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Theorem iscau4 25313
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 25312 . . . 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 2851 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑍) → 𝑗 ∈ (ℤ𝑀))
7 eluzelz 12888 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ𝑀) → 𝑗 ∈ ℤ)
8 uzid 12893 . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → ((𝐹𝑘)𝐷(𝐹𝑚)) = ((𝐹𝑗)𝐷(𝐹𝑚)))
1312breq1d 5153 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 ↔ ((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥))
1410, 13raleqbidv 3346 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 ↔ ∀𝑚 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥))
1514rspcv 3618 . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . 14 (𝑚 = 𝑘 → ((𝐹𝑗)𝐷(𝐹𝑚)) = ((𝐹𝑗)𝐷(𝐹𝑘)))
2019breq1d 5153 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥 ↔ ((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥))
2120cbvralvw 3237 . . . . . . . . . . . 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 3083 . . . . . . . . . . . . . . 15 (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑋)
2411eleq1d 2826 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → ((𝐹𝑘) ∈ 𝑋 ↔ (𝐹𝑗) ∈ 𝑋))
2524rspcv 3618 . . . . . . . . . . . . . . 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 3111 . . . . . . . . . . . . . . . 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 24357 . . . . . . . . . . . . . . . . . . . . 21 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹𝑗) ∈ 𝑋 ∧ (𝐹𝑘) ∈ 𝑋) → ((𝐹𝑗)𝐷(𝐹𝑘)) = ((𝐹𝑘)𝐷(𝐹𝑗)))
3329, 30, 31, 32syl3anc 1373 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → ((𝐹𝑗)𝐷(𝐹𝑘)) = ((𝐹𝑘)𝐷(𝐹𝑗)))
3433breq1d 5153 . . . . . . . . . . . . . . . . . . 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 3169 . . . . . . . . . . . . . . . 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 3111 . . . . . . . . 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 3111 . . . . . . . . 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 1089 . . . . . . . . 9 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥))
4948ralbii 3093 . . . . . . . 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 1089 . . . . . . . . 9 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
5150ralbii 3093 . . . . . . . 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 3168 . . . . . 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 3169 . . . . 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 12899 . . . . . . . . 9 (ℤ𝑀) ⊆ ℤ
581, 57eqsstri 4030 . . . . . . . 8 𝑍 ⊆ ℤ
59 ssrexv 4053 . . . . . . . 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 3083 . . . . . 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 25311 . . . . 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 12897 . . . . . . . . 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 2826 . . . . . . . . 9 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝐹𝑘) ∈ 𝑋𝐴𝑋))
73 iscau4.6 . . . . . . . . . . . 12 ((𝜑𝑗𝑍) → (𝐹𝑗) = 𝐵)
7473adantrr 717 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (𝐹𝑗) = 𝐵)
7571, 74oveq12d 7449 . . . . . . . . . 10 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝐹𝑘)𝐷(𝐹𝑗)) = (𝐴𝐷𝐵))
7675breq1d 5153 . . . . . . . . 9 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥 ↔ (𝐴𝐷𝐵) < 𝑥))
7772, 763anbi23d 1441 . . . . . . . 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 3176 . . . . 5 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8180rexbidva 3177 . . . 4 (𝜑 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8281ralbidv 3178 . . 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 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951   class class class wbr 5143  dom cdm 5685  cfv 6561  (class class class)co 7431  pm cpm 8867  cc 11153   < clt 11295  cz 12613  cuz 12878  +crp 13034  ∞Metcxmet 21349  Cauccau 25287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-2 12329  df-z 12614  df-uz 12879  df-rp 13035  df-xneg 13154  df-xadd 13155  df-psmet 21356  df-xmet 21357  df-bl 21359  df-cau 25290
This theorem is referenced by:  iscauf  25314  cmetcaulem  25322  caures  37767  caushft  37768
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