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Theorem iscau4 22980
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 22979 . . . 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 477 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → 𝑗𝑍)
65, 1syl6eleq 2714 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑍) → 𝑗 ∈ (ℤ𝑀))
7 eluzelz 11641 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ𝑀) → 𝑗 ∈ ℤ)
8 uzid 11646 . . . . . . . . . . . . . 14 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
96, 7, 83syl 18 . . . . . . . . . . . . 13 ((𝜑𝑗𝑍) → 𝑗 ∈ (ℤ𝑗))
10 fveq2 6150 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (ℤ𝑘) = (ℤ𝑗))
11 fveq2 6150 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
1211oveq1d 6620 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → ((𝐹𝑘)𝐷(𝐹𝑚)) = ((𝐹𝑗)𝐷(𝐹𝑚)))
1312breq1d 4628 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 ↔ ((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥))
1410, 13raleqbidv 3146 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 ↔ ∀𝑚 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥))
1514rspcv 3296 . . . . . . . . . . . . 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 481 . . . . . . . . . . 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 6150 . . . . . . . . . . . . . . 15 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
1918oveq2d 6621 . . . . . . . . . . . . . 14 (𝑚 = 𝑘 → ((𝐹𝑗)𝐷(𝐹𝑚)) = ((𝐹𝑗)𝐷(𝐹𝑘)))
2019breq1d 4628 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥 ↔ ((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥))
2120cbvralv 3164 . . . . . . . . . . . 12 (∀𝑚 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥)
22 simpr 477 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → (𝐹𝑘) ∈ 𝑋)
2322ralimi 2952 . . . . . . . . . . . . . . 15 (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑋)
2411eleq1d 2688 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → ((𝐹𝑘) ∈ 𝑋 ↔ (𝐹𝑗) ∈ 𝑋))
2524rspcv 3296 . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (𝐹𝑗) ∈ 𝑋)
28 r19.26 3062 . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → 𝐷 ∈ (∞Met‘𝑋))
30 simplr 791 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (𝐹𝑗) ∈ 𝑋)
31 simprr 795 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (𝐹𝑘) ∈ 𝑋)
32 xmetsym 22057 . . . . . . . . . . . . . . . . . . . . 21 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹𝑗) ∈ 𝑋 ∧ (𝐹𝑘) ∈ 𝑋) → ((𝐹𝑗)𝐷(𝐹𝑘)) = ((𝐹𝑘)𝐷(𝐹𝑗)))
3329, 30, 31, 32syl3anc 1323 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → ((𝐹𝑗)𝐷(𝐹𝑘)) = ((𝐹𝑘)𝐷(𝐹𝑗)))
3433breq1d 4628 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥 ↔ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3534biimpd 219 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥 → ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3635expimpd 628 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → (((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥) → ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3736ralimdv 2962 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → (∀𝑘 ∈ (ℤ𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥) → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3828, 37syl5bir 233 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → ((∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥) → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3938expd 452 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥 → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)))
4039impancom 456 . . . . . . . . . . . . 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, 41syl5bi 232 . . . . . . . . . . 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 728 . . . . . . . . 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 3062 . . . . . . . . 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 3062 . . . . . . . . 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 285 . . . . . . . 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 1038 . . . . . . . . 9 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥))
4948ralbii 2979 . . . . . . . 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 1038 . . . . . . . . 9 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
5150ralbii 2979 . . . . . . . 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 285 . . . . . . 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 3016 . . . . . 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 2962 . . . . 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 588 . . . 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 230 . . 3 (𝜑 → (𝐹 ∈ (Cau‘𝐷) → (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))))
57 uzssz 11651 . . . . . . . . 9 (ℤ𝑀) ⊆ ℤ
581, 57eqsstri 3619 . . . . . . . 8 𝑍 ⊆ ℤ
59 ssrexv 3651 . . . . . . . 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 2952 . . . . . 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 592 . . . . 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 22978 . . . . 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, 63syl5ibr 236 . . . 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 202 . 2 (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))))
67 simpl 473 . . . . . . . . 9 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑗𝑍)
681uztrn2 11649 . . . . . . . . 9 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
6967, 68jca 554 . . . . . . . 8 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → (𝑗𝑍𝑘𝑍))
70 iscau4.5 . . . . . . . . . . 11 ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)
7170adantrl 751 . . . . . . . . . 10 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (𝐹𝑘) = 𝐴)
7271eleq1d 2688 . . . . . . . . 9 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝐹𝑘) ∈ 𝑋𝐴𝑋))
73 iscau4.6 . . . . . . . . . . . 12 ((𝜑𝑗𝑍) → (𝐹𝑗) = 𝐵)
7473adantrr 752 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (𝐹𝑗) = 𝐵)
7571, 74oveq12d 6623 . . . . . . . . . 10 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝐹𝑘)𝐷(𝐹𝑗)) = (𝐴𝐷𝐵))
7675breq1d 4628 . . . . . . . . 9 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥 ↔ (𝐴𝐷𝐵) < 𝑥))
7772, 763anbi23d 1399 . . . . . . . 8 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
7869, 77sylan2 491 . . . . . . 7 ((𝜑 ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
7978anassrs 679 . . . . . 6 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8079ralbidva 2984 . . . . 5 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8180rexbidva 3047 . . . 4 (𝜑 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8281ralbidv 2985 . . 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 739 . 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 268 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  wrex 2913  wss 3560   class class class wbr 4618  dom cdm 5079  cfv 5850  (class class class)co 6605  pm cpm 7804  cc 9879   < clt 10019  cz 11322  cuz 11631  +crp 11776  ∞Metcxmt 19645  Caucca 22954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-2 11024  df-z 11323  df-uz 11632  df-rp 11777  df-xneg 11890  df-xadd 11891  df-psmet 19652  df-xmet 19653  df-bl 19655  df-cau 22957
This theorem is referenced by:  iscauf  22981  cmetcaulem  22989  caures  33174  caushft  33175
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