ruclem12 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ruclem12		Structured version Visualization version GIF version

Theorem ruclem12 16289

Description: Lemma for ruc 16291. 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 16288	. . . . 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 11290	. . . . 5 ⊢ 1 ∈ ℝ
10	6	simp3d 1144	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1)
11		brralrspcev 5226	. . . . 5 ⊢ ((1 ∈ ℝ ∧ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 1) → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 𝑛)
12	9, 10, 11	sylancr 586	. . . 4 ⊢ (𝜑 → ∃𝑛 ∈ ℝ ∀𝑧 ∈ ran (1^st ∘ 𝐺)𝑧 ≤ 𝑛)
13	7, 8, 12	suprcld 12258	. . 3 ⊢ (𝜑 → sup(ran (1^st ∘ 𝐺), ℝ, < ) ∈ ℝ)
14	1, 13	eqeltrid 2848	. 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 16283	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℕ₀⟶(ℝ × ℝ))
18		nnm1nn0 12594	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
19		ffvelcdm 7115	. . . . . . . . . . 11 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ (𝑛 − 1) ∈ ℕ₀) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
20	17, 18, 19	syl2an 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘(𝑛 − 1)) ∈ (ℝ × ℝ))
21		xp1st 8062	. . . . . . . . . 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 8063	. . . . . . . . . 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 7118	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℝ)
26		eqid 2740	. . . . . . . . 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 2740	. . . . . . . . 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 16285	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 − 1) ∈ ℕ₀) → (1^st ‘(𝐺‘(𝑛 − 1))) < (2^nd ‘(𝐺‘(𝑛 − 1))))
29	18, 28	sylan2 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘(𝑛 − 1))) < (2^nd ‘(𝐺‘(𝑛 − 1))))
30	15, 16, 22, 24, 25, 26, 27, 29	ruclem3 16281	. . . . . . . 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 16284	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 − 1) ∈ ℕ₀) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
32	18, 31	sylan2 592	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = ((𝐺‘(𝑛 − 1))𝐷(𝐹‘((𝑛 − 1) + 1))))
33		nncn 12301	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
34	33	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
35		ax-1cn 11242	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
36		npcan 11545	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
37	34, 35, 36	sylancl 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 − 1) + 1) = 𝑛)
38	37	fveq2d 6924	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘((𝑛 − 1) + 1)) = (𝐺‘𝑛))
39		1st2nd2 8069	. . . . . . . . . . . . . 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 6924	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘((𝑛 − 1) + 1)) = (𝐹‘𝑛))
42	40, 41	oveq12d 7466	. . . . . . . . . . . 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 2788	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = (⟨(1^st ‘(𝐺‘(𝑛 − 1))), (2^nd ‘(𝐺‘(𝑛 − 1)))⟩𝐷(𝐹‘𝑛)))
44	43	fveq2d 6924	. . . . . . . . . 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 5178	. . . . . . . . 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 6924	. . . . . . . . . 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 5176	. . . . . . . . 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 917	. . . . . . . 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 12560	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
54		fvco3 7021	. . . . . . . . . . . . 13 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) = (1^st ‘(𝐺‘𝑛)))
55	17, 53, 54	syl2an 595	. . . . . . . . . . . 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 8060	. . . . . . . . . . . . . 14 ⊢ (𝐺:ℕ₀⟶(ℝ × ℝ) → (1^st ∘ 𝐺):ℕ₀⟶ℝ)
58		ffn 6747	. . . . . . . . . . . . . 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 7114	. . . . . . . . . . . . 13 ⊢ (((1^st ∘ 𝐺) Fn ℕ₀ ∧ 𝑛 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑛) ∈ ran (1^st ∘ 𝐺))
62	59, 60, 61	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1^st ∘ 𝐺)‘𝑛) ∈ ran (1^st ∘ 𝐺))
63	55, 62	eqeltrrd 2845	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ∈ ran (1^st ∘ 𝐺))
64	50, 51, 52, 63	suprubd 12257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ≤ sup(ran (1^st ∘ 𝐺), ℝ, < ))
65	64, 1	breqtrrdi 5208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐺‘𝑛)) ≤ 𝑆)
66		ffvelcdm 7115	. . . . . . . . . . . 12 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑛 ∈ ℕ₀) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
67	17, 53, 66	syl2an 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ (ℝ × ℝ))
68		xp1st 8062	. . . . . . . . . . 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 11382	. . . . . . . . . 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 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) ∧ (1^st ‘(𝐺‘𝑛)) ≤ 𝑆) → (𝐹‘𝑛) < 𝑆))
73	65, 72	mpan2d 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) < (1^st ‘(𝐺‘𝑛)) → (𝐹‘𝑛) < 𝑆))
74		fvco3 7021	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:ℕ₀⟶(ℝ × ℝ) ∧ 𝑘 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑘) = (1^st ‘(𝐺‘𝑘)))
75	56, 74	sylan 579	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑘) = (1^st ‘(𝐺‘𝑘)))
76	56	ffvelcdmda 7118	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) ∈ (ℝ × ℝ))
77		xp1st 8062	. . . . . . . . . . . . . . . 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 8063	. . . . . . . . . . . . . . . . 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 16287	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑘)) < (2^nd ‘(𝐺‘𝑛)))
87	78, 81, 86	ltled 11438	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (1^st ‘(𝐺‘𝑘)) ≤ (2^nd ‘(𝐺‘𝑛)))
88	75, 87	eqbrtrd 5188	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛)))
89	88	ralrimiva 3152	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ ℕ₀ ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛)))
90		breq1 5169	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((1^st ∘ 𝐺)‘𝑘) → (𝑧 ≤ (2^nd ‘(𝐺‘𝑛)) ↔ ((1^st ∘ 𝐺)‘𝑘) ≤ (2^nd ‘(𝐺‘𝑛))))
91	90	ralrn 7122	. . . . . . . . . . . . 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 12261	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 5201	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ≤ (2^nd ‘(𝐺‘𝑛)))
98		lelttr 11380	. . . . . . . . . 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 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑆 ≤ (2^nd ‘(𝐺‘𝑛)) ∧ (2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛)) → 𝑆 < (𝐹‘𝑛)))
100	97, 99	mpand 694	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2^nd ‘(𝐺‘𝑛)) < (𝐹‘𝑛) → 𝑆 < (𝐹‘𝑛)))
101	73, 100	orim12d 965	. . . . . . 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 11429	. . . . . 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 2951	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (𝐹‘𝑛) = 𝑆)
106	105	nrexdv 3155	. . 3 ⊢ (𝜑 → ¬ ∃𝑛 ∈ ℕ (𝐹‘𝑛) = 𝑆)
107		risset 3239	. . . 4 ⊢ (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ ran 𝐹 𝑧 = 𝑆)
108		ffn 6747	. . . . 5 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
109		eqeq1 2744	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑛) → (𝑧 = 𝑆 ↔ (𝐹‘𝑛) = 𝑆))
110	109	rexrn 7121	. . . . 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 3987	1 ⊢ (𝜑 → 𝑆 ∈ (ℝ ∖ ran 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 ⦋csb 3921 ∖ cdif 3973 ∪ cun 3974 ⊆ wss 3976 ∅c0 4352 ifcif 4548 {csn 4648 ⟨cop 4654 class class class wbr 5166 × cxp 5698 ran crn 5701 ∘ ccom 5704 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 1^st c1st 8028 2^nd c2nd 8029 supcsup 9509 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 seqcseq 14052

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-seq 14053

This theorem is referenced by: ruclem13 16290

W3C validator