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Theorem ruclem12 16185

Description: Lemma for ruc 16187. 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 16184	. . . . 5 ⊢ (𝜑 → (ran (1^st ∘ 𝐺) ⊆ ℝ ∧ ran (1^st ∘ 𝐺) ≠ ∅ ∧ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1))
7	6	simp1d 1142	. . . 4 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ⊆ ℝ)
8	6	simp2d 1143	. . . 4 ⊢ (𝜑 → ran (1^st ∘ 𝐺) ≠ ∅)
9		1re 11150	. . . . 5 ⊢ 1 ∈ ℝ
10	6	simp3d 1144	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1)
11		brralrspcev 5162	. . . . 5 ⊢ ((1 ∈ ℝ ∧ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 𝑛)
12	9, 10, 11	sylancr 587	. . . 4 ⊢ (𝜑 → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 𝑛)
13	7, 8, 12	suprcld 12122	. . 3 ⊢ (𝜑 → sup(ran (1^st ∘ 𝐺), ℝ, < ) ∈ ℝ)
14	1, 13	eqeltrid 2832	. 2 ⊢ (𝜑 → 𝑆 ∈ ℝ)
15	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶ℝ)
16	3	adantr 480	. . . . . . . . 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 16179	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
18		nnm1nn0 12459	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
19		ffvelcdm 7035	. . . . . . . . . . 11 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ (𝑛 − 1) ∈ ℕ₀) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
20	17, 18, 19	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
21		xp1st 7979	. . . . . . . . . 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 7980	. . . . . . . . . 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	ffvelcdmda 7038	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℝ)
26		eqid 2729	. . . . . . . . 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 2729	. . . . . . . . 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 16181	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 − 1) ∈ ℕ₀) → (1^st ‘(𝐺‘(𝑛 − 1))) < (2^nd ‘(𝐺‘(𝑛 − 1))))
29	18, 28	sylan2 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘(𝑛 − 1))) < (2^nd ‘(𝐺‘(𝑛 − 1))))
30	15, 16, 22, 24, 25, 26, 27, 29	ruclem3 16177	. . . . . . . 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 16180	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 − 1) ∈ ℕ₀) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
32	18, 31	sylan2 593	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
33		nncn 12170	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
34	33	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
35		ax-1cn 11102	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
36		npcan 11406	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
37	34, 35, 36	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 − 1) + 1) = 𝑛)
38	37	fveq2d 6844	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = (𝐺‘𝑛))
39		1st2nd2 7986	. . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘((𝑛 − 1) + 1)) = (𝐹‘𝑛))
42	40, 41	oveq12d 7387	. . . . . . . . . . . 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 2772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = (⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛)))
44	43	fveq2d 6844	. . . . . . . . . 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 5114	. . . . . . . . 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 6844	. . . . . . . . . 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 5112	. . . . . . . . 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 918	. . . . . . . 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 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) ∨ (2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛)))
50	7	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (1^st ∘ 𝐺) ⊆ ℝ)
51	8	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (1^st ∘ 𝐺) ≠ ∅)
52	12	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 𝑛)
53		nnnn0 12425	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
54		fvco3 6942	. . . . . . . . . . . . 13 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
55	17, 53, 54	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
56	17	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:ℕ₀⟶(ℝ × ℝ))
57		1stcof 7977	. . . . . . . . . . . . . 14 ⊢ (𝐺:ℕ₀⟶(ℝ × ℝ) → (1^st ∘ 𝐺):ℕ₀⟶ℝ)
58		ffn 6670	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
61		fnfvelrn 7034	. . . . . . . . . . . . 13 ⊢ (((1^st ∘ 𝐺) Fn ℕ₀ ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) ∈ ran (1^st ∘ 𝐺))
62	59, 60, 61	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1^st ∘ 𝐺)‘𝑛) ∈ ran (1^st ∘ 𝐺))
63	55, 62	eqeltrrd 2829	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ∈ ran (1^st ∘ 𝐺))
64	50, 51, 52, 63	suprubd 12121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ≤ sup(ran (1^st ∘ 𝐺), ℝ, < ))
65	64, 1	breqtrrdi 5144	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ≤ 𝑆)
66		ffvelcdm 7035	. . . . . . . . . . . 12 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
67	17, 53, 66	syl2an 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
68		xp1st 7979	. . . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
71		ltletr 11242	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) ∧ (1^st ‘(𝐺‘𝑛)) ≤ 𝑆) → (𝐹‘𝑛) < 𝑆))
73	65, 72	mpan2d 694	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) → (𝐹‘𝑛) < 𝑆))
74		fvco3 6942	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑘 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑘) = (1^st ‘(𝐺‘𝑘)))
75	56, 74	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑘) = (1^st ‘(𝐺‘𝑘)))
76	56	ffvelcdmda 7038	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) ∈ (ℝ × ℝ))
77		xp1st 7979	. . . . . . . . . . . . . . . 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 7980	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (2^nd ‘(𝐺‘𝑛)) ∈ ℝ)
82	15	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝐹:ℕ⟶ℝ)
83	16	adantr 480	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
85	60	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
86	82, 83, 4, 5, 84, 85	ruclem10 16183	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑛)))
87	78, 81, 86	ltled 11298	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑛)))
88	75, 87	eqbrtrd 5124	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛)))
89	88	ralrimiva 3125	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛)))
90		breq1 5105	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((1^st ∘ 𝐺)‘𝑘) → (𝑧 ≤ (2^nd ‘(𝐺‘𝑛)) ↔ ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛))))
91	90	ralrn 7042	. . . . . . . . . . . . 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 257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ (2^nd ‘(𝐺‘𝑛)))
94		suprleub 12125	. . . . . . . . . . . 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 1375	. . . . . . . . . . 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 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → sup(ran (1^st ∘ 𝐺), ℝ, < ) ≤ (2^nd ‘(𝐺‘𝑛)))
97	1, 96	eqbrtrid 5137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ≤ (2^nd ‘(𝐺‘𝑛)))
98		lelttr 11240	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑆 ≤ (2^nd ‘(𝐺‘𝑛)) ∧ (2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛)) → 𝑆 < (𝐹‘𝑛)))
100	97, 99	mpand 695	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛) → 𝑆 < (𝐹‘𝑛)))
101	73, 100	orim12d 966	. . . . . . 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 11289	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) ≠ 𝑆 ↔ ((𝐹‘𝑛) < 𝑆 ∨ 𝑆 < (𝐹‘𝑛))))
104	102, 103	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ≠ 𝑆)
105	104	neneqd 2930	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (𝐹‘𝑛) = 𝑆)
106	105	nrexdv 3128	. . 3 ⊢ (𝜑 → ¬ ∃𝑛 ∈ ℕ (𝐹‘𝑛) = 𝑆)
107		risset 3210	. . . 4 ⊢ (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆)
108		ffn 6670	. . . . 5 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
109		eqeq1 2733	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑛) → (𝑧 = 𝑆 ↔ (𝐹‘𝑛) = 𝑆))
110	109	rexrn 7041	. . . . 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	bitrid 283	. . 3 ⊢ (𝜑 → (𝑆 ∈ ran 𝐹 ↔ ∃𝑛 ∈ ℕ (𝐹‘𝑛) = 𝑆))
113	106, 112	mtbird 325	. 2 ⊢ (𝜑 → ¬ 𝑆 ∈ ran 𝐹)
114	14, 113	eldifd 3922	1 ⊢ (𝜑 → 𝑆 ∈ (ℝ ∖ ran 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ⦋csb 3859 ∖ cdif 3908 ∪ cun 3909 ⊆ wss 3911 ∅c0 4292 ifcif 4484 {csn 4585 ⟨cop 4591 class class class wbr 5102 × cxp 5629 ran crn 5632 ∘ ccom 5635 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 1^st c1st 7945 2^nd c2nd 7946 supcsup 9367 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 ℕcn 12162 2c2 12217 ℕ₀cn0 12418 seqcseq 13942

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9369 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-seq 13943

This theorem is referenced by: ruclem13 16186

W3C validator