bfplem1 - Mathbox for Jeff Madsen

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Theorem bfplem1 38533

Description: Lemma for bfp 38535. 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 25498	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4		nnuz 12919	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
5		bfp.10	. . . . 5 ⊢ 𝐺 = seq1((𝐹 ∘ 1^st ), (ℕ × {𝐴}))
6		1zzd 12642	. . . . 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 16655	. . . 4 ⊢ (𝜑 → 𝐺:ℕ⟶𝑋)
10	8, 7	ffvelcdmd 7084	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑋)
11		metcl 24542	. . . . . 6 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ (𝐹‘𝐴) ∈ 𝑋) → (𝐴𝐷(𝐹‘𝐴)) ∈ ℝ)
12	3, 7, 10, 11	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (𝐴𝐷(𝐹‘𝐴)) ∈ ℝ)
13		bfp.4	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
14	12, 13	rerpdivcld 13109	. . . 4 ⊢ (𝜑 → ((𝐴𝐷(𝐹‘𝐴)) / 𝐾) ∈ ℝ)
15		bfp.5	. . . 4 ⊢ (𝜑 → 𝐾 < 1)
16		fveq2 6885	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝐺‘𝑗) = (𝐺‘1))
17		fvoveq1 7442	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝐺‘(𝑗 + 1)) = (𝐺‘(1 + 1)))
18	16, 17	oveq12d 7437	. . . . . . . 8 ⊢ (𝑗 = 1 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) = ((𝐺‘1)𝐷(𝐺‘(1 + 1))))
19		oveq2 7427	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝐾↑𝑗) = (𝐾↑1))
20	19	oveq2d 7435	. . . . . . . 8 ⊢ (𝑗 = 1 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)))
21	18, 20	breq12d 5124	. . . . . . 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 343	. . . . . 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 6885	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐺‘𝑗) = (𝐺‘𝑘))
24		fvoveq1 7442	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐺‘(𝑗 + 1)) = (𝐺‘(𝑘 + 1)))
25	23, 24	oveq12d 7437	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) = ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))))
26		oveq2 7427	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐾↑𝑗) = (𝐾↑𝑘))
27	26	oveq2d 7435	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)))
28	25, 27	breq12d 5124	. . . . . . 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 343	. . . . . 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 6885	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝐺‘𝑗) = (𝐺‘(𝑘 + 1)))
31		fvoveq1 7442	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝐺‘(𝑗 + 1)) = (𝐺‘((𝑘 + 1) + 1)))
32	30, 31	oveq12d 7437	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) = ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))))
33		oveq2 7427	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝐾↑𝑗) = (𝐾↑(𝑘 + 1)))
34	33	oveq2d 7435	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))
35	32, 34	breq12d 5124	. . . . . . 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 343	. . . . . 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 11797	. . . . . . 7 ⊢ (𝜑 → (𝐴𝐷(𝐹‘𝐴)) ≤ (𝐴𝐷(𝐹‘𝐴)))
38	4, 5, 6, 7	algr0 16654	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘1) = 𝐴)
39		1nn 12261	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
40	4, 5, 6, 7, 8	algrp1 16656	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝐺‘(1 + 1)) = (𝐹‘(𝐺‘1)))
41	39, 40	mpan2 704	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘(1 + 1)) = (𝐹‘(𝐺‘1)))
42	38	fveq2d 6889	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝐺‘1)) = (𝐹‘𝐴))
43	41, 42	eqtrd 2800	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘(1 + 1)) = (𝐹‘𝐴))
44	38, 43	oveq12d 7437	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘1)𝐷(𝐺‘(1 + 1))) = (𝐴𝐷(𝐹‘𝐴)))
45	13	rpred 13078	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℝ)
46	45	recnd 11254	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℂ)
47	46	exp1d 14197	. . . . . . . . 9 ⊢ (𝜑 → (𝐾↑1) = 𝐾)
48	47	oveq2d 7435	. . . . . . . 8 ⊢ (𝜑 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · 𝐾))
49	12	recnd 11254	. . . . . . . . 9 ⊢ (𝜑 → (𝐴𝐷(𝐹‘𝐴)) ∈ ℂ)
50	13	rpne0d 13083	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ≠ 0)
51	49, 46, 50	divcan1d 12009	. . . . . . . 8 ⊢ (𝜑 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · 𝐾) = (𝐴𝐷(𝐹‘𝐴)))
52	48, 51	eqtrd 2800	. . . . . . 7 ⊢ (𝜑 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)) = (𝐴𝐷(𝐹‘𝐴)))
53	37, 44, 52	3brtr4d 5145	. . . . . 6 ⊢ (𝜑 → ((𝐺‘1)𝐷(𝐺‘(1 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)))
54	9	ffvelcdmda 7083	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ 𝑋)
55		peano2nn 12262	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
56		ffvelcdm 7080	. . . . . . . . . . . . 13 ⊢ ((𝐺:ℕ⟶𝑋 ∧ (𝑘 + 1) ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ 𝑋)
57	9, 55, 56	syl2an 608	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ 𝑋)
58	54, 57	jca 521	. . . . . . . . . . 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 3210	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))
61	60	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))
62		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐺‘𝑘) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑘)))
63	62	oveq1d 7434	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐺‘𝑘) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) = ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)))
64		oveq1 7426	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐺‘𝑘) → (𝑥𝐷𝑦) = ((𝐺‘𝑘)𝐷𝑦))
65	64	oveq2d 7435	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐺‘𝑘) → (𝐾 · (𝑥𝐷𝑦)) = (𝐾 · ((𝐺‘𝑘)𝐷𝑦)))
66	63, 65	breq12d 5124	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺‘𝑘) → (((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)) ↔ ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)) ≤ (𝐾 · ((𝐺‘𝑘)𝐷𝑦))))
67		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → (𝐹‘𝑦) = (𝐹‘(𝐺‘(𝑘 + 1))))
68	67	oveq2d 7435	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)) = ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))))
69		oveq2 7427	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → ((𝐺‘𝑘)𝐷𝑦) = ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))))
70	69	oveq2d 7435	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → (𝐾 · ((𝐺‘𝑘)𝐷𝑦)) = (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))))
71	68, 70	breq12d 5124	. . . . . . . . . . . 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 3594	. . . . . . . . . . 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 66	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))))
74	3	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐷 ∈ (Met‘𝑋))
75	8	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:𝑋⟶𝑋)
76	75, 54	ffvelcdmd 7084	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝐺‘𝑘)) ∈ 𝑋)
77	75, 57	ffvelcdmd 7084	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝐺‘(𝑘 + 1))) ∈ 𝑋)
78		metcl 24542	. . . . . . . . . . . 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 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ∈ ℝ)
80	45	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐾 ∈ ℝ)
81		metcl 24542	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ (𝐺‘(𝑘 + 1)) ∈ 𝑋) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℝ)
82	74, 54, 57, 81	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℝ)
83	80, 82	remulcld 11256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ∈ ℝ)
84	14	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝐴)) / 𝐾) ∈ ℝ)
85	55	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
86	85	nnnn0d 12582	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ₀)
87	80, 86	reexpcld 14219	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑(𝑘 + 1)) ∈ ℝ)
88	84, 87	remulcld 11256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) ∈ ℝ)
89		letr 11321	. . . . . . . . . . 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 1398	. . . . . . . . . 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 708	. . . . . . . . 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 12528	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
93		reexpcl 14134	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℝ ∧ 𝑘 ∈ ℕ₀) → (𝐾↑𝑘) ∈ ℝ)
94	45, 92, 93	syl2an 608	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑𝑘) ∈ ℝ)
95	84, 94	remulcld 11256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ∈ ℝ)
96	13	rpgt0d 13081	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐾)
97	96	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < 𝐾)
98		lemul1 12084	. . . . . . . . . . 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 1401	. . . . . . . . . 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 11254	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℂ)
101	46	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐾 ∈ ℂ)
102	100, 101	mulcomd 11247	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) · 𝐾) = (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))))
103	84	recnd 11254	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝐴)) / 𝐾) ∈ ℂ)
104	94	recnd 11254	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑𝑘) ∈ ℂ)
105	103, 104, 101	mulassd 11249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · ((𝐾↑𝑘) · 𝐾)))
106		expp1 14124	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐾↑(𝑘 + 1)) = ((𝐾↑𝑘) · 𝐾))
107	46, 92, 106	syl2an 608	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑(𝑘 + 1)) = ((𝐾↑𝑘) · 𝐾))
108	107	oveq2d 7435	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · ((𝐾↑𝑘) · 𝐾)))
109	105, 108	eqtr4d 2803	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))
110	102, 109	breq12d 5124	. . . . . . . . . 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 282	. . . . . . . . 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 16656	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) = (𝐹‘(𝐺‘𝑘)))
113	4, 5, 6, 7, 8	algrp1 16656	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ ℕ) → (𝐺‘((𝑘 + 1) + 1)) = (𝐹‘(𝐺‘(𝑘 + 1))))
114	55, 113	sylan2 605	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘((𝑘 + 1) + 1)) = (𝐹‘(𝐺‘(𝑘 + 1))))
115	112, 114	oveq12d 7437	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) = ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))))
116	115	breq1d 5121	. . . . . . . . 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 297	. . . . . . . 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 419	. . . . . . 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 30	. . . . . 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 12268	. . . . 5 ⊢ (𝑘 ∈ ℕ → (𝜑 → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘))))
121	120	impcom 413	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)))
122	3, 9, 14, 13, 15, 121	geomcau 38470	. . 3 ⊢ (𝜑 → 𝐺 ∈ (Cau‘𝐷))
123		bfp.8	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐷)
124	123	cmetcau 25501	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐺 ∈ (Cau‘𝐷)) → 𝐺 ∈ dom (⇝_𝑡‘𝐽))
125	1, 122, 124	syl2anc 596	. 2 ⊢ (𝜑 → 𝐺 ∈ dom (⇝_𝑡‘𝐽))
126		metxmet 24544	. . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
127	123	methaus 24730	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus)
128	3, 126, 127	3syl 19	. . 3 ⊢ (𝜑 → 𝐽 ∈ Haus)
129		lmfun 23590	. . 3 ⊢ (𝐽 ∈ Haus → Fun (⇝_𝑡‘𝐽))
130		funfvbrb 7050	. . 3 ⊢ (Fun (⇝_𝑡‘𝐽) → (𝐺 ∈ dom (⇝_𝑡‘𝐽) ↔ 𝐺(⇝_𝑡‘𝐽)((⇝_𝑡‘𝐽)‘𝐺)))
131	128, 129, 130	3syl 19	. 2 ⊢ (𝜑 → (𝐺 ∈ dom (⇝_𝑡‘𝐽) ↔ 𝐺(⇝_𝑡‘𝐽)((⇝_𝑡‘𝐽)‘𝐺)))
132	125, 131	mpbid 235	1 ⊢ (𝜑 → 𝐺(⇝_𝑡‘𝐽)((⇝_𝑡‘𝐽)‘𝐺))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ∅c0 4286 {csn 4591 class class class wbr 5111 × cxp 5661 dom cdm 5663 ∘ ccom 5667 Fun wfun 6534 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 1^st c1st 7990 ℂcc 11115 ℝcr 11116 0cc0 11117 1c1 11118 + caddc 11120 · cmul 11122 < clt 11260 ≤ cle 11261 / cdiv 11888 ℕcn 12250 ℕ₀cn0 12521 ℝ⁺crp 13034 seqcseq 14057 ↑cexp 14117 ∞Metcxmet 21559 Metcmet 21560 MetOpencmopn 21564 ⇝_𝑡clm 23435 Hauscha 23517 Cauccau 25465 CMetccmet 25466

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-pm 8833 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-n0 12522 df-z 12609 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-rlim 15566 df-sum 15764 df-rest 17499 df-topgen 17520 df-psmet 21566 df-xmet 21567 df-met 21568 df-bl 21569 df-mopn 21570 df-fbas 21571 df-fg 21572 df-top 23103 df-topon 23120 df-bases 23155 df-ntr 23229 df-nei 23307 df-lm 23438 df-haus 23524 df-fil 24056 df-fm 24148 df-flim 24149 df-flf 24150 df-cfil 25467 df-cau 25468 df-cmet 25469

This theorem is used by: bfplem2 38534

W3C validator