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Theorem iscau4 25326
Description: Express the property "𝐹 is a Cauchy sequence of metric 𝐷 " using an arbitrary upper set of integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
Hypotheses
Ref Expression
iscau3.2 𝑍 = (ℤ𝑀)
iscau3.3 (𝜑𝐷 ∈ (∞Met‘𝑋))
iscau3.4 (𝜑𝑀 ∈ ℤ)
iscau4.5 ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)
iscau4.6 ((𝜑𝑗𝑍) → (𝐹𝑗) = 𝐵)
Assertion
Ref Expression
iscau4 (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥))))
Distinct variable groups:   𝑗,𝑘,𝑥,𝐷   𝑗,𝐹,𝑘,𝑥   𝜑,𝑗,𝑘,𝑥   𝑗,𝑋,𝑘,𝑥   𝑗,𝑀   𝑗,𝑍,𝑘,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑗,𝑘)   𝐵(𝑥,𝑗,𝑘)   𝑀(𝑥,𝑘)

Proof of Theorem iscau4
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 iscau3.2 . . . . 5 𝑍 = (ℤ𝑀)
2 iscau3.3 . . . . 5 (𝜑𝐷 ∈ (∞Met‘𝑋))
3 iscau3.4 . . . . 5 (𝜑𝑀 ∈ ℤ)
41, 2, 3iscau3 25325 . . . 4 (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥))))
5 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → 𝑗𝑍)
65, 1eleqtrdi 2848 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑍) → 𝑗 ∈ (ℤ𝑀))
7 eluzelz 12885 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ𝑀) → 𝑗 ∈ ℤ)
8 uzid 12890 . . . . . . . . . . . . . 14 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
96, 7, 83syl 18 . . . . . . . . . . . . 13 ((𝜑𝑗𝑍) → 𝑗 ∈ (ℤ𝑗))
10 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (ℤ𝑘) = (ℤ𝑗))
11 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
1211oveq1d 7445 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → ((𝐹𝑘)𝐷(𝐹𝑚)) = ((𝐹𝑗)𝐷(𝐹𝑚)))
1312breq1d 5157 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 ↔ ((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥))
1410, 13raleqbidv 3343 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 ↔ ∀𝑚 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥))
1514rspcv 3617 . . . . . . . . . . . . 13 (𝑗 ∈ (ℤ𝑗) → (∀𝑘 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 → ∀𝑚 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥))
169, 15syl 17 . . . . . . . . . . . 12 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 → ∀𝑚 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥))
1716adantr 480 . . . . . . . . . . 11 (((𝜑𝑗𝑍) ∧ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (∀𝑘 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 → ∀𝑚 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥))
18 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
1918oveq2d 7446 . . . . . . . . . . . . . 14 (𝑚 = 𝑘 → ((𝐹𝑗)𝐷(𝐹𝑚)) = ((𝐹𝑗)𝐷(𝐹𝑘)))
2019breq1d 5157 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥 ↔ ((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥))
2120cbvralvw 3234 . . . . . . . . . . . 12 (∀𝑚 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥)
22 simpr 484 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → (𝐹𝑘) ∈ 𝑋)
2322ralimi 3080 . . . . . . . . . . . . . . 15 (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑋)
2411eleq1d 2823 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → ((𝐹𝑘) ∈ 𝑋 ↔ (𝐹𝑗) ∈ 𝑋))
2524rspcv 3617 . . . . . . . . . . . . . . 15 (𝑗 ∈ (ℤ𝑗) → (∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑋 → (𝐹𝑗) ∈ 𝑋))
269, 23, 25syl2im 40 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → (𝐹𝑗) ∈ 𝑋))
2726imp 406 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (𝐹𝑗) ∈ 𝑋)
28 r19.26 3108 . . . . . . . . . . . . . . . 16 (∀𝑘 ∈ (ℤ𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥) ↔ (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥))
292ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → 𝐷 ∈ (∞Met‘𝑋))
30 simplr 769 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (𝐹𝑗) ∈ 𝑋)
31 simprr 773 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (𝐹𝑘) ∈ 𝑋)
32 xmetsym 24372 . . . . . . . . . . . . . . . . . . . . 21 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹𝑗) ∈ 𝑋 ∧ (𝐹𝑘) ∈ 𝑋) → ((𝐹𝑗)𝐷(𝐹𝑘)) = ((𝐹𝑘)𝐷(𝐹𝑗)))
3329, 30, 31, 32syl3anc 1370 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → ((𝐹𝑗)𝐷(𝐹𝑘)) = ((𝐹𝑘)𝐷(𝐹𝑗)))
3433breq1d 5157 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥 ↔ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3534biimpd 229 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥 → ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3635expimpd 453 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → (((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥) → ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3736ralimdv 3166 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → (∀𝑘 ∈ (ℤ𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥) → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3828, 37biimtrrid 243 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → ((∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥) → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3938expd 415 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥 → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)))
4039impancom 451 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → ((𝐹𝑗) ∈ 𝑋 → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥 → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)))
4127, 40mpd 15 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥 → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
4221, 41biimtrid 242 . . . . . . . . . . 11 (((𝜑𝑗𝑍) ∧ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (∀𝑚 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥 → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
4317, 42syld 47 . . . . . . . . . 10 (((𝜑𝑗𝑍) ∧ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (∀𝑘 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
4443imdistanda 571 . . . . . . . . 9 ((𝜑𝑗𝑍) → ((∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥) → (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)))
45 r19.26 3108 . . . . . . . . 9 (∀𝑘 ∈ (ℤ𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥) ↔ (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥))
46 r19.26 3108 . . . . . . . . 9 (∀𝑘 ∈ (ℤ𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
4744, 45, 463imtr4g 296 . . . . . . . 8 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥) → ∀𝑘 ∈ (ℤ𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)))
48 df-3an 1088 . . . . . . . . 9 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥))
4948ralbii 3090 . . . . . . . 8 (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥))
50 df-3an 1088 . . . . . . . . 9 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
5150ralbii 3090 . . . . . . . 8 (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
5247, 49, 513imtr4g 296 . . . . . . 7 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥) → ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)))
5352reximdva 3165 . . . . . 6 (𝜑 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)))
5453ralimdv 3166 . . . . 5 (𝜑 → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥) → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)))
5554anim2d 612 . . . 4 (𝜑 → ((𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥)) → (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))))
564, 55sylbid 240 . . 3 (𝜑 → (𝐹 ∈ (Cau‘𝐷) → (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))))
57 uzssz 12896 . . . . . . . . 9 (ℤ𝑀) ⊆ ℤ
581, 57eqsstri 4029 . . . . . . . 8 𝑍 ⊆ ℤ
59 ssrexv 4064 . . . . . . . 8 (𝑍 ⊆ ℤ → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)))
6058, 59ax-mp 5 . . . . . . 7 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
6160ralimi 3080 . . . . . 6 (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) → ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
6261anim2i 617 . . . . 5 ((𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)) → (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)))
63 iscau2 25324 . . . . 5 (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))))
6462, 63imbitrrid 246 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → ((𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)) → 𝐹 ∈ (Cau‘𝐷)))
652, 64syl 17 . . 3 (𝜑 → ((𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)) → 𝐹 ∈ (Cau‘𝐷)))
6656, 65impbid 212 . 2 (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))))
67 simpl 482 . . . . . . . . 9 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑗𝑍)
681uztrn2 12894 . . . . . . . . 9 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
6967, 68jca 511 . . . . . . . 8 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → (𝑗𝑍𝑘𝑍))
70 iscau4.5 . . . . . . . . . . 11 ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)
7170adantrl 716 . . . . . . . . . 10 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (𝐹𝑘) = 𝐴)
7271eleq1d 2823 . . . . . . . . 9 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝐹𝑘) ∈ 𝑋𝐴𝑋))
73 iscau4.6 . . . . . . . . . . . 12 ((𝜑𝑗𝑍) → (𝐹𝑗) = 𝐵)
7473adantrr 717 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (𝐹𝑗) = 𝐵)
7571, 74oveq12d 7448 . . . . . . . . . 10 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝐹𝑘)𝐷(𝐹𝑗)) = (𝐴𝐷𝐵))
7675breq1d 5157 . . . . . . . . 9 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥 ↔ (𝐴𝐷𝐵) < 𝑥))
7772, 763anbi23d 1438 . . . . . . . 8 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
7869, 77sylan2 593 . . . . . . 7 ((𝜑 ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
7978anassrs 467 . . . . . 6 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8079ralbidva 3173 . . . . 5 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8180rexbidva 3174 . . . 4 (𝜑 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8281ralbidv 3175 . . 3 (𝜑 → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8382anbi2d 630 . 2 (𝜑 → ((𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥))))
8466, 83bitrd 279 1 (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  wss 3962   class class class wbr 5147  dom cdm 5688  cfv 6562  (class class class)co 7430  pm cpm 8865  cc 11150   < clt 11292  cz 12610  cuz 12875  +crp 13031  ∞Metcxmet 21366  Cauccau 25300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-2 12326  df-z 12611  df-uz 12876  df-rp 13032  df-xneg 13151  df-xadd 13152  df-psmet 21373  df-xmet 21374  df-bl 21376  df-cau 25303
This theorem is referenced by:  iscauf  25327  cmetcaulem  25335  caures  37746  caushft  37747
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