . . . . . 6 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
5 ruc.5 . . . . . 6 𝐺 = seq0(𝐷, 𝐶)
62, 3, 4, 5ruclem11 15581 . . . . 5 (𝜑 → (ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1))
76simp1d 1134 . . . 4 (𝜑 → ran (1st𝐺) ⊆ ℝ)
86simp2d 1135 . . . 4 (𝜑 → ran (1st𝐺) ≠ ∅)
9 1re 10629 . . . . 5 1 ∈ ℝ
106simp3d 1136 . . . . 5 (𝜑 → ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1)
11 brralrspcev 5117 . . . . 5 ((1 ∈ ℝ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛)
129, 10, 11sylancr 587 . . . 4 (𝜑 → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛)
137, 8, 12suprcld 11592 . . 3 (𝜑 → sup(ran (1st𝐺), ℝ, < ) ∈ ℝ)
141, 13eqeltrid 2914 . 2 (𝜑𝑆 ∈ ℝ)
152adantr 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶ℝ)
163adantr 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
172, 3, 4, 5ruclem6 15576 . . . . . . . . . . 11 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
18 nnm1nn0 11926 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
19 ffvelrn 6841 . . . . . . . . . . 11 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ (𝑛 − 1) ∈ ℕ0) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
2017, 18, 19syl2an 595 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
21 xp1st 7710 . . . . . . . . . 10 ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
2220, 21syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
23 xp2nd 7711 . . . . . . . . . 10 ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
2420, 23syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
252ffvelrnda 6843 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ℝ)
26 eqid 2818 . . . . . . . . 9 (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) = (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
27 eqid 2818 . . . . . . . . 9 (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) = (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
282, 3, 4, 5ruclem8 15578 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 − 1) ∈ ℕ0) → (1st ‘(𝐺‘(𝑛 − 1))) < (2nd ‘(𝐺‘(𝑛 − 1))))
2918, 28sylan2 592 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺‘(𝑛 − 1))) < (2nd ‘(𝐺‘(𝑛 − 1))))
3015, 16, 22, 24, 25, 26, 27, 29ruclem3 15574 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) ∨ (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) < (𝐹𝑛)))
312, 3, 4, 5ruclem7 15577 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 − 1) ∈ ℕ0) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
3218, 31sylan2 592 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
33 nncn 11634 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
3433adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
35 ax-1cn 10583 . . . . . . . . . . . . . 14 1 ∈ ℂ
36 npcan 10883 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
3734, 35, 36sylancl 586 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((𝑛 − 1) + 1) = 𝑛)
3837fveq2d 6667 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = (𝐺𝑛))
39 1st2nd2 7717 . . . . . . . . . . . . . 14 ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (𝐺‘(𝑛 − 1)) = ⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩)
4020, 39syl 17 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐺‘(𝑛 − 1)) = ⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩)
4137fveq2d 6667 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐹‘((𝑛 − 1) + 1)) = (𝐹𝑛))
4240, 41oveq12d 7163 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))) = (⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
4332, 38, 423eqtr3d 2861 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = (⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
4443fveq2d 6667 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))))
4544breq2d 5069 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < (1st ‘(𝐺𝑛)) ↔ (𝐹𝑛) < (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))))
4643fveq2d 6667 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))))
4746breq1d 5067 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) < (𝐹𝑛) ↔ (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) < (𝐹𝑛)))
4845, 47orbi12d 912 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∨ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)) ↔ ((𝐹𝑛) < (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) ∨ (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) < (𝐹𝑛))))
4930, 48mpbird 258 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∨ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)))
507adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ran (1st𝐺) ⊆ ℝ)
518adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ran (1st𝐺) ≠ ∅)
5212adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛)
53 nnnn0 11892 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
54 fvco3 6753 . . . . . . . . . . . . 13 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
5517, 53, 54syl2an 595 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
5617adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℕ0⟶(ℝ × ℝ))
57 1stcof 7708 . . . . . . . . . . . . . 14 (𝐺:ℕ0⟶(ℝ × ℝ) → (1st𝐺):ℕ0⟶ℝ)
58 ffn 6507 . . . . . . . . . . . . . 14 ((1st𝐺):ℕ0⟶ℝ → (1st𝐺) Fn ℕ0)
5956, 57, 583syl 18 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (1st𝐺) Fn ℕ0)
6053adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
61 fnfvelrn 6840 . . . . . . . . . . . . 13 (((1st𝐺) Fn ℕ0𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) ∈ ran (1st𝐺))
6259, 60, 61syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((1st𝐺)‘𝑛) ∈ ran (1st𝐺))
6355, 62eqeltrrd 2911 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ∈ ran (1st𝐺))
6450, 51, 52, 63suprubd 11591 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ≤ sup(ran (1st𝐺), ℝ, < ))
6564, 1breqtrrdi 5099 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ≤ 𝑆)
66 ffvelrn 6841 . . . . . . . . . . . 12 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
6717, 53, 66syl2an 595 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ (ℝ × ℝ))
68 xp1st 7710 . . . . . . . . . . 11 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
6967, 68syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
7014adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
71 ltletr 10720 . . . . . . . . . 10 (((𝐹𝑛) ∈ ℝ ∧ (1st ‘(𝐺𝑛)) ∈ ℝ ∧ 𝑆 ∈ ℝ) → (((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∧ (1st ‘(𝐺𝑛)) ≤ 𝑆) → (𝐹𝑛) < 𝑆))
7225, 69, 70, 71syl3anc 1363 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∧ (1st ‘(𝐺𝑛)) ≤ 𝑆) → (𝐹𝑛) < 𝑆))
7365, 72mpan2d 690 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < (1st ‘(𝐺𝑛)) → (𝐹𝑛) < 𝑆))
74 fvco3 6753 . . . . . . . . . . . . . . 15 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑘 ∈ ℕ0) → ((1st𝐺)‘𝑘) = (1st ‘(𝐺𝑘)))
7556, 74sylan 580 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((1st𝐺)‘𝑘) = (1st ‘(𝐺𝑘)))
7656ffvelrnda 6843 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (𝐺𝑘) ∈ (ℝ × ℝ))
77 xp1st 7710 . . . . . . . . . . . . . . . 16 ((𝐺𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑘)) ∈ ℝ)
7876, 77syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (1st ‘(𝐺𝑘)) ∈ ℝ)
79 xp2nd 7711 . . . . . . . . . . . . . . . . 17 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
8067, 79syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
8180adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
8215adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝐹:ℕ⟶ℝ)
8316adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
84 simpr 485 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
8560adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝑛 ∈ ℕ0)
8682, 83, 4, 5, 84, 85ruclem10 15580 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑛)))
8778, 81, 86ltled 10776 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (1st ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑛)))
8875, 87eqbrtrd 5079 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛)))
8988ralrimiva 3179 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ∀𝑘 ∈ ℕ0 ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛)))
90 breq1 5060 . . . . . . . . . . . . . 14 (𝑧 = ((1st𝐺)‘𝑘) → (𝑧 ≤ (2nd ‘(𝐺𝑛)) ↔ ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛))))
9190ralrn 6846 . . . . . . . . . . . . 13 ((1st𝐺) Fn ℕ0 → (∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛)) ↔ ∀𝑘 ∈ ℕ0 ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛))))
9259, 91syl 17 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛)) ↔ ∀𝑘 ∈ ℕ0 ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛))))
9389, 92mpbird 258 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛)))
94 suprleub 11595 . . . . . . . . . . . 12 (((ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛) ∧ (2nd ‘(𝐺𝑛)) ∈ ℝ) → (sup(ran (1st𝐺), ℝ, < ) ≤ (2nd ‘(𝐺𝑛)) ↔ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛))))
9550, 51, 52, 80, 94syl31anc 1365 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (sup(ran (1st𝐺), ℝ, < ) ≤ (2nd ‘(𝐺𝑛)) ↔ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛))))
9693, 95mpbird 258 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → sup(ran (1st𝐺), ℝ, < ) ≤ (2nd ‘(𝐺𝑛)))
971, 96eqbrtrid 5092 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑆 ≤ (2nd ‘(𝐺𝑛)))
98 lelttr 10719 . . . . . . . . . 10 ((𝑆 ∈ ℝ ∧ (2nd ‘(𝐺𝑛)) ∈ ℝ ∧ (𝐹𝑛) ∈ ℝ) → ((𝑆 ≤ (2nd ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)) → 𝑆 < (𝐹𝑛)))
9970, 80, 25, 98syl3anc 1363 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝑆 ≤ (2nd ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)) → 𝑆 < (𝐹𝑛)))
10097, 99mpand 691 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) < (𝐹𝑛) → 𝑆 < (𝐹𝑛)))
10173, 100orim12d 958 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∨ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)) → ((𝐹𝑛) < 𝑆𝑆 < (𝐹𝑛))))
10249, 101mpd 15 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < 𝑆𝑆 < (𝐹𝑛)))
10325, 70lttri2d 10767 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ≠ 𝑆 ↔ ((𝐹𝑛) < 𝑆𝑆 < (𝐹𝑛))))
104102, 103mpbird 258 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ≠ 𝑆)
105104neneqd 3018 . . . 4 ((𝜑𝑛 ∈ ℕ) → ¬ (𝐹𝑛) = 𝑆)
106105nrexdv 3267 . . 3 (𝜑 → ¬ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆)
107 risset 3264 . . . 4 (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆)
108 ffn 6507 . . . . 5 (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
109 eqeq1 2822 . . . . . 6 (𝑧 = (𝐹𝑛) → (𝑧 = 𝑆 ↔ (𝐹𝑛) = 𝑆))
110109rexrn 6845 . . . . 5 (𝐹 Fn ℕ → (∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆 ↔ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆))
1112, 108, 1103syl 18 . . . 4 (𝜑 → (∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆 ↔ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆))
112107, 111syl5bb 284 . . 3 (𝜑 → (𝑆 ∈ ran 𝐹 ↔ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆))
113106, 112mtbird 326 . 2 (𝜑 → ¬ 𝑆 ∈ ran 𝐹)
11414, 113eldifd 3944 1 (𝜑𝑆 ∈ (ℝ ∖ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  csb 3880  cdif 3930  cun 3931  wss 3933  c0 4288  ifcif 4463  {csn 4557  cop 4563   class class class wbr 5057   × cxp 5546  ran crn 5549  ccom 5552   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  supcsup 8892  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  cle 10664  cmin 10858   / cdiv 11285  cn 11626  2c2 11680  0cn0 11885  seqcseq 13357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-seq 13358
This theorem is referenced by:  ruclem13  15583
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