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Theorem iscau4 23565
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 23564 . . . 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 485 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → 𝑗𝑍)
65, 1syl6eleq 2892 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑍) → 𝑗 ∈ (ℤ𝑀))
7 eluzelz 12103 . . . . . . . . . . . . . 14 (𝑗 ∈ (ℤ𝑀) → 𝑗 ∈ ℤ)
8 uzid 12108 . . . . . . . . . . . . . 14 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
96, 7, 83syl 18 . . . . . . . . . . . . 13 ((𝜑𝑗𝑍) → 𝑗 ∈ (ℤ𝑗))
10 fveq2 6541 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (ℤ𝑘) = (ℤ𝑗))
11 fveq2 6541 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
1211oveq1d 7034 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → ((𝐹𝑘)𝐷(𝐹𝑚)) = ((𝐹𝑗)𝐷(𝐹𝑚)))
1312breq1d 4974 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 ↔ ((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥))
1410, 13raleqbidv 3360 . . . . . . . . . . . . . 14 (𝑘 = 𝑗 → (∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥 ↔ ∀𝑚 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥))
1514rspcv 3553 . . . . . . . . . . . . 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 6541 . . . . . . . . . . . . . . 15 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
1918oveq2d 7035 . . . . . . . . . . . . . 14 (𝑚 = 𝑘 → ((𝐹𝑗)𝐷(𝐹𝑚)) = ((𝐹𝑗)𝐷(𝐹𝑘)))
2019breq1d 4974 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥 ↔ ((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥))
2120cbvralv 3402 . . . . . . . . . . . 12 (∀𝑚 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑚)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥)
22 simpr 485 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → (𝐹𝑘) ∈ 𝑋)
2322ralimi 3126 . . . . . . . . . . . . . . 15 (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑋)
2411eleq1d 2866 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → ((𝐹𝑘) ∈ 𝑋 ↔ (𝐹𝑗) ∈ 𝑋))
2524rspcv 3553 . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (𝐹𝑗) ∈ 𝑋)
28 r19.26 3136 . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → 𝐷 ∈ (∞Met‘𝑋))
30 simplr 765 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (𝐹𝑗) ∈ 𝑋)
31 simprr 769 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (𝐹𝑘) ∈ 𝑋)
32 xmetsym 22640 . . . . . . . . . . . . . . . . . . . . 21 ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹𝑗) ∈ 𝑋 ∧ (𝐹𝑘) ∈ 𝑋) → ((𝐹𝑗)𝐷(𝐹𝑘)) = ((𝐹𝑘)𝐷(𝐹𝑗)))
3329, 30, 31, 32syl3anc 1364 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → ((𝐹𝑗)𝐷(𝐹𝑘)) = ((𝐹𝑘)𝐷(𝐹𝑗)))
3433breq1d 4974 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥 ↔ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3534biimpd 230 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋)) → (((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥 → ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3635expimpd 454 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → (((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥) → ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3736ralimdv 3144 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → (∀𝑘 ∈ (ℤ𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥) → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3828, 37syl5bir 244 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → ((∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥) → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
3938expd 416 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ (𝐹𝑗) ∈ 𝑋) → (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑗)𝐷(𝐹𝑘)) < 𝑥 → ∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥)))
4039impancom 452 . . . . . . . . . . . . 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 243 . . . . . . . . . . 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 572 . . . . . . . . 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 3136 . . . . . . . . 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 3136 . . . . . . . . 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 297 . . . . . . . 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 1082 . . . . . . . . 9 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥))
4948ralbii 3131 . . . . . . . 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 1082 . . . . . . . . 9 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋) ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))
5150ralbii 3131 . . . . . . . 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 297 . . . . . . 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 3236 . . . . . 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 3144 . . . . 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 611 . . . 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 241 . . 3 (𝜑 → (𝐹 ∈ (Cau‘𝐷) → (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))))
57 uzssz 12113 . . . . . . . . 9 (ℤ𝑀) ⊆ ℤ
581, 57eqsstri 3924 . . . . . . . 8 𝑍 ⊆ ℤ
59 ssrexv 3957 . . . . . . . 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 3126 . . . . . 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 616 . . . . 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 23563 . . . . 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 247 . . . 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 213 . 2 (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))))
67 simpl 483 . . . . . . . . 9 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑗𝑍)
681uztrn2 12111 . . . . . . . . 9 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
6967, 68jca 512 . . . . . . . 8 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → (𝑗𝑍𝑘𝑍))
70 iscau4.5 . . . . . . . . . . 11 ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)
7170adantrl 712 . . . . . . . . . 10 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (𝐹𝑘) = 𝐴)
7271eleq1d 2866 . . . . . . . . 9 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝐹𝑘) ∈ 𝑋𝐴𝑋))
73 iscau4.6 . . . . . . . . . . . 12 ((𝜑𝑗𝑍) → (𝐹𝑗) = 𝐵)
7473adantrr 713 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (𝐹𝑗) = 𝐵)
7571, 74oveq12d 7037 . . . . . . . . . 10 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝐹𝑘)𝐷(𝐹𝑗)) = (𝐴𝐷𝐵))
7675breq1d 4974 . . . . . . . . 9 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → (((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥 ↔ (𝐴𝐷𝐵) < 𝑥))
7772, 763anbi23d 1431 . . . . . . . 8 ((𝜑 ∧ (𝑗𝑍𝑘𝑍)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
7869, 77sylan2 592 . . . . . . 7 ((𝜑 ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
7978anassrs 468 . . . . . 6 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8079ralbidva 3162 . . . . 5 ((𝜑𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8180rexbidva 3258 . . . 4 (𝜑 → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥) ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
8281ralbidv 3163 . . 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 628 . 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 280 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 207  wa 396  w3a 1080   = wceq 1522  wcel 2080  wral 3104  wrex 3105  wss 3861   class class class wbr 4964  dom cdm 5446  cfv 6228  (class class class)co 7019  pm cpm 8260  cc 10384   < clt 10524  cz 11831  cuz 12093  +crp 12239  ∞Metcxmet 20212  Cauccau 23539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322  ax-cnex 10442  ax-resscn 10443  ax-1cn 10444  ax-icn 10445  ax-addcl 10446  ax-addrcl 10447  ax-mulcl 10448  ax-mulrcl 10449  ax-mulcom 10450  ax-addass 10451  ax-mulass 10452  ax-distr 10453  ax-i2m1 10454  ax-1ne0 10455  ax-1rid 10456  ax-rnegex 10457  ax-rrecex 10458  ax-cnre 10459  ax-pre-lttri 10460  ax-pre-lttrn 10461  ax-pre-ltadd 10462  ax-pre-mulgt0 10463
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-nel 3090  df-ral 3109  df-rex 3110  df-reu 3111  df-rmo 3112  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-po 5365  df-so 5366  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-riota 6980  df-ov 7022  df-oprab 7023  df-mpo 7024  df-1st 7548  df-2nd 7549  df-er 8142  df-map 8261  df-pm 8262  df-en 8361  df-dom 8362  df-sdom 8363  df-pnf 10526  df-mnf 10527  df-xr 10528  df-ltxr 10529  df-le 10530  df-sub 10721  df-neg 10722  df-div 11148  df-2 11550  df-z 11832  df-uz 12094  df-rp 12240  df-xneg 12357  df-xadd 12358  df-psmet 20219  df-xmet 20220  df-bl 20222  df-cau 23542
This theorem is referenced by:  iscauf  23566  cmetcaulem  23574  caures  34580  caushft  34581
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