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Theorem ruclem12 15198
Description: Lemma for ruc 15200. 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 15197 . . . . 5 (𝜑 → (ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1))
76simp1d 1163 . . . 4 (𝜑 → ran (1st𝐺) ⊆ ℝ)
86simp2d 1164 . . . 4 (𝜑 → ran (1st𝐺) ≠ ∅)
9 1re 10262 . . . . 5 1 ∈ ℝ
106simp3d 1165 . . . . 5 (𝜑 → ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1)
11 brralrspcev 4857 . . . . 5 ((1 ∈ ℝ ∧ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ 1) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛)
129, 10, 11sylancr 576 . . . 4 (𝜑 → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛)
13 suprcl 11206 . . . 4 ((ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛) → sup(ran (1st𝐺), ℝ, < ) ∈ ℝ)
147, 8, 12, 13syl3anc 1480 . . 3 (𝜑 → sup(ran (1st𝐺), ℝ, < ) ∈ ℝ)
151, 14syl5eqel 2857 . 2 (𝜑𝑆 ∈ ℝ)
162adantr 467 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℕ⟶ℝ)
173adantr 467 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
182, 3, 4, 5ruclem6 15192 . . . . . . . . . . 11 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
19 nnm1nn0 11558 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
20 ffvelrn 6517 . . . . . . . . . . 11 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ (𝑛 − 1) ∈ ℕ0) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
2118, 19, 20syl2an 584 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
22 xp1st 7368 . . . . . . . . . 10 ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
2321, 22syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
24 xp2nd 7369 . . . . . . . . . 10 ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
2521, 24syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
262ffvelrnda 6519 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ℝ)
27 eqid 2774 . . . . . . . . 9 (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) = (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
28 eqid 2774 . . . . . . . . 9 (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) = (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
292, 3, 4, 5ruclem8 15194 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 − 1) ∈ ℕ0) → (1st ‘(𝐺‘(𝑛 − 1))) < (2nd ‘(𝐺‘(𝑛 − 1))))
3019, 29sylan2 581 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺‘(𝑛 − 1))) < (2nd ‘(𝐺‘(𝑛 − 1))))
3116, 17, 23, 25, 26, 27, 28, 30ruclem3 15190 . . . . . . . 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)))
322, 3, 4, 5ruclem7 15193 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 − 1) ∈ ℕ0) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
3319, 32sylan2 581 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
34 nncn 11251 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
3534adantl 468 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
36 ax-1cn 10217 . . . . . . . . . . . . . 14 1 ∈ ℂ
37 npcan 10513 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
3835, 36, 37sylancl 575 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((𝑛 − 1) + 1) = 𝑛)
3938fveq2d 6352 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = (𝐺𝑛))
40 1st2nd2 7375 . . . . . . . . . . . . . 14 ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (𝐺‘(𝑛 − 1)) = ⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩)
4121, 40syl 17 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐺‘(𝑛 − 1)) = ⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩)
4238fveq2d 6352 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐹‘((𝑛 − 1) + 1)) = (𝐹𝑛))
4341, 42oveq12d 6830 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))) = (⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
4433, 39, 433eqtr3d 2816 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = (⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))
4544fveq2d 6352 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) = (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))))
4645breq2d 4809 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < (1st ‘(𝐺𝑛)) ↔ (𝐹𝑛) < (1st ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛)))))
4744fveq2d 6352 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) = (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))))
4847breq1d 4807 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) < (𝐹𝑛) ↔ (2nd ‘(⟨(1st ‘(𝐺‘(𝑛 − 1))), (2nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹𝑛))) < (𝐹𝑛)))
4946, 48orbi12d 931 . . . . . . . 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))))
5031, 49mpbird 248 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∨ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)))
517adantr 467 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ran (1st𝐺) ⊆ ℝ)
528adantr 467 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ran (1st𝐺) ≠ ∅)
5312adantr 467 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛)
54 nnnn0 11523 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
55 fvco3 6434 . . . . . . . . . . . . 13 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
5618, 54, 55syl2an 584 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((1st𝐺)‘𝑛) = (1st ‘(𝐺𝑛)))
5718adantr 467 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → 𝐺:ℕ0⟶(ℝ × ℝ))
58 1stcof 7366 . . . . . . . . . . . . . 14 (𝐺:ℕ0⟶(ℝ × ℝ) → (1st𝐺):ℕ0⟶ℝ)
59 ffn 6196 . . . . . . . . . . . . . 14 ((1st𝐺):ℕ0⟶ℝ → (1st𝐺) Fn ℕ0)
6057, 58, 593syl 18 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (1st𝐺) Fn ℕ0)
6154adantl 468 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
62 fnfvelrn 6516 . . . . . . . . . . . . 13 (((1st𝐺) Fn ℕ0𝑛 ∈ ℕ0) → ((1st𝐺)‘𝑛) ∈ ran (1st𝐺))
6360, 61, 62syl2anc 574 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((1st𝐺)‘𝑛) ∈ ran (1st𝐺))
6456, 63eqeltrrd 2854 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ∈ ran (1st𝐺))
65 suprub 11207 . . . . . . . . . . 11 (((ran (1st𝐺) ⊆ ℝ ∧ ran (1st𝐺) ≠ ∅ ∧ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1st𝐺)𝑧𝑛) ∧ (1st ‘(𝐺𝑛)) ∈ ran (1st𝐺)) → (1st ‘(𝐺𝑛)) ≤ sup(ran (1st𝐺), ℝ, < ))
6651, 52, 53, 64, 65syl31anc 1482 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ≤ sup(ran (1st𝐺), ℝ, < ))
6766, 1syl6breqr 4839 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ≤ 𝑆)
68 ffvelrn 6517 . . . . . . . . . . . 12 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ0) → (𝐺𝑛) ∈ (ℝ × ℝ))
6918, 54, 68syl2an 584 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ (ℝ × ℝ))
70 xp1st 7368 . . . . . . . . . . 11 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
7169, 70syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
7215adantr 467 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
73 ltletr 10352 . . . . . . . . . 10 (((𝐹𝑛) ∈ ℝ ∧ (1st ‘(𝐺𝑛)) ∈ ℝ ∧ 𝑆 ∈ ℝ) → (((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∧ (1st ‘(𝐺𝑛)) ≤ 𝑆) → (𝐹𝑛) < 𝑆))
7426, 71, 72, 73syl3anc 1480 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∧ (1st ‘(𝐺𝑛)) ≤ 𝑆) → (𝐹𝑛) < 𝑆))
7567, 74mpan2d 675 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < (1st ‘(𝐺𝑛)) → (𝐹𝑛) < 𝑆))
76 fvco3 6434 . . . . . . . . . . . . . . 15 ((𝐺:ℕ0⟶(ℝ × ℝ) ∧ 𝑘 ∈ ℕ0) → ((1st𝐺)‘𝑘) = (1st ‘(𝐺𝑘)))
7757, 76sylan 570 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((1st𝐺)‘𝑘) = (1st ‘(𝐺𝑘)))
7857ffvelrnda 6519 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (𝐺𝑘) ∈ (ℝ × ℝ))
79 xp1st 7368 . . . . . . . . . . . . . . . 16 ((𝐺𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑘)) ∈ ℝ)
8078, 79syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (1st ‘(𝐺𝑘)) ∈ ℝ)
81 xp2nd 7369 . . . . . . . . . . . . . . . . 17 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
8269, 81syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
8382adantr 467 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
8416adantr 467 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝐹:ℕ⟶ℝ)
8517adantr 467 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
86 simpr 472 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
8761adantr 467 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝑛 ∈ ℕ0)
8884, 85, 4, 5, 86, 87ruclem10 15196 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (1st ‘(𝐺𝑘)) < (2nd ‘(𝐺𝑛)))
8980, 83, 88ltled 10408 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (1st ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑛)))
9077, 89eqbrtrd 4819 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛)))
9190ralrimiva 3118 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ∀𝑘 ∈ ℕ0 ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛)))
92 breq1 4800 . . . . . . . . . . . . . 14 (𝑧 = ((1st𝐺)‘𝑘) → (𝑧 ≤ (2nd ‘(𝐺𝑛)) ↔ ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛))))
9392ralrn 6522 . . . . . . . . . . . . 13 ((1st𝐺) Fn ℕ0 → (∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛)) ↔ ∀𝑘 ∈ ℕ0 ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛))))
9460, 93syl 17 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛)) ↔ ∀𝑘 ∈ ℕ0 ((1st𝐺)‘𝑘) ≤ (2nd ‘(𝐺𝑛))))
9591, 94mpbird 248 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛)))
96 suprleub 11212 . . . . . . . . . . . 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))))
9751, 52, 53, 82, 96syl31anc 1482 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (sup(ran (1st𝐺), ℝ, < ) ≤ (2nd ‘(𝐺𝑛)) ↔ ∀𝑧 ∈ ran (1st𝐺)𝑧 ≤ (2nd ‘(𝐺𝑛))))
9895, 97mpbird 248 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → sup(ran (1st𝐺), ℝ, < ) ≤ (2nd ‘(𝐺𝑛)))
991, 98syl5eqbr 4832 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑆 ≤ (2nd ‘(𝐺𝑛)))
100 lelttr 10351 . . . . . . . . . 10 ((𝑆 ∈ ℝ ∧ (2nd ‘(𝐺𝑛)) ∈ ℝ ∧ (𝐹𝑛) ∈ ℝ) → ((𝑆 ≤ (2nd ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)) → 𝑆 < (𝐹𝑛)))
10172, 82, 26, 100syl3anc 1480 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝑆 ≤ (2nd ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)) → 𝑆 < (𝐹𝑛)))
10299, 101mpand 676 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → ((2nd ‘(𝐺𝑛)) < (𝐹𝑛) → 𝑆 < (𝐹𝑛)))
10375, 102orim12d 976 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (((𝐹𝑛) < (1st ‘(𝐺𝑛)) ∨ (2nd ‘(𝐺𝑛)) < (𝐹𝑛)) → ((𝐹𝑛) < 𝑆𝑆 < (𝐹𝑛))))
10450, 103mpd 15 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) < 𝑆𝑆 < (𝐹𝑛)))
10526, 72lttri2d 10399 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ≠ 𝑆 ↔ ((𝐹𝑛) < 𝑆𝑆 < (𝐹𝑛))))
106104, 105mpbird 248 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ≠ 𝑆)
107106neneqd 2951 . . . 4 ((𝜑𝑛 ∈ ℕ) → ¬ (𝐹𝑛) = 𝑆)
108107nrexdv 3152 . . 3 (𝜑 → ¬ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆)
109 risset 3214 . . . 4 (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆)
110 ffn 6196 . . . . 5 (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
111 eqeq1 2778 . . . . . 6 (𝑧 = (𝐹𝑛) → (𝑧 = 𝑆 ↔ (𝐹𝑛) = 𝑆))
112111rexrn 6521 . . . . 5 (𝐹 Fn ℕ → (∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆 ↔ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆))
1132, 110, 1123syl 18 . . . 4 (𝜑 → (∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆 ↔ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆))
114109, 113syl5bb 273 . . 3 (𝜑 → (𝑆 ∈ ran 𝐹 ↔ ∃𝑛 ∈ ℕ (𝐹𝑛) = 𝑆))
115108, 114mtbird 315 . 2 (𝜑 → ¬ 𝑆 ∈ ran 𝐹)
11615, 115eldifd 3740 1 (𝜑𝑆 ∈ (ℝ ∖ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 383  wo 863   = wceq 1634  wcel 2148  wne 2946  wral 3064  wrex 3065  csb 3688  cdif 3726  cun 3727  wss 3729  c0 4073  ifcif 4235  {csn 4326  cop 4332   class class class wbr 4797   × cxp 5261  ran crn 5264  ccom 5267   Fn wfn 6037  wf 6038  cfv 6042  (class class class)co 6812  cmpt2 6814  1st c1st 7334  2nd c2nd 7335  supcsup 8523  cc 10157  cr 10158  0cc0 10159  1c1 10160   + caddc 10162   < clt 10297  cle 10298  cmin 10489   / cdiv 10907  cn 11243  2c2 11293  0cn0 11516  seqcseq 13030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-cnex 10215  ax-resscn 10216  ax-1cn 10217  ax-icn 10218  ax-addcl 10219  ax-addrcl 10220  ax-mulcl 10221  ax-mulrcl 10222  ax-mulcom 10223  ax-addass 10224  ax-mulass 10225  ax-distr 10226  ax-i2m1 10227  ax-1ne0 10228  ax-1rid 10229  ax-rnegex 10230  ax-rrecex 10231  ax-cnre 10232  ax-pre-lttri 10233  ax-pre-lttrn 10234  ax-pre-ltadd 10235  ax-pre-mulgt0 10236  ax-pre-sup 10237
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-fal 1640  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-om 7234  df-1st 7336  df-2nd 7337  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-er 7917  df-en 8131  df-dom 8132  df-sdom 8133  df-sup 8525  df-pnf 10299  df-mnf 10300  df-xr 10301  df-ltxr 10302  df-le 10303  df-sub 10491  df-neg 10492  df-div 10908  df-nn 11244  df-2 11302  df-n0 11517  df-z 11602  df-uz 11911  df-fz 12556  df-seq 13031
This theorem is referenced by:  ruclem13  15199
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