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Theorem ruclem12 15593

Description: Lemma for ruc 15595. 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 15592	. . . . 5 ⊢ (𝜑 → (ran (1^st ∘ 𝐺) ⊆ ℝ ∧ ran (1^st ∘ 𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1))
7	6	simp1d 1138	. . . 4 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ⊆ ℝ)
8	6	simp2d 1139	. . . 4 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ≠ ∅)
9		1re 10640	. . . . 5 ⊢ 1 ∈ ℝ
10	6	simp3d 1140	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1)
11		brralrspcev 5125	. . . . 5 ⊢ ((1 ∈ ℝ ∧ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 𝑛)
12	9, 10, 11	sylancr 589	. . . 4 ⊢ (𝜑 → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 𝑛)
13	7, 8, 12	suprcld 11603	. . 3 ⊢ (𝜑 → sup(ran (1^st ∘ 𝐺), ℝ, < ) ∈ ℝ)
14	1, 13	eqeltrid 2917	. 2 ⊢ (𝜑 → 𝑆 ∈ ℝ)
15	2	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶ℝ)
16	3	adantr 483	. . . . . . . . 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 15587	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
18		nnm1nn0 11937	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
19		ffvelrn 6848	. . . . . . . . . . 11 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ (𝑛 − 1) ∈ ℕ₀) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
20	17, 18, 19	syl2an 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
21		xp1st 7720	. . . . . . . . . 10 ⊢ ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
22	20, 21	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
23		xp2nd 7721	. . . . . . . . . 10 ⊢ ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (2^nd ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
24	20, 23	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝐺‘(𝑛 − 1))) ∈ ℝ)
25	2	ffvelrnda 6850	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℝ)
26		eqid 2821	. . . . . . . . 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 2821	. . . . . . . . 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 15589	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 − 1) ∈ ℕ₀) → (1^st ‘(𝐺‘(𝑛 − 1))) < (2^nd ‘(𝐺‘(𝑛 − 1))))
29	18, 28	sylan2 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘(𝑛 − 1))) < (2^nd ‘(𝐺‘(𝑛 − 1))))
30	15, 16, 22, 24, 25, 26, 27, 29	ruclem3 15585	. . . . . . . 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 15588	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 − 1) ∈ ℕ₀) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
32	18, 31	sylan2 594	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
33		nncn 11645	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
34	33	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
35		ax-1cn 10594	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
36		npcan 10894	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
37	34, 35, 36	sylancl 588	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 − 1) + 1) = 𝑛)
38	37	fveq2d 6673	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = (𝐺‘𝑛))
39		1st2nd2 7727	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ) → (𝐺‘(𝑛 − 1)) = ⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩)
40	20, 39	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘(𝑛 − 1)) = ⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩)
41	37	fveq2d 6673	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘((𝑛 − 1) + 1)) = (𝐹‘𝑛))
42	40, 41	oveq12d 7173	. . . . . . . . . . . 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 2864	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = (⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛)))
44	43	fveq2d 6673	. . . . . . . . . 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 5077	. . . . . . . . 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 6673	. . . . . . . . . 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 5075	. . . . . . . . 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 915	. . . . . . . 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 259	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) ∨ (2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛)))
50	7	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (1^st ∘ 𝐺) ⊆ ℝ)
51	8	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (1^st ∘ 𝐺) ≠ ∅)
52	12	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 𝑛)
53		nnnn0 11903	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
54		fvco3 6759	. . . . . . . . . . . . 13 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
55	17, 53, 54	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
56	17	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:ℕ₀⟶(ℝ × ℝ))
57		1stcof 7718	. . . . . . . . . . . . . 14 ⊢ (𝐺:ℕ₀⟶(ℝ × ℝ) → (1^st ∘ 𝐺):ℕ₀⟶ℝ)
58		ffn 6513	. . . . . . . . . . . . . 14 ⊢ ((1^st ∘ 𝐺):ℕ₀⟶ℝ → (1^st ∘ 𝐺) Fn ℕ₀)
59	56, 57, 58	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ∘ 𝐺) Fn ℕ₀)
60	53	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
61		fnfvelrn 6847	. . . . . . . . . . . . 13 ⊢ (((1^st ∘ 𝐺) Fn ℕ₀ ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) ∈ ran (1^st ∘ 𝐺))
62	59, 60, 61	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1^st ∘ 𝐺)‘𝑛) ∈ ran (1^st ∘ 𝐺))
63	55, 62	eqeltrrd 2914	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ∈ ran (1^st ∘ 𝐺))
64	50, 51, 52, 63	suprubd 11602	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ≤ sup(ran (1^st ∘ 𝐺), ℝ, < ))
65	64, 1	breqtrrdi 5107	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ≤ 𝑆)
66		ffvelrn 6848	. . . . . . . . . . . 12 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
67	17, 53, 66	syl2an 597	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
68		xp1st 7720	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
69	67, 68	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ∈ ℝ)
70	14	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
71		ltletr 10731	. . . . . . . . . 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 1367	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) ∧ (1^st ‘(𝐺‘𝑛)) ≤ 𝑆) → (𝐹‘𝑛) < 𝑆))
73	65, 72	mpan2d 692	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) → (𝐹‘𝑛) < 𝑆))
74		fvco3 6759	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑘 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑘) = (1^st ‘(𝐺‘𝑘)))
75	56, 74	sylan 582	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑘) = (1^st ‘(𝐺‘𝑘)))
76	56	ffvelrnda 6850	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) ∈ (ℝ × ℝ))
77		xp1st 7720	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑘) ∈ (ℝ × ℝ) → (1^st ‘(𝐺‘𝑘)) ∈ ℝ)
78	76, 77	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑘)) ∈ ℝ)
79		xp2nd 7721	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
80	67, 79	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
81	80	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
82	15	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝐹:ℕ⟶ℝ)
83	16	adantr 483	. . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
85	60	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
86	82, 83, 4, 5, 84, 85	ruclem10 15591	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑛)))
87	78, 81, 86	ltled 10787	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑛)))
88	75, 87	eqbrtrd 5087	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛)))
89	88	ralrimiva 3182	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛)))
90		breq1 5068	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((1^st ∘ 𝐺)‘𝑘) → (𝑧 ≤ (2^nd ‘(𝐺‘𝑛)) ↔ ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛))))
91	90	ralrn 6853	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 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 259	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ (2^nd ‘(𝐺‘𝑛)))
94		suprleub 11606	. . . . . . . . . . . 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 1369	. . . . . . . . . . 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 259	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → sup(ran (1^st ∘ 𝐺), ℝ, < ) ≤ (2^nd ‘(𝐺‘𝑛)))
97	1, 96	eqbrtrid 5100	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ≤ (2^nd ‘(𝐺‘𝑛)))
98		lelttr 10730	. . . . . . . . . 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 1367	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑆 ≤ (2^nd ‘(𝐺‘𝑛)) ∧ (2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛)) → 𝑆 < (𝐹‘𝑛)))
100	97, 99	mpand 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛) → 𝑆 < (𝐹‘𝑛)))
101	73, 100	orim12d 961	. . . . . . 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 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) < 𝑆 ∨ 𝑆 < (𝐹‘𝑛)))
103	25, 70	lttri2d 10778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) ≠ 𝑆 ↔ ((𝐹‘𝑛) < 𝑆 ∨ 𝑆 < (𝐹‘𝑛))))
104	102, 103	mpbird 259	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ≠ 𝑆)
105	104	neneqd 3021	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (𝐹‘𝑛) = 𝑆)
106	105	nrexdv 3270	. . 3 ⊢ (𝜑 → ¬ ∃𝑛 ∈ ℕ (𝐹‘𝑛) = 𝑆)
107		risset 3267	. . . 4 ⊢ (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆)
108		ffn 6513	. . . . 5 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
109		eqeq1 2825	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑛) → (𝑧 = 𝑆 ↔ (𝐹‘𝑛) = 𝑆))
110	109	rexrn 6852	. . . . 5 ⊢ (𝐹 Fn ℕ → (∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆 ↔ ∃𝑛 ∈ ℕ (𝐹‘𝑛) = 𝑆))
111	2, 108, 110	3syl 18	. . . 4 ⊢ (𝜑 → (∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆 ↔ ∃𝑛 ∈ ℕ (𝐹‘𝑛) = 𝑆))
112	107, 111	syl5bb 285	. . 3 ⊢ (𝜑 → (𝑆 ∈ ran 𝐹 ↔ ∃𝑛 ∈ ℕ (𝐹‘𝑛) = 𝑆))
113	106, 112	mtbird 327	. 2 ⊢ (𝜑 → ¬ 𝑆 ∈ ran 𝐹)
114	14, 113	eldifd 3946	1 ⊢ (𝜑 → 𝑆 ∈ (ℝ ∖ ran 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ⦋csb 3882 ∖ cdif 3932 ∪ cun 3933 ⊆ wss 3935 ∅c0 4290 ifcif 4466 {csn 4566 ⟨cop 4572 class class class wbr 5065 × cxp 5552 ran crn 5555 ∘ ccom 5558 Fn wfn 6349 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 1^st c1st 7686 2^nd c2nd 7687 supcsup 8903 ℂcc 10534 ℝcr 10535 0cc0 10536 1c1 10537 + caddc 10539 < clt 10674 ≤ cle 10675 − cmin 10869 / cdiv 11296 ℕcn 11637 2c2 11691 ℕ₀cn0 11896 seqcseq 13368

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-seq 13369

This theorem is referenced by: ruclem13 15594

W3C validator