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Theorem ruclem12 16421

Description: Lemma for ruc 16423. The supremum of the increasing sequence 1^st ∘ 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	⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
ruc.2	⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1^st ‘𝑥) + (2^nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1^st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2^nd ‘𝑥)) / 2), (2^nd ‘𝑥)⟩)))
ruc.4	⊢ 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
ruc.5	⊢ 𝐺 = seq0(𝐷, 𝐶)
ruc.6	⊢ 𝑆 = sup(ran (1^st ∘ 𝐺), ℝ, < )

**Assertion**
Ref	Expression
ruclem12	⊢ (𝜑 → 𝑆 ∈ (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.

Step	Hyp	Ref	Expression
1		ruc.6	. . 3 ⊢ 𝑆 = sup(ran (1^st ∘ 𝐺), ℝ, < )
2		ruc.1	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
3		ruc.2	. . . . . 6 ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1^st ‘𝑥) + (2^nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1^st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2^nd ‘𝑥)) / 2), (2^nd ‘𝑥)⟩)))
4		ruc.4	. . . . . 6 ⊢ 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
5		ruc.5	. . . . . 6 ⊢ 𝐺 = seq0(𝐷, 𝐶)
6	2, 3, 4, 5	ruclem11 16420	. . . . 5 ⊢ (𝜑 → (ran (1^st ∘ 𝐺) ⊆ ℝ ∧ ran (1^st ∘ 𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1))
7	6	simp1d 1160	. . . 4 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ⊆ ℝ)
8	6	simp2d 1161	. . . 4 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ≠ ∅)
9		1re 11318	. . . . 5 ⊢ 1 ∈ ℝ
10	6	simp3d 1162	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1)
11		brralrspcev 5167	. . . . 5 ⊢ ((1 ∈ ℝ ∧ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 𝑛)
12	9, 10, 11	sylancr 599	. . . 4 ⊢ (𝜑 → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 𝑛)
13	7, 8, 12	suprcld 12290	. . 3 ⊢ (𝜑 → sup(ran (1^st ∘ 𝐺), ℝ, < ) ∈ ℝ)
14	1, 13	eqeltrid 2867	. 2 ⊢ (𝜑 → 𝑆 ∈ ℝ)
15	2	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶ℝ)
16	3	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1^st ‘𝑥) + (2^nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1^st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2^nd ‘𝑥)) / 2), (2^nd ‘𝑥)⟩)))
17	2, 3, 4, 5	ruclem6 16415	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
18		nnm1nn0 12657	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
19		ffvelcdm 7089	. . . . . . . . . . 11 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ (𝑛 − 1) ∈ ℕ₀) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
20	17, 18, 19	syl2an 608	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
21		xp1st 8040	. . . . . . . . . 10 ⊢ ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
22	20, 21	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
23		xp2nd 8041	. . . . . . . . . 10 ⊢ ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (2^nd ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
24	20, 23	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
25	2	ffvelcdmda 7092	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℝ)
26		eqid 2763	. . . . . . . . 9 ⊢ (1^st ‘(⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛))) = (1^st ‘(⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛)))
27		eqid 2763	. . . . . . . . 9 ⊢ (2^nd ‘(⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛))) = (2^nd ‘(⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛)))
28	2, 3, 4, 5	ruclem8 16417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 − 1) ∈ ℕ₀) → (1^st ‘(𝐺‘(𝑛 − 1))) < (2^nd ‘(𝐺‘(𝑛 − 1))))
29	18, 28	sylan2 605	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘(𝑛 − 1))) < (2^nd ‘(𝐺‘(𝑛 − 1))))
30	15, 16, 22, 24, 25, 26, 27, 29	ruclem3 16413	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) < (1^st ‘(⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛))) ∨ (2^nd ‘(⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛))) < (𝐹‘𝑛)))
31	2, 3, 4, 5	ruclem7 16416	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 − 1) ∈ ℕ₀) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
32	18, 31	sylan2 605	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
33		nncn 12353	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
34	33	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
35		ax-1cn 11268	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
36		npcan 11576	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
37	34, 35, 36	sylancl 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 − 1) + 1) = 𝑛)
38	37	fveq2d 6897	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = (𝐺‘𝑛))
39		1st2nd2 8047	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (𝐺‘(𝑛 − 1)) = ⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩)
40	20, 39	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘(𝑛 − 1)) = ⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩)
41	37	fveq2d 6897	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘((𝑛 − 1) + 1)) = (𝐹‘𝑛))
42	40, 41	oveq12d 7445	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))) = (⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛)))
43	32, 38, 42	3eqtr3d 2806	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = (⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛)))
44	43	fveq2d 6897	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) = (1^st ‘(⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛))))
45	44	breq2d 5117	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) ↔ (𝐹‘𝑛) < (1^st ‘(⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛)))))
46	43	fveq2d 6897	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝐺‘𝑛)) = (2^nd ‘(⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛))))
47	46	breq1d 5115	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛) ↔ (2^nd ‘(⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛))) < (𝐹‘𝑛)))
48	45, 47	orbi12d 932	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) ∨ (2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛)) ↔ ((𝐹‘𝑛) < (1^st ‘(⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛))) ∨ (2^nd ‘(⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛))) < (𝐹‘𝑛))))
49	30, 48	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) ∨ (2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛)))
50	7	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (1^st ∘ 𝐺) ⊆ ℝ)
51	8	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (1^st ∘ 𝐺) ≠ ∅)
52	12	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 𝑛)
53		nnnn0 12623	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
54		fvco3 6993	. . . . . . . . . . . . 13 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
55	17, 53, 54	syl2an 608	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
56	17	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:ℕ₀⟶(ℝ × ℝ))
57		1stcof 8038	. . . . . . . . . . . . . 14 ⊢ (𝐺:ℕ₀⟶(ℝ × ℝ) → (1^st ∘ 𝐺):ℕ₀⟶ℝ)
58		ffn 6717	. . . . . . . . . . . . . 14 ⊢ ((1^st ∘ 𝐺):ℕ₀⟶ℝ → (1^st ∘ 𝐺) Fn ℕ₀)
59	56, 57, 58	3syl 19	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ∘ 𝐺) Fn ℕ₀)
60	53	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
61		fnfvelrn 7088	. . . . . . . . . . . . 13 ⊢ (((1^st ∘ 𝐺) Fn ℕ₀ ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) ∈ ran (1^st ∘ 𝐺))
62	59, 60, 61	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1^st ∘ 𝐺)‘𝑛) ∈ ran (1^st ∘ 𝐺))
63	55, 62	eqeltrrd 2864	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ∈ ran (1^st ∘ 𝐺))
64	50, 51, 52, 63	suprubd 12289	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ≤ sup(ran (1^st ∘ 𝐺), ℝ, < ))
65	64, 1	breqtrrdi 5149	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ≤ 𝑆)
66		ffvelcdm 7089	. . . . . . . . . . . 12 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
67	17, 53, 66	syl2an 608	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
68		xp1st 8040	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
69	67, 68	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
70	14	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
71		ltletr 11412	. . . . . . . . . 10 ⊢ (((𝐹‘𝑛) ∈ ℝ ∧ (1^st ‘(𝐺‘𝑛)) ∈ ℝ ∧ 𝑆 ∈ ℝ) → (((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) ∧ (1^st ‘(𝐺‘𝑛)) ≤ 𝑆) → (𝐹‘𝑛) < 𝑆))
72	25, 69, 70, 71	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) ∧ (1^st ‘(𝐺‘𝑛)) ≤ 𝑆) → (𝐹‘𝑛) < 𝑆))
73	65, 72	mpan2d 707	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) → (𝐹‘𝑛) < 𝑆))
74		fvco3 6993	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑘 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑘) = (1^st ‘(𝐺‘𝑘)))
75	56, 74	sylan 592	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑘) = (1^st ‘(𝐺‘𝑘)))
76	56	ffvelcdmda 7092	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) ∈ (ℝ × ℝ))
77		xp1st 8040	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑘) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘𝑘)) ∈ ℝ)
78	76, 77	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑘)) ∈ ℝ)
79		xp2nd 8041	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
80	67, 79	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
81	80	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
82	15	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝐹:ℕ⟶ℝ)
83	16	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ ⦋(((1^st ‘𝑥) + (2^nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, ⟨(1^st ‘𝑥), 𝑚⟩, ⟨((𝑚 + (2^nd ‘𝑥)) / 2), (2^nd ‘𝑥)⟩)))
84		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
85	60	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
86	82, 83, 4, 5, 84, 85	ruclem10 16419	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑛)))
87	78, 81, 86	ltled 11468	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑛)))
88	75, 87	eqbrtrd 5129	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛)))
89	88	ralrimiva 3157	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛)))
90		breq1 5108	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((1^st ∘ 𝐺)‘𝑘) → (𝑧 ≤ (2^nd ‘(𝐺‘𝑛)) ↔ ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛))))
91	90	ralrn 7096	. . . . . . . . . . . . 13 ⊢ ((1^st ∘ 𝐺) Fn ℕ₀ → (∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ (2^nd ‘(𝐺‘𝑛)) ↔ ∀𝑘 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛))))
92	59, 91	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ (2^nd ‘(𝐺‘𝑛)) ↔ ∀𝑘 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛))))
93	89, 92	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ (2^nd ‘(𝐺‘𝑛)))
94		suprleub 12293	. . . . . . . . . . . 12 ⊢ (((ran (1^st ∘ 𝐺) ⊆ ℝ ∧ ran (1^st ∘ 𝐺) ≠ ∅ ∧ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 𝑛) ∧ (2^nd ‘(𝐺‘𝑛)) ∈ ℝ) → (sup(ran (1^st ∘ 𝐺), ℝ, < ) ≤ (2^nd ‘(𝐺‘𝑛)) ↔ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ (2^nd ‘(𝐺‘𝑛))))
95	50, 51, 52, 80, 94	syl31anc 1400	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (sup(ran (1^st ∘ 𝐺), ℝ, < ) ≤ (2^nd ‘(𝐺‘𝑛)) ↔ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ (2^nd ‘(𝐺‘𝑛))))
96	93, 95	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → sup(ran (1^st ∘ 𝐺), ℝ, < ) ≤ (2^nd ‘(𝐺‘𝑛)))
97	1, 96	eqbrtrid 5142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ≤ (2^nd ‘(𝐺‘𝑛)))
98		lelttr 11410	. . . . . . . . . 10 ⊢ ((𝑆 ∈ ℝ ∧ (2^nd ‘(𝐺‘𝑛)) ∈ ℝ ∧ (𝐹‘𝑛) ∈ ℝ) → ((𝑆 ≤ (2^nd ‘(𝐺‘𝑛)) ∧ (2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛)) → 𝑆 < (𝐹‘𝑛)))
99	70, 80, 25, 98	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑆 ≤ (2^nd ‘(𝐺‘𝑛)) ∧ (2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛)) → 𝑆 < (𝐹‘𝑛)))
100	97, 99	mpand 708	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛) → 𝑆 < (𝐹‘𝑛)))
101	73, 100	orim12d 979	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) ∨ (2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛)) → ((𝐹‘𝑛) < 𝑆 ∨ 𝑆 < (𝐹‘𝑛))))
102	49, 101	mpd 16	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) < 𝑆 ∨ 𝑆 < (𝐹‘𝑛)))
103	25, 70	lttri2d 11459	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) ≠ 𝑆 ↔ ((𝐹‘𝑛) < 𝑆 ∨ 𝑆 < (𝐹‘𝑛))))
104	102, 103	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ≠ 𝑆)
105	104	neneqd 2963	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (𝐹‘𝑛) = 𝑆)
106	105	nrexdv 3160	. . 3 ⊢ (𝜑 → ¬ ∃𝑛 ∈ ℕ (𝐹‘𝑛) = 𝑆)
107		risset 3240	. . . 4 ⊢ (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆)
108		ffn 6717	. . . . 5 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
109		eqeq1 2767	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑛) → (𝑧 = 𝑆 ↔ (𝐹‘𝑛) = 𝑆))
110	109	rexrn 7095	. . . . 5 ⊢ (𝐹 Fn ℕ → (∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆 ↔ ∃𝑛 ∈ ℕ (𝐹‘𝑛) = 𝑆))
111	2, 108, 110	3syl 19	. . . 4 ⊢ (𝜑 → (∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆 ↔ ∃𝑛 ∈ ℕ (𝐹‘𝑛) = 𝑆))
112	107, 111	bitrid 286	. . 3 ⊢ (𝜑 → (𝑆 ∈ ran 𝐹 ↔ ∃𝑛 ∈ ℕ (𝐹‘𝑛) = 𝑆))
113	106, 112	mtbird 328	. 2 ⊢ (𝜑 → ¬ 𝑆 ∈ ran 𝐹)
114	14, 113	eldifd 4052	1 ⊢ (𝜑 → 𝑆 ∈ (ℝ ∖ ran 𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2146 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 ⦋csb 3850 ⊆ wss 3899 ∖ cdif 4047 ∪ cun 4151 ∅c0 4280 ifcif 4484 {csn 4587 ⟨cop 4837 class class class wbr 5105 × cxp 5652 ran crn 5655 ∘ ccom 5658 Fn wfn 6541 ⟶wf 6542 ‘cfv 6546 (class class class)co 7427 ∈ cmpo 7429 1^st c1st 8006 2^nd c2nd 8007 supcsup 9442 ℂcc 11208 ℝcr 11209 0cc0 11210 1c1 11211 + caddc 11213 < clt 11353 ≤ cle 11354 − cmin 11551 / cdiv 11983 ℕcn 12345 2c2 12407 ℕ₀cn0 12616 seqcseq 14154

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-1st 8008 df-2nd 8009 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-er 8728 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9444 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-n0 12617 df-z 12704 df-uz 12976 df-fz 13650 df-seq 14155

This theorem is used by: ruclem13 16422

W3C validator