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Theorem ruclem12 16123
Description: Lemma for ruc 16125. 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 16122 . . . . 5 (𝜑 → (ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1))
76simp1d 1142 . . . 4 (𝜑 → ran (1st𝐺) ⊆ ℝ)
86simp2d 1143 . . . 4 (𝜑 → ran (1st𝐺) ≠ ∅)
9 1re 11155 . . . . 5 1 ∈ ℝ
106simp3d 1144 . . . . 5 (𝜑 → ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1)
11 brralrspcev 5165 . . . . 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 12118 . . 3 (𝜑 → sup(ran (1st𝐺), ℝ, < ) ∈ ℝ)
141, 13eqeltrid 2842 . 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 16117 . . . . . . . . . . 11 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
18 nnm1nn0 12454 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
19 ffvelcdm 7032 . . . . . . . . . . 11 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ (𝑛 − 1) ∈ ℕ0) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
2017, 18, 19syl2an 596 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
21 xp1st 7953 . . . . . . . . . 10 ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
2220, 21syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
23 xp2nd 7954 . . . . . . . . . 10 ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
2420, 23syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
252ffvelcdmda 7035 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ℝ)
26 eqid 2736 . . . . . . . . 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 2736 . . . . . . . . 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 16119 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 − 1) ∈ ℕ0) → (1st ‘(𝐺‘(𝑛 − 1))) < (2nd ‘(𝐺‘(𝑛 − 1))))
2918, 28sylan2 593 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺‘(𝑛 − 1))) < (2nd ‘(𝐺‘(𝑛 − 1))))
3015, 16, 22, 24, 25, 26, 27, 29ruclem3 16115 . . . . . . . 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 16118 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 − 1) ∈ ℕ0) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
3218, 31sylan2 593 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
33 nncn 12161 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
3433adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
35 ax-1cn 11109 . . . . . . . . . . . . . 14 1 ∈ ℂ
36 npcan 11410 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
3734, 35, 36sylancl 586 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((𝑛 − 1) + 1) = 𝑛)
3837fveq2d 6846 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = (𝐺𝑛))
39 1st2nd2 7960 . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐹‘((𝑛 − 1) + 1)) = (𝐹𝑛))
4240, 41oveq12d 7375 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))) = (⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
4332, 38, 423eqtr3d 2784 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = (⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
4443fveq2d 6846 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))))
4544breq2d 5117 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < (1st ‘(𝐺𝑛)) ↔ (𝐹𝑛) < (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))))
4643fveq2d 6846 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))))
4746breq1d 5115 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) < (𝐹𝑛) ↔ (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) < (𝐹𝑛)))
4845, 47orbi12d 917 . . . . . . . 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 256 . . . . . . 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 12420 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
54 fvco3 6940 . . . . . . . . . . . . 13 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
5517, 53, 54syl2an 596 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
5617adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℕ0⟶(ℝ × ℝ))
57 1stcof 7951 . . . . . . . . . . . . . 14 (𝐺:ℕ0⟶(ℝ × ℝ) → (1st𝐺):ℕ0⟶ℝ)
58 ffn 6668 . . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . 13 (((1st𝐺) Fn ℕ0𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) ∈ ran (1st𝐺))
6259, 60, 61syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((1st𝐺)‘𝑛) ∈ ran (1st𝐺))
6355, 62eqeltrrd 2839 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ∈ ran (1st𝐺))
6450, 51, 52, 63suprubd 12117 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ≤ sup(ran (1st𝐺), ℝ, < ))
6564, 1breqtrrdi 5147 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ≤ 𝑆)
66 ffvelcdm 7032 . . . . . . . . . . . 12 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
6717, 53, 66syl2an 596 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ (ℝ × ℝ))
68 xp1st 7953 . . . . . . . . . . 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 11247 . . . . . . . . . 10 (((𝐹𝑛) ∈ ℝ ∧ (1st ‘(𝐺𝑛)) ∈ ℝ ∧ 𝑆 ∈ ℝ) → (((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∧ (1st ‘(𝐺𝑛)) ≤ 𝑆) → (𝐹𝑛) < 𝑆))
7225, 69, 70, 71syl3anc 1371 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∧ (1st ‘(𝐺𝑛)) ≤ 𝑆) → (𝐹𝑛) < 𝑆))
7365, 72mpan2d 692 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < (1st ‘(𝐺𝑛)) → (𝐹𝑛) < 𝑆))
74 fvco3 6940 . . . . . . . . . . . . . . 15 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑘 ∈ ℕ0) → ((1st𝐺)‘𝑘) = (1st ‘(𝐺𝑘)))
7556, 74sylan 580 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((1st𝐺)‘𝑘) = (1st ‘(𝐺𝑘)))
7656ffvelcdmda 7035 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (𝐺𝑘) ∈ (ℝ × ℝ))
77 xp1st 7953 . . . . . . . . . . . . . . . 16 ((𝐺𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑘)) ∈ ℝ)
7876, 77syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (1st ‘(𝐺𝑘)) ∈ ℝ)
79 xp2nd 7954 . . . . . . . . . . . . . . . . 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 16121 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑛)))
8778, 81, 86ltled 11303 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (1st ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑛)))
8875, 87eqbrtrd 5127 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛)))
8988ralrimiva 3143 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ∀𝑘 ∈ ℕ0 ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛)))
90 breq1 5108 . . . . . . . . . . . . . 14 (𝑧 = ((1st𝐺)‘𝑘) → (𝑧 ≤ (2nd ‘(𝐺𝑛)) ↔ ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛))))
9190ralrn 7038 . . . . . . . . . . . . 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 256 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛)))
94 suprleub 12121 . . . . . . . . . . . 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 1373 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (sup(ran (1st𝐺), ℝ, < ) ≤ (2nd ‘(𝐺𝑛)) ↔ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛))))
9693, 95mpbird 256 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → sup(ran (1st𝐺), ℝ, < ) ≤ (2nd ‘(𝐺𝑛)))
971, 96eqbrtrid 5140 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑆 ≤ (2nd ‘(𝐺𝑛)))
98 lelttr 11245 . . . . . . . . . 10 ((𝑆 ∈ ℝ ∧ (2nd ‘(𝐺𝑛)) ∈ ℝ ∧ (𝐹𝑛) ∈ ℝ) → ((𝑆 ≤ (2nd ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)) → 𝑆 < (𝐹𝑛)))
9970, 80, 25, 98syl3anc 1371 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝑆 ≤ (2nd ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)) → 𝑆 < (𝐹𝑛)))
10097, 99mpand 693 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) < (𝐹𝑛) → 𝑆 < (𝐹𝑛)))
10173, 100orim12d 963 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∨ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)) → ((𝐹𝑛) < 𝑆𝑆 < (𝐹𝑛))))
10249, 101mpd 15 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < 𝑆𝑆 < (𝐹𝑛)))
10325, 70lttri2d 11294 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ≠ 𝑆 ↔ ((𝐹𝑛) < 𝑆𝑆 < (𝐹𝑛))))
104102, 103mpbird 256 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ≠ 𝑆)
105104neneqd 2948 . . . 4 ((𝜑𝑛 ∈ ℕ) → ¬ (𝐹𝑛) = 𝑆)
106105nrexdv 3146 . . 3 (𝜑 → ¬ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆)
107 risset 3221 . . . 4 (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆)
108 ffn 6668 . . . . 5 (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
109 eqeq1 2740 . . . . . 6 (𝑧 = (𝐹𝑛) → (𝑧 = 𝑆 ↔ (𝐹𝑛) = 𝑆))
110109rexrn 7037 . . . . 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, 111bitrid 282 . . 3 (𝜑 → (𝑆 ∈ ran 𝐹 ↔ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆))
113106, 112mtbird 324 . 2 (𝜑 → ¬ 𝑆 ∈ ran 𝐹)
11414, 113eldifd 3921 1 (𝜑𝑆 ∈ (ℝ ∖ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  csb 3855  cdif 3907  cun 3908  wss 3910  c0 4282  ifcif 4486  {csn 4586  cop 4592   class class class wbr 5105   × cxp 5631  ran crn 5634  ccom 5637   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  supcsup 9376  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  2c2 12208  0cn0 12413  seqcseq 13906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907
This theorem is referenced by:  ruclem13  16124
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