Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscau4 Structured version   Visualization version   GIF version

Theorem iscau4 23277
 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 23276 . . . 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 479 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → 𝑗𝑍)
65, 1syl6eleq 2849 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑍) → 𝑗 ∈ (ℤ𝑀))
7 eluzelz 11889 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ𝑀) → 𝑗 ∈ ℤ)
8 uzid 11894 . . . . . . . . . . . . . 14 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
96, 7, 83syl 18 . . . . . . . . . . . . 13 ((𝜑𝑗𝑍) → 𝑗 ∈ (ℤ𝑗))
10 fveq2 6352 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (ℤ𝑘) = (ℤ𝑗))
11 fveq2 6352 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
1211oveq1d 6828 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → ((𝐹𝑘)𝐷(𝐹𝑚)) = ((𝐹𝑗)𝐷(𝐹𝑚)))
1312breq1d 4814 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 ↔ ((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥))
1410, 13raleqbidv 3291 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 ↔ ∀𝑚 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥))
1514rspcv 3445 . . . . . . . . . . . . 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 472 . . . . . . . . . . 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 6352 . . . . . . . . . . . . . . 15 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
1918oveq2d 6829 . . . . . . . . . . . . . 14 (𝑚 = 𝑘 → ((𝐹𝑗)𝐷(𝐹𝑚)) = ((𝐹𝑗)𝐷(𝐹𝑘)))
2019breq1d 4814 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥 ↔ ((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥))
2120cbvralv 3310 . . . . . . . . . . . 12 (∀𝑚 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥)
22 simpr 479 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → (𝐹𝑘) ∈ 𝑋)
2322ralimi 3090 . . . . . . . . . . . . . . 15 (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑋)
2411eleq1d 2824 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → ((𝐹𝑘) ∈ 𝑋 ↔ (𝐹𝑗) ∈ 𝑋))
2524rspcv 3445 . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (𝐹𝑗) ∈ 𝑋)
28 r19.26 3202 . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → 𝐷 ∈ (∞Met‘𝑋))
30 simplr 809 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (𝐹𝑗) ∈ 𝑋)
31 simprr 813 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (𝐹𝑘) ∈ 𝑋)
32 xmetsym 22353 . . . . . . . . . . . . . . . . . . . . 21 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹𝑗) ∈ 𝑋 ∧ (𝐹𝑘) ∈ 𝑋) → ((𝐹𝑗)𝐷(𝐹𝑘)) = ((𝐹𝑘)𝐷(𝐹𝑗)))
3329, 30, 31, 32syl3anc 1477 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → ((𝐹𝑗)𝐷(𝐹𝑘)) = ((𝐹𝑘)𝐷(𝐹𝑗)))
3433breq1d 4814 . . . . . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → (((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥) → ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3736ralimdv 3101 . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥 → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)))
4039impancom 455 . . . . . . . . . . . . 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 731 . . . . . . . . 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 3202 . . . . . . . . 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 3202 . . . . . . . . 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 1074 . . . . . . . . 9 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥))
4948ralbii 3118 . . . . . . . 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 1074 . . . . . . . . 9 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
5150ralbii 3118 . . . . . . . 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 3155 . . . . . 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 3101 . . . . 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 590 . . . 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 11899 . . . . . . . . 9 (ℤ𝑀) ⊆ ℤ
581, 57eqsstri 3776 . . . . . . . 8 𝑍 ⊆ ℤ
59 ssrexv 3808 . . . . . . . 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 3090 . . . . . 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 594 . . . . 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 23275 . . . . 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 474 . . . . . . . . 9 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑗𝑍)
681uztrn2 11897 . . . . . . . . 9 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
6967, 68jca 555 . . . . . . . 8 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → (𝑗𝑍𝑘𝑍))
70 iscau4.5 . . . . . . . . . . 11 ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)
7170adantrl 754 . . . . . . . . . 10 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (𝐹𝑘) = 𝐴)
7271eleq1d 2824 . . . . . . . . 9 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝐹𝑘) ∈ 𝑋𝐴𝑋))
73 iscau4.6 . . . . . . . . . . . 12 ((𝜑𝑗𝑍) → (𝐹𝑗) = 𝐵)
7473adantrr 755 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (𝐹𝑗) = 𝐵)
7571, 74oveq12d 6831 . . . . . . . . . 10 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝐹𝑘)𝐷(𝐹𝑗)) = (𝐴𝐷𝐵))
7675breq1d 4814 . . . . . . . . 9 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥 ↔ (𝐴𝐷𝐵) < 𝑥))
7772, 763anbi23d 1551 . . . . . . . 8 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
7869, 77sylan2 492 . . . . . . 7 ((𝜑 ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
7978anassrs 683 . . . . . 6 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8079ralbidva 3123 . . . . 5 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8180rexbidva 3187 . . . 4 (𝜑 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8281ralbidv 3124 . . 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 742 . 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 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   ⊆ wss 3715   class class class wbr 4804  dom cdm 5266  ‘cfv 6049  (class class class)co 6813   ↑pm cpm 8024  ℂcc 10126   < clt 10266  ℤcz 11569  ℤ≥cuz 11879  ℝ+crp 12025  ∞Metcxmt 19933  Caucca 23251 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-2 11271  df-z 11570  df-uz 11880  df-rp 12026  df-xneg 12139  df-xadd 12140  df-psmet 19940  df-xmet 19941  df-bl 19943  df-cau 23254 This theorem is referenced by:  iscauf  23278  cmetcaulem  23286  caures  33869  caushft  33870
 Copyright terms: Public domain W3C validator