Theorem iscau3 24347

Description: Express the Cauchy sequence property in the more conventional three-quantifier form. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

Hypotheses

Ref	Expression
iscau3.2	⊢ 𝑍 = (ℤ≥‘𝑀)
iscau3.3	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
iscau3.4	⊢ (𝜑 → 𝑀 ∈ ℤ)

Assertion

Ref	Expression
iscau3	⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))))

Distinct variable groups:   𝑗,𝑘,𝑚,𝑥,𝐷   𝑗,𝐹,𝑘,𝑚,𝑥   𝜑,𝑗,𝑘,𝑥   𝑗,𝑋,𝑘,𝑚,𝑥   𝑗,𝑀   𝑗,𝑍,𝑘,𝑥

Allowed substitution hints:   𝜑(𝑚)   𝑀(𝑥,𝑘,𝑚)   𝑍(𝑚)

Proof of Theorem iscau3

Step	Hyp	Ref	Expression
1		iscau3.3	. . 3 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
2		iscau2 24346	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))))
4	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋 ↑pm ℂ)) → 𝐷 ∈ (∞Met‘𝑋))
5		ssid 3939	. . . . . . 7 ⊢ ℤ ⊆ ℤ
6		simpr 484	. . . . . . 7 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) → (𝐹‘𝑘) ∈ 𝑋)
7		eleq1 2826	. . . . . . 7 ⊢ ((𝐹‘𝑘) = (𝐹‘𝑗) → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑗) ∈ 𝑋))
8		eleq1 2826	. . . . . . 7 ⊢ ((𝐹‘𝑘) = (𝐹‘𝑚) → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑚) ∈ 𝑋))
9		xmetsym 23408	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑗) ∈ 𝑋 ∧ (𝐹‘𝑘) ∈ 𝑋) → ((𝐹‘𝑗)𝐷(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(𝐹‘𝑗)))
10	9	fveq2d 6760	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑗) ∈ 𝑋 ∧ (𝐹‘𝑘) ∈ 𝑋) → ( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑘))) = ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))))
11		xmetsym 23408	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑚) ∈ 𝑋 ∧ (𝐹‘𝑗) ∈ 𝑋) → ((𝐹‘𝑚)𝐷(𝐹‘𝑗)) = ((𝐹‘𝑗)𝐷(𝐹‘𝑚)))
12	11	fveq2d 6760	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑚) ∈ 𝑋 ∧ (𝐹‘𝑗) ∈ 𝑋) → ( I ‘((𝐹‘𝑚)𝐷(𝐹‘𝑗))) = ( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))))
13		simp1 1134	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → 𝐷 ∈ (∞Met‘𝑋))
14		simp2l 1197	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → (𝐹‘𝑘) ∈ 𝑋)
15		simp3l 1199	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → (𝐹‘𝑗) ∈ 𝑋)
16		xmetcl 23392	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑗) ∈ 𝑋) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) ∈ ℝ*)
17	13, 14, 15, 16	syl3anc 1369	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) ∈ ℝ*)
18		simp2r 1198	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → (𝐹‘𝑚) ∈ 𝑋)
19		xmetcl 23392	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑗) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) → ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) ∈ ℝ*)
20	13, 15, 18, 19	syl3anc 1369	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) ∈ ℝ*)
21		simp3r 1200	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → 𝑥 ∈ ℝ)
22	21	rehalfcld 12150	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → (𝑥 / 2) ∈ ℝ)
23	22	rexrd 10956	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → (𝑥 / 2) ∈ ℝ*)
24		xlt2add 12923	. . . . . . . . . 10 ⊢ (((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) ∈ ℝ* ∧ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) ∈ ℝ*) ∧ ((𝑥 / 2) ∈ ℝ* ∧ (𝑥 / 2) ∈ ℝ*)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < (𝑥 / 2) ∧ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < (𝑥 / 2)) → (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < ((𝑥 / 2) +𝑒 (𝑥 / 2))))
25	17, 20, 23, 23, 24	syl22anc 835	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < (𝑥 / 2) ∧ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < (𝑥 / 2)) → (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < ((𝑥 / 2) +𝑒 (𝑥 / 2))))
26	22, 22	rexaddd 12897	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝑥 / 2) +𝑒 (𝑥 / 2)) = ((𝑥 / 2) + (𝑥 / 2)))
27	21	recnd 10934	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → 𝑥 ∈ ℂ)
28	27	2halvesd 12149	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝑥 / 2) + (𝑥 / 2)) = 𝑥)
29	26, 28	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝑥 / 2) +𝑒 (𝑥 / 2)) = 𝑥)
30	29	breq2d 5082	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < ((𝑥 / 2) +𝑒 (𝑥 / 2)) ↔ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < 𝑥))
31		xmettri 23412	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋 ∧ (𝐹‘𝑗) ∈ 𝑋)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ≤ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))))
32	13, 14, 18, 15, 31	syl13anc 1370	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ≤ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))))
33		xmetcl 23392	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ∈ ℝ*)
34	13, 14, 18, 33	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ∈ ℝ*)
35	17, 20	xaddcld 12964	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) ∈ ℝ*)
36	21	rexrd 10956	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → 𝑥 ∈ ℝ*)
37		xrlelttr 12819	. . . . . . . . . . . 12 ⊢ ((((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ∈ ℝ* ∧ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) ∈ ℝ* ∧ 𝑥 ∈ ℝ*) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ≤ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) ∧ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < 𝑥) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
38	34, 35, 36, 37	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ≤ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) ∧ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < 𝑥) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
39	32, 38	mpand 691	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < 𝑥 → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
40	30, 39	sylbid 239	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < ((𝑥 / 2) +𝑒 (𝑥 / 2)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
41	25, 40	syld 47	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < (𝑥 / 2) ∧ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < (𝑥 / 2)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
42		ovex 7288	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) ∈ V
43		fvi 6826	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) ∈ V → ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) = ((𝐹‘𝑘)𝐷(𝐹‘𝑗)))
44	42, 43	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) = ((𝐹‘𝑘)𝐷(𝐹‘𝑗))
45	44	breq1i 5077	. . . . . . . . 9 ⊢ (( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < (𝑥 / 2) ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < (𝑥 / 2))
46		ovex 7288	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) ∈ V
47		fvi 6826	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑗)𝐷(𝐹‘𝑚)) ∈ V → ( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))) = ((𝐹‘𝑗)𝐷(𝐹‘𝑚)))
48	46, 47	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))) = ((𝐹‘𝑗)𝐷(𝐹‘𝑚))
49	48	breq1i 5077	. . . . . . . . 9 ⊢ (( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < (𝑥 / 2) ↔ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < (𝑥 / 2))
50	45, 49	anbi12i 626	. . . . . . . 8 ⊢ ((( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < (𝑥 / 2) ∧ ( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < (𝑥 / 2)) ↔ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < (𝑥 / 2) ∧ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < (𝑥 / 2)))
51		ovex 7288	. . . . . . . . . 10 ⊢ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ∈ V
52		fvi 6826	. . . . . . . . . 10 ⊢ (((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ∈ V → ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) = ((𝐹‘𝑘)𝐷(𝐹‘𝑚)))
53	51, 52	ax-mp 5	. . . . . . . . 9 ⊢ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) = ((𝐹‘𝑘)𝐷(𝐹‘𝑚))
54	53	breq1i 5077	. . . . . . . 8 ⊢ (( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥 ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)
55	41, 50, 54	3imtr4g 295	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < (𝑥 / 2) ∧ ( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < (𝑥 / 2)) → ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥))
56	5, 6, 7, 8, 10, 12, 55	cau3lem 14994	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥)))
57	4, 56	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋 ↑pm ℂ)) → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥)))
58	44	breq1i 5077	. . . . . . . . . 10 ⊢ (( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥 ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)
59	58	anbi2i 622	. . . . . . . . 9 ⊢ (((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
60		df-3an 1087	. . . . . . . . 9 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
61	59, 60	bitr4i 277	. . . . . . . 8 ⊢ (((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
62	61	ralbii 3090	. . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
63	62	rexbii 3177	. . . . . 6 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
64	63	ralbii 3090	. . . . 5 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
65	54	ralbii 3090	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ (ℤ≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)
66	65	anbi2i 622	. . . . . . . . 9 ⊢ (((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
67		df-3an 1087	. . . . . . . . 9 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
68	66, 67	bitr4i 277	. . . . . . . 8 ⊢ (((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥) ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
69	68	ralbii 3090	. . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
70	69	rexbii 3177	. . . . . 6 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
71	70	ralbii 3090	. . . . 5 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
72	57, 64, 71	3bitr3g 312	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋 ↑pm ℂ)) → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
73		iscau3.4	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
74	73	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋 ↑pm ℂ)) → 𝑀 ∈ ℤ)
75		iscau3.2	. . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀)
76	75	rexuz3 14988	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
77	74, 76	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋 ↑pm ℂ)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
78	77	ralbidv 3120	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋 ↑pm ℂ)) → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
79	72, 78	bitr4d 281	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋 ↑pm ℂ)) → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
80	79	pm5.32da 578	. 2 ⊢ (𝜑 → ((𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))))
81	3, 80	bitrd 278	1 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   class class class wbr 5070   I cid 5479  dom cdm 5580  ‘cfv 6418  (class class class)co 7255   ↑_pm cpm 8574  ℂcc 10800  ℝcr 10801   + caddc 10805  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941   / cdiv 11562  2c2 11958  ℤcz 12249  ℤ_≥cuz 12511  ℝ⁺crp 12659   +_𝑒 cxad 12775  ∞Metcxmet 20495  Cauccau 24322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-2 11966  df-z 12250  df-uz 12512  df-rp 12660  df-xneg 12777  df-xadd 12778  df-psmet 20502  df-xmet 20503  df-bl 20505  df-cau 24325

This theorem is referenced by:  iscau4  24348  caucfil  24352  cmetcaulem  24357  heibor1lem  35894