bfplem1 - Mathbox for Jeff Madsen

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Theorem bfplem1 34542

Description: Lemma for bfp 34544. The sequence 𝐺, which simply starts from any point in the space and iterates 𝐹, satisfies the property that the distance from 𝐺(𝑛) to 𝐺(𝑛 + 1) decreases by at least 𝐾 after each step. Thus, the total distance from any 𝐺(𝑖) to 𝐺(𝑗) is bounded by a geometric series, and the sequence is Cauchy. Therefore, it converges to a point ((⇝_𝑡‘𝐽)‘𝐺) since the space is complete. (Contributed by Jeff Madsen, 17-Jun-2014.)

**Hypotheses**
Ref	Expression
bfp.2	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
bfp.3	⊢ (𝜑 → 𝑋 ≠ ∅)
bfp.4	⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
bfp.5	⊢ (𝜑 → 𝐾 < 1)
bfp.6	⊢ (𝜑 → 𝐹:𝑋⟶𝑋)
bfp.7	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))
bfp.8	⊢ 𝐽 = (MetOpen‘𝐷)
bfp.9	⊢ (𝜑 → 𝐴 ∈ 𝑋)
bfp.10	⊢ 𝐺 = seq1((𝐹 ∘ 1^st ), (ℕ × {𝐴}))

**Assertion**
Ref	Expression
bfplem1	⊢ (𝜑 → 𝐺(⇝_𝑡‘𝐽)((⇝_𝑡‘𝐽)‘𝐺))

Distinct variable groups: 𝑥,𝑦,𝐷 𝑥,𝐺,𝑦 𝑥,𝐽,𝑦 𝜑,𝑥,𝑦 𝑥,𝐹,𝑦 𝑥,𝐾,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: 𝐴(𝑥,𝑦)

Proof of Theorem bfplem1 Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bfp.2	. . 3 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
2		cmetmet 23586	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4		nnuz 12089	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
5		bfp.10	. . . . 5 ⊢ 𝐺 = seq1((𝐹 ∘ 1^st ), (ℕ × {𝐴}))
6		1zzd 11820	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
7		bfp.9	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
8		bfp.6	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶𝑋)
9	4, 5, 6, 7, 8	algrf 15767	. . . 4 ⊢ (𝜑 → 𝐺:ℕ⟶𝑋)
10	8, 7	ffvelrnd 6671	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑋)
11		metcl 22639	. . . . . 6 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ (𝐹‘𝐴) ∈ 𝑋) → (𝐴𝐷(𝐹‘𝐴)) ∈ ℝ)
12	3, 7, 10, 11	syl3anc 1351	. . . . 5 ⊢ (𝜑 → (𝐴𝐷(𝐹‘𝐴)) ∈ ℝ)
13		bfp.4	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
14	12, 13	rerpdivcld 12273	. . . 4 ⊢ (𝜑 → ((𝐴𝐷(𝐹‘𝐴)) / 𝐾) ∈ ℝ)
15		bfp.5	. . . 4 ⊢ (𝜑 → 𝐾 < 1)
16		fveq2 6493	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝐺‘𝑗) = (𝐺‘1))
17		fvoveq1 6993	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝐺‘(𝑗 + 1)) = (𝐺‘(1 + 1)))
18	16, 17	oveq12d 6988	. . . . . . . 8 ⊢ (𝑗 = 1 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) = ((𝐺‘1)𝐷(𝐺‘(1 + 1))))
19		oveq2 6978	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝐾↑𝑗) = (𝐾↑1))
20	19	oveq2d 6986	. . . . . . . 8 ⊢ (𝑗 = 1 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)))
21	18, 20	breq12d 4936	. . . . . . 7 ⊢ (𝑗 = 1 → (((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) ↔ ((𝐺‘1)𝐷(𝐺‘(1 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1))))
22	21	imbi2d 333	. . . . . 6 ⊢ (𝑗 = 1 → ((𝜑 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗))) ↔ (𝜑 → ((𝐺‘1)𝐷(𝐺‘(1 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)))))
23		fveq2 6493	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐺‘𝑗) = (𝐺‘𝑘))
24		fvoveq1 6993	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐺‘(𝑗 + 1)) = (𝐺‘(𝑘 + 1)))
25	23, 24	oveq12d 6988	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) = ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))))
26		oveq2 6978	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐾↑𝑗) = (𝐾↑𝑘))
27	26	oveq2d 6986	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)))
28	25, 27	breq12d 4936	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) ↔ ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘))))
29	28	imbi2d 333	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝜑 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗))) ↔ (𝜑 → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)))))
30		fveq2 6493	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝐺‘𝑗) = (𝐺‘(𝑘 + 1)))
31		fvoveq1 6993	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝐺‘(𝑗 + 1)) = (𝐺‘((𝑘 + 1) + 1)))
32	30, 31	oveq12d 6988	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) = ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))))
33		oveq2 6978	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝐾↑𝑗) = (𝐾↑(𝑘 + 1)))
34	33	oveq2d 6986	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))
35	32, 34	breq12d 4936	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → (((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) ↔ ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
36	35	imbi2d 333	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → ((𝜑 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗))) ↔ (𝜑 → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))))
37	12	leidd 11001	. . . . . . 7 ⊢ (𝜑 → (𝐴𝐷(𝐹‘𝐴)) ≤ (𝐴𝐷(𝐹‘𝐴)))
38	4, 5, 6, 7	algr0 15766	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘1) = 𝐴)
39		1nn 11446	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
40	4, 5, 6, 7, 8	algrp1 15768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝐺‘(1 + 1)) = (𝐹‘(𝐺‘1)))
41	39, 40	mpan2 678	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘(1 + 1)) = (𝐹‘(𝐺‘1)))
42	38	fveq2d 6497	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝐺‘1)) = (𝐹‘𝐴))
43	41, 42	eqtrd 2808	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘(1 + 1)) = (𝐹‘𝐴))
44	38, 43	oveq12d 6988	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘1)𝐷(𝐺‘(1 + 1))) = (𝐴𝐷(𝐹‘𝐴)))
45	13	rpred 12242	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℝ)
46	45	recnd 10462	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℂ)
47	46	exp1d 13314	. . . . . . . . 9 ⊢ (𝜑 → (𝐾↑1) = 𝐾)
48	47	oveq2d 6986	. . . . . . . 8 ⊢ (𝜑 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · 𝐾))
49	12	recnd 10462	. . . . . . . . 9 ⊢ (𝜑 → (𝐴𝐷(𝐹‘𝐴)) ∈ ℂ)
50	13	rpne0d 12247	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ≠ 0)
51	49, 46, 50	divcan1d 11212	. . . . . . . 8 ⊢ (𝜑 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · 𝐾) = (𝐴𝐷(𝐹‘𝐴)))
52	48, 51	eqtrd 2808	. . . . . . 7 ⊢ (𝜑 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)) = (𝐴𝐷(𝐹‘𝐴)))
53	37, 44, 52	3brtr4d 4955	. . . . . 6 ⊢ (𝜑 → ((𝐺‘1)𝐷(𝐺‘(1 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)))
54	9	ffvelrnda 6670	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ 𝑋)
55		peano2nn 11447	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
56		ffvelrn 6668	. . . . . . . . . . . . 13 ⊢ ((𝐺:ℕ⟶𝑋 ∧ (𝑘 + 1) ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ 𝑋)
57	9, 55, 56	syl2an 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ 𝑋)
58	54, 57	jca 504	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘) ∈ 𝑋 ∧ (𝐺‘(𝑘 + 1)) ∈ 𝑋))
59		bfp.7	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))
60	59	ralrimivva 3135	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))
61	60	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))
62		fveq2 6493	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐺‘𝑘) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑘)))
63	62	oveq1d 6985	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐺‘𝑘) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) = ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)))
64		oveq1 6977	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐺‘𝑘) → (𝑥𝐷𝑦) = ((𝐺‘𝑘)𝐷𝑦))
65	64	oveq2d 6986	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐺‘𝑘) → (𝐾 · (𝑥𝐷𝑦)) = (𝐾 · ((𝐺‘𝑘)𝐷𝑦)))
66	63, 65	breq12d 4936	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺‘𝑘) → (((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)) ↔ ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)) ≤ (𝐾 · ((𝐺‘𝑘)𝐷𝑦))))
67		fveq2 6493	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → (𝐹‘𝑦) = (𝐹‘(𝐺‘(𝑘 + 1))))
68	67	oveq2d 6986	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)) = ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))))
69		oveq2 6978	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → ((𝐺‘𝑘)𝐷𝑦) = ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))))
70	69	oveq2d 6986	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → (𝐾 · ((𝐺‘𝑘)𝐷𝑦)) = (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))))
71	68, 70	breq12d 4936	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → (((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)) ≤ (𝐾 · ((𝐺‘𝑘)𝐷𝑦)) ↔ ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))))))
72	66, 71	rspc2v 3542	. . . . . . . . . . 11 ⊢ (((𝐺‘𝑘) ∈ 𝑋 ∧ (𝐺‘(𝑘 + 1)) ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))))))
73	58, 61, 72	sylc 65	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))))
74	3	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐷 ∈ (Met‘𝑋))
75	8	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:𝑋⟶𝑋)
76	75, 54	ffvelrnd 6671	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝐺‘𝑘)) ∈ 𝑋)
77	75, 57	ffvelrnd 6671	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝐺‘(𝑘 + 1))) ∈ 𝑋)
78		metcl 22639	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘(𝐺‘𝑘)) ∈ 𝑋 ∧ (𝐹‘(𝐺‘(𝑘 + 1))) ∈ 𝑋) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ∈ ℝ)
79	74, 76, 77, 78	syl3anc 1351	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ∈ ℝ)
80	45	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐾 ∈ ℝ)
81		metcl 22639	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ (𝐺‘(𝑘 + 1)) ∈ 𝑋) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℝ)
82	74, 54, 57, 81	syl3anc 1351	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℝ)
83	80, 82	remulcld 10464	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ∈ ℝ)
84	14	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝐴)) / 𝐾) ∈ ℝ)
85	55	adantl 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
86	85	nnnn0d 11761	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ₀)
87	80, 86	reexpcld 13336	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑(𝑘 + 1)) ∈ ℝ)
88	84, 87	remulcld 10464	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) ∈ ℝ)
89		letr 10528	. . . . . . . . . . 11 ⊢ ((((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ∈ ℝ ∧ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ∈ ℝ ∧ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) ∈ ℝ) → ((((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ∧ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
90	79, 83, 88, 89	syl3anc 1351	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ∧ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
91	73, 90	mpand 682	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
92		nnnn0 11709	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
93		reexpcl 13255	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℝ ∧ 𝑘 ∈ ℕ₀) → (𝐾↑𝑘) ∈ ℝ)
94	45, 92, 93	syl2an 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑𝑘) ∈ ℝ)
95	84, 94	remulcld 10464	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ∈ ℝ)
96	13	rpgt0d 12245	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐾)
97	96	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < 𝐾)
98		lemul1 11287	. . . . . . . . . . 11 ⊢ ((((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℝ ∧ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ∈ ℝ ∧ (𝐾 ∈ ℝ ∧ 0 < 𝐾)) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ↔ (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) · 𝐾) ≤ ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾)))
99	82, 95, 80, 97, 98	syl112anc 1354	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ↔ (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) · 𝐾) ≤ ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾)))
100	82	recnd 10462	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℂ)
101	46	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐾 ∈ ℂ)
102	100, 101	mulcomd 10455	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) · 𝐾) = (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))))
103	84	recnd 10462	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝐴)) / 𝐾) ∈ ℂ)
104	94	recnd 10462	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑𝑘) ∈ ℂ)
105	103, 104, 101	mulassd 10457	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · ((𝐾↑𝑘) · 𝐾)))
106		expp1 13245	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐾↑(𝑘 + 1)) = ((𝐾↑𝑘) · 𝐾))
107	46, 92, 106	syl2an 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑(𝑘 + 1)) = ((𝐾↑𝑘) · 𝐾))
108	107	oveq2d 6986	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · ((𝐾↑𝑘) · 𝐾)))
109	105, 108	eqtr4d 2811	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))
110	102, 109	breq12d 4936	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) · 𝐾) ≤ ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾) ↔ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
111	99, 110	bitrd 271	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ↔ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
112	4, 5, 6, 7, 8	algrp1 15768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) = (𝐹‘(𝐺‘𝑘)))
113	4, 5, 6, 7, 8	algrp1 15768	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ ℕ) → (𝐺‘((𝑘 + 1) + 1)) = (𝐹‘(𝐺‘(𝑘 + 1))))
114	55, 113	sylan2 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘((𝑘 + 1) + 1)) = (𝐹‘(𝐺‘(𝑘 + 1))))
115	112, 114	oveq12d 6988	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) = ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))))
116	115	breq1d 4933	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) ↔ ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
117	91, 111, 116	3imtr4d 286	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
118	117	expcom 406	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝜑 → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))))
119	118	a2d 29	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ((𝜑 → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘))) → (𝜑 → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))))
120	22, 29, 36, 29, 53, 119	nnind 11453	. . . . 5 ⊢ (𝑘 ∈ ℕ → (𝜑 → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘))))
121	120	impcom 399	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)))
122	3, 9, 14, 13, 15, 121	geomcau 34476	. . 3 ⊢ (𝜑 → 𝐺 ∈ (Cau‘𝐷))
123		bfp.8	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐷)
124	123	cmetcau 23589	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐺 ∈ (Cau‘𝐷)) → 𝐺 ∈ dom (⇝_𝑡‘𝐽))
125	1, 122, 124	syl2anc 576	. 2 ⊢ (𝜑 → 𝐺 ∈ dom (⇝_𝑡‘𝐽))
126		metxmet 22641	. . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
127	123	methaus 22827	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus)
128	3, 126, 127	3syl 18	. . 3 ⊢ (𝜑 → 𝐽 ∈ Haus)
129		lmfun 21687	. . 3 ⊢ (𝐽 ∈ Haus → Fun (⇝_𝑡‘𝐽))
130		funfvbrb 6640	. . 3 ⊢ (Fun (⇝_𝑡‘𝐽) → (𝐺 ∈ dom (⇝_𝑡‘𝐽) ↔ 𝐺(⇝_𝑡‘𝐽)((⇝_𝑡‘𝐽)‘𝐺)))
131	128, 129, 130	3syl 18	. 2 ⊢ (𝜑 → (𝐺 ∈ dom (⇝_𝑡‘𝐽) ↔ 𝐺(⇝_𝑡‘𝐽)((⇝_𝑡‘𝐽)‘𝐺)))
132	125, 131	mpbid 224	1 ⊢ (𝜑 → 𝐺(⇝_𝑡‘𝐽)((⇝_𝑡‘𝐽)‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 ∀wral 3082 ∅c0 4172 {csn 4435 class class class wbr 4923 × cxp 5399 dom cdm 5401 ∘ ccom 5405 Fun wfun 6176 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 1^st c1st 7493 ℂcc 10327 ℝcr 10328 0cc0 10329 1c1 10330 + caddc 10332 · cmul 10334 < clt 10468 ≤ cle 10469 / cdiv 11092 ℕcn 11433 ℕ₀cn0 11701 ℝ⁺crp 12198 seqcseq 13178 ↑cexp 13238 ∞Metcxmet 20226 Metcmet 20227 MetOpencmopn 20231 ⇝_𝑡clm 21532 Hauscha 21614 Cauccau 23553 CMetccmet 23554

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8892 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-pre-sup 10407

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-se 5361 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-map 8202 df-pm 8203 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-sup 8695 df-inf 8696 df-oi 8763 df-card 9156 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-n0 11702 df-z 11788 df-uz 12053 df-q 12157 df-rp 12199 df-xneg 12318 df-xadd 12319 df-xmul 12320 df-ico 12554 df-icc 12555 df-fz 12703 df-fzo 12844 df-fl 12971 df-seq 13179 df-exp 13239 df-hash 13500 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-clim 14700 df-rlim 14701 df-sum 14898 df-rest 16546 df-topgen 16567 df-psmet 20233 df-xmet 20234 df-met 20235 df-bl 20236 df-mopn 20237 df-fbas 20238 df-fg 20239 df-top 21200 df-topon 21217 df-bases 21252 df-ntr 21326 df-nei 21404 df-lm 21535 df-haus 21621 df-fil 22152 df-fm 22244 df-flim 22245 df-flf 22246 df-cfil 23555 df-cau 23556 df-cmet 23557

This theorem is referenced by: bfplem2 34543

W3C validator