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Theorem ruclem12 14755
Description: Lemma for ruc 14757. 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 14754 . . . . 5 (𝜑 → (ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1))
76simp1d 1065 . . . 4 (𝜑 → ran (1st𝐺) ⊆ ℝ)
86simp2d 1066 . . . 4 (𝜑 → ran (1st𝐺) ≠ ∅)
9 1re 9895 . . . . 5 1 ∈ ℝ
106simp3d 1067 . . . . 5 (𝜑 → ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1)
11 breq2 4581 . . . . . . 7 (𝑛 = 1 → (𝑧𝑛𝑧 ≤ 1))
1211ralbidv 2968 . . . . . 6 (𝑛 = 1 → (∀𝑧 ∈ ran (1st𝐺)𝑧𝑛 ↔ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1))
1312rspcev 3281 . . . . 5 ((1 ∈ ℝ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛)
149, 10, 13sylancr 693 . . . 4 (𝜑 → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛)
15 suprcl 10832 . . . 4 ((ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛) → sup(ran (1st𝐺), ℝ, < ) ∈ ℝ)
167, 8, 14, 15syl3anc 1317 . . 3 (𝜑 → sup(ran (1st𝐺), ℝ, < ) ∈ ℝ)
171, 16syl5eqel 2691 . 2 (𝜑𝑆 ∈ ℝ)
182adantr 479 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶ℝ)
193adantr 479 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
202, 3, 4, 5ruclem6 14749 . . . . . . . . . . 11 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
21 nnm1nn0 11181 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
22 ffvelrn 6250 . . . . . . . . . . 11 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ (𝑛 − 1) ∈ ℕ0) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
2320, 21, 22syl2an 492 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
24 xp1st 7066 . . . . . . . . . 10 ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
2523, 24syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
26 xp2nd 7067 . . . . . . . . . 10 ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
2723, 26syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
282ffvelrnda 6252 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ℝ)
29 eqid 2609 . . . . . . . . 9 (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) = (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
30 eqid 2609 . . . . . . . . 9 (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) = (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
312, 3, 4, 5ruclem8 14751 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 − 1) ∈ ℕ0) → (1st ‘(𝐺‘(𝑛 − 1))) < (2nd ‘(𝐺‘(𝑛 − 1))))
3221, 31sylan2 489 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺‘(𝑛 − 1))) < (2nd ‘(𝐺‘(𝑛 − 1))))
3318, 19, 25, 27, 28, 29, 30, 32ruclem3 14747 . . . . . . . 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)))
342, 3, 4, 5ruclem7 14750 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 − 1) ∈ ℕ0) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
3521, 34sylan2 489 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
36 nncn 10875 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
3736adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
38 ax-1cn 9850 . . . . . . . . . . . . . 14 1 ∈ ℂ
39 npcan 10141 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
4037, 38, 39sylancl 692 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((𝑛 − 1) + 1) = 𝑛)
4140fveq2d 6092 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = (𝐺𝑛))
42 1st2nd2 7073 . . . . . . . . . . . . . 14 ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (𝐺‘(𝑛 − 1)) = ⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩)
4323, 42syl 17 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐺‘(𝑛 − 1)) = ⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩)
4440fveq2d 6092 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐹‘((𝑛 − 1) + 1)) = (𝐹𝑛))
4543, 44oveq12d 6545 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))) = (⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
4635, 41, 453eqtr3d 2651 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = (⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
4746fveq2d 6092 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))))
4847breq2d 4589 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < (1st ‘(𝐺𝑛)) ↔ (𝐹𝑛) < (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))))
4946fveq2d 6092 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))))
5049breq1d 4587 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) < (𝐹𝑛) ↔ (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) < (𝐹𝑛)))
5148, 50orbi12d 741 . . . . . . . 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))))
5233, 51mpbird 245 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∨ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)))
537adantr 479 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ran (1st𝐺) ⊆ ℝ)
548adantr 479 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ran (1st𝐺) ≠ ∅)
5514adantr 479 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛)
56 nnnn0 11146 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
57 fvco3 6170 . . . . . . . . . . . . 13 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
5820, 56, 57syl2an 492 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
5920adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℕ0⟶(ℝ × ℝ))
60 1stcof 7064 . . . . . . . . . . . . . 14 (𝐺:ℕ0⟶(ℝ × ℝ) → (1st𝐺):ℕ0⟶ℝ)
61 ffn 5944 . . . . . . . . . . . . . 14 ((1st𝐺):ℕ0⟶ℝ → (1st𝐺) Fn ℕ0)
6259, 60, 613syl 18 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (1st𝐺) Fn ℕ0)
6356adantl 480 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
64 fnfvelrn 6249 . . . . . . . . . . . . 13 (((1st𝐺) Fn ℕ0𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) ∈ ran (1st𝐺))
6562, 63, 64syl2anc 690 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((1st𝐺)‘𝑛) ∈ ran (1st𝐺))
6658, 65eqeltrrd 2688 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ∈ ran (1st𝐺))
67 suprub 10833 . . . . . . . . . . 11 (((ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛) ∧ (1st ‘(𝐺𝑛)) ∈ ran (1st𝐺)) → (1st ‘(𝐺𝑛)) ≤ sup(ran (1st𝐺), ℝ, < ))
6853, 54, 55, 66, 67syl31anc 1320 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ≤ sup(ran (1st𝐺), ℝ, < ))
6968, 1syl6breqr 4619 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ≤ 𝑆)
70 ffvelrn 6250 . . . . . . . . . . . 12 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
7120, 56, 70syl2an 492 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ (ℝ × ℝ))
72 xp1st 7066 . . . . . . . . . . 11 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
7371, 72syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
7417adantr 479 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
75 ltletr 9980 . . . . . . . . . 10 (((𝐹𝑛) ∈ ℝ ∧ (1st ‘(𝐺𝑛)) ∈ ℝ ∧ 𝑆 ∈ ℝ) → (((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∧ (1st ‘(𝐺𝑛)) ≤ 𝑆) → (𝐹𝑛) < 𝑆))
7628, 73, 74, 75syl3anc 1317 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∧ (1st ‘(𝐺𝑛)) ≤ 𝑆) → (𝐹𝑛) < 𝑆))
7769, 76mpan2d 705 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < (1st ‘(𝐺𝑛)) → (𝐹𝑛) < 𝑆))
78 fvco3 6170 . . . . . . . . . . . . . . 15 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑘 ∈ ℕ0) → ((1st𝐺)‘𝑘) = (1st ‘(𝐺𝑘)))
7959, 78sylan 486 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((1st𝐺)‘𝑘) = (1st ‘(𝐺𝑘)))
8059ffvelrnda 6252 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (𝐺𝑘) ∈ (ℝ × ℝ))
81 xp1st 7066 . . . . . . . . . . . . . . . 16 ((𝐺𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑘)) ∈ ℝ)
8280, 81syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (1st ‘(𝐺𝑘)) ∈ ℝ)
83 xp2nd 7067 . . . . . . . . . . . . . . . . 17 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
8471, 83syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
8584adantr 479 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
8618adantr 479 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝐹:ℕ⟶ℝ)
8719adantr 479 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
88 simpr 475 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
8963adantr 479 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝑛 ∈ ℕ0)
9086, 87, 4, 5, 88, 89ruclem10 14753 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑛)))
9182, 85, 90ltled 10036 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (1st ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑛)))
9279, 91eqbrtrd 4599 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛)))
9392ralrimiva 2948 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ∀𝑘 ∈ ℕ0 ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛)))
94 breq1 4580 . . . . . . . . . . . . . 14 (𝑧 = ((1st𝐺)‘𝑘) → (𝑧 ≤ (2nd ‘(𝐺𝑛)) ↔ ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛))))
9594ralrn 6255 . . . . . . . . . . . . 13 ((1st𝐺) Fn ℕ0 → (∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛)) ↔ ∀𝑘 ∈ ℕ0 ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛))))
9662, 95syl 17 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛)) ↔ ∀𝑘 ∈ ℕ0 ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛))))
9793, 96mpbird 245 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛)))
98 suprleub 10836 . . . . . . . . . . . 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))))
9953, 54, 55, 84, 98syl31anc 1320 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (sup(ran (1st𝐺), ℝ, < ) ≤ (2nd ‘(𝐺𝑛)) ↔ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛))))
10097, 99mpbird 245 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → sup(ran (1st𝐺), ℝ, < ) ≤ (2nd ‘(𝐺𝑛)))
1011, 100syl5eqbr 4612 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑆 ≤ (2nd ‘(𝐺𝑛)))
102 lelttr 9979 . . . . . . . . . 10 ((𝑆 ∈ ℝ ∧ (2nd ‘(𝐺𝑛)) ∈ ℝ ∧ (𝐹𝑛) ∈ ℝ) → ((𝑆 ≤ (2nd ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)) → 𝑆 < (𝐹𝑛)))
10374, 84, 28, 102syl3anc 1317 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝑆 ≤ (2nd ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)) → 𝑆 < (𝐹𝑛)))
104101, 103mpand 706 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) < (𝐹𝑛) → 𝑆 < (𝐹𝑛)))
10577, 104orim12d 878 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∨ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)) → ((𝐹𝑛) < 𝑆𝑆 < (𝐹𝑛))))
10652, 105mpd 15 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < 𝑆𝑆 < (𝐹𝑛)))
10728, 74lttri2d 10027 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ≠ 𝑆 ↔ ((𝐹𝑛) < 𝑆𝑆 < (𝐹𝑛))))
108106, 107mpbird 245 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ≠ 𝑆)
109108neneqd 2786 . . . 4 ((𝜑𝑛 ∈ ℕ) → ¬ (𝐹𝑛) = 𝑆)
110109nrexdv 2983 . . 3 (𝜑 → ¬ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆)
111 risset 3043 . . . 4 (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆)
112 ffn 5944 . . . . 5 (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
113 eqeq1 2613 . . . . . 6 (𝑧 = (𝐹𝑛) → (𝑧 = 𝑆 ↔ (𝐹𝑛) = 𝑆))
114113rexrn 6254 . . . . 5 (𝐹 Fn ℕ → (∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆 ↔ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆))
1152, 112, 1143syl 18 . . . 4 (𝜑 → (∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆 ↔ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆))
116111, 115syl5bb 270 . . 3 (𝜑 → (𝑆 ∈ ran 𝐹 ↔ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆))
117110, 116mtbird 313 . 2 (𝜑 → ¬ 𝑆 ∈ ran 𝐹)
11817, 117eldifd 3550 1 (𝜑𝑆 ∈ (ℝ ∖ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  csb 3498  cdif 3536  cun 3537  wss 3539  c0 3873  ifcif 4035  {csn 4124  cop 4130   class class class wbr 4577   × cxp 5026  ran crn 5029  ccom 5032   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  cmpt2 6529  1st c1st 7034  2nd c2nd 7035  supcsup 8206  cc 9790  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   < clt 9930  cle 9931  cmin 10117   / cdiv 10533  cn 10867  2c2 10917  0cn0 11139  seqcseq 12618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-sup 8208  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-seq 12619
This theorem is referenced by:  ruclem13  14756
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