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Theorem cfilucfil 24587
Description: Given a metric 𝐷 and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 25312. (Contributed by Thierry Arnoux, 29-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
metust.1 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
Assertion
Ref Expression
cfilucfil ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
Distinct variable groups:   𝐷,𝑎   𝑋,𝑎   𝐹,𝑎,𝑥   𝑥,𝐷,𝑦   𝑥,𝐹,𝑦   𝑥,𝑋,𝑦,𝑎   𝑦,𝐷   𝐶,𝑎,𝑥,𝑦

Proof of Theorem cfilucfil
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metust.1 . . . . 5 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
21metust 24586 . . . 4 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
3 cfilufbas 24313 . . . 4 ((((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) → 𝐶 ∈ (fBas‘𝑋))
42, 3sylan 580 . . 3 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) → 𝐶 ∈ (fBas‘𝑋))
5 simpllr 776 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋))
6 psmetf 24331 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
7 ffun 6739 . . . . . 6 (𝐷:(𝑋 × 𝑋)⟶ℝ* → Fun 𝐷)
85, 6, 73syl 18 . . . . 5 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → Fun 𝐷)
92ad2antrr 726 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
10 simplr 769 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹)))
111metustfbas 24585 . . . . . . . 8 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
1211ad2antrr 726 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
13 cnvimass 6101 . . . . . . . 8 (𝐷 “ (0[,)𝑥)) ⊆ dom 𝐷
14 fdm 6745 . . . . . . . . 9 (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
155, 6, 143syl 18 . . . . . . . 8 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → dom 𝐷 = (𝑋 × 𝑋))
1613, 15sseqtrid 4047 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → (𝐷 “ (0[,)𝑥)) ⊆ (𝑋 × 𝑋))
17 simpr 484 . . . . . . . . . . 11 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
1817rphalfcld 13086 . . . . . . . . . 10 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → (𝑥 / 2) ∈ ℝ+)
19 eqidd 2735 . . . . . . . . . 10 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → (𝐷 “ (0[,)(𝑥 / 2))) = (𝐷 “ (0[,)(𝑥 / 2))))
20 oveq2 7438 . . . . . . . . . . . 12 (𝑎 = (𝑥 / 2) → (0[,)𝑎) = (0[,)(𝑥 / 2)))
2120imaeq2d 6079 . . . . . . . . . . 11 (𝑎 = (𝑥 / 2) → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)(𝑥 / 2))))
2221rspceeqv 3644 . . . . . . . . . 10 (((𝑥 / 2) ∈ ℝ+ ∧ (𝐷 “ (0[,)(𝑥 / 2))) = (𝐷 “ (0[,)(𝑥 / 2)))) → ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)(𝑥 / 2))) = (𝐷 “ (0[,)𝑎)))
2318, 19, 22syl2anc 584 . . . . . . . . 9 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)(𝑥 / 2))) = (𝐷 “ (0[,)𝑎)))
241metustel 24578 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → ((𝐷 “ (0[,)(𝑥 / 2))) ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)(𝑥 / 2))) = (𝐷 “ (0[,)𝑎))))
2524biimpar 477 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)(𝑥 / 2))) = (𝐷 “ (0[,)𝑎))) → (𝐷 “ (0[,)(𝑥 / 2))) ∈ 𝐹)
265, 23, 25syl2anc 584 . . . . . . . 8 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → (𝐷 “ (0[,)(𝑥 / 2))) ∈ 𝐹)
27 0xr 11305 . . . . . . . . . . 11 0 ∈ ℝ*
2827a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → 0 ∈ ℝ*)
29 rpxr 13041 . . . . . . . . . 10 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
30 0le0 12364 . . . . . . . . . . 11 0 ≤ 0
3130a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → 0 ≤ 0)
32 rpre 13040 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
3332rehalfcld 12510 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ)
34 rphalflt 13061 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑥 / 2) < 𝑥)
3533, 32, 34ltled 11406 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (𝑥 / 2) ≤ 𝑥)
36 icossico 13453 . . . . . . . . . 10 (((0 ∈ ℝ*𝑥 ∈ ℝ*) ∧ (0 ≤ 0 ∧ (𝑥 / 2) ≤ 𝑥)) → (0[,)(𝑥 / 2)) ⊆ (0[,)𝑥))
3728, 29, 31, 35, 36syl22anc 839 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (0[,)(𝑥 / 2)) ⊆ (0[,)𝑥))
38 imass2 6122 . . . . . . . . 9 ((0[,)(𝑥 / 2)) ⊆ (0[,)𝑥) → (𝐷 “ (0[,)(𝑥 / 2))) ⊆ (𝐷 “ (0[,)𝑥)))
3917, 37, 383syl 18 . . . . . . . 8 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → (𝐷 “ (0[,)(𝑥 / 2))) ⊆ (𝐷 “ (0[,)𝑥)))
40 sseq1 4020 . . . . . . . . 9 (𝑤 = (𝐷 “ (0[,)(𝑥 / 2))) → (𝑤 ⊆ (𝐷 “ (0[,)𝑥)) ↔ (𝐷 “ (0[,)(𝑥 / 2))) ⊆ (𝐷 “ (0[,)𝑥))))
4140rspcev 3621 . . . . . . . 8 (((𝐷 “ (0[,)(𝑥 / 2))) ∈ 𝐹 ∧ (𝐷 “ (0[,)(𝑥 / 2))) ⊆ (𝐷 “ (0[,)𝑥))) → ∃𝑤𝐹 𝑤 ⊆ (𝐷 “ (0[,)𝑥)))
4226, 39, 41syl2anc 584 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → ∃𝑤𝐹 𝑤 ⊆ (𝐷 “ (0[,)𝑥)))
43 elfg 23894 . . . . . . . 8 (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → ((𝐷 “ (0[,)𝑥)) ∈ ((𝑋 × 𝑋)filGen𝐹) ↔ ((𝐷 “ (0[,)𝑥)) ⊆ (𝑋 × 𝑋) ∧ ∃𝑤𝐹 𝑤 ⊆ (𝐷 “ (0[,)𝑥)))))
4443biimpar 477 . . . . . . 7 ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ ((𝐷 “ (0[,)𝑥)) ⊆ (𝑋 × 𝑋) ∧ ∃𝑤𝐹 𝑤 ⊆ (𝐷 “ (0[,)𝑥)))) → (𝐷 “ (0[,)𝑥)) ∈ ((𝑋 × 𝑋)filGen𝐹))
4512, 16, 42, 44syl12anc 837 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → (𝐷 “ (0[,)𝑥)) ∈ ((𝑋 × 𝑋)filGen𝐹))
46 cfiluexsm 24314 . . . . . 6 ((((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹)) ∧ (𝐷 “ (0[,)𝑥)) ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑦𝐶 (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑥)))
479, 10, 45, 46syl3anc 1370 . . . . 5 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → ∃𝑦𝐶 (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑥)))
48 funimass2 6650 . . . . . . 7 ((Fun 𝐷 ∧ (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑥))) → (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
4948ex 412 . . . . . 6 (Fun 𝐷 → ((𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑥)) → (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
5049reximdv 3167 . . . . 5 (Fun 𝐷 → (∃𝑦𝐶 (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑥)) → ∃𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
518, 47, 50sylc 65 . . . 4 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → ∃𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
5251ralrimiva 3143 . . 3 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) → ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
534, 52jca 511 . 2 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) → (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
54 simprl 771 . . 3 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → 𝐶 ∈ (fBas‘𝑋))
55 oveq2 7438 . . . . . . . . . 10 (𝑥 = 𝑎 → (0[,)𝑥) = (0[,)𝑎))
5655sseq2d 4027 . . . . . . . . 9 (𝑥 = 𝑎 → ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎)))
5756rexbidv 3176 . . . . . . . 8 (𝑥 = 𝑎 → (∃𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∃𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎)))
58 simp-4r 784 . . . . . . . . 9 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
5958simprd 495 . . . . . . . 8 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
60 simplr 769 . . . . . . . 8 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → 𝑎 ∈ ℝ+)
6157, 59, 60rspcdva 3622 . . . . . . 7 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∃𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎))
62 nfv 1911 . . . . . . . . . . . 12 𝑦(𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋))
63 nfv 1911 . . . . . . . . . . . . 13 𝑦 𝐶 ∈ (fBas‘𝑋)
64 nfcv 2902 . . . . . . . . . . . . . 14 𝑦+
65 nfre1 3282 . . . . . . . . . . . . . 14 𝑦𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)
6664, 65nfralw 3308 . . . . . . . . . . . . 13 𝑦𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)
6763, 66nfan 1896 . . . . . . . . . . . 12 𝑦(𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
6862, 67nfan 1896 . . . . . . . . . . 11 𝑦((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
69 nfv 1911 . . . . . . . . . . 11 𝑦 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)
7068, 69nfan 1896 . . . . . . . . . 10 𝑦(((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
71 nfv 1911 . . . . . . . . . 10 𝑦 𝑎 ∈ ℝ+
7270, 71nfan 1896 . . . . . . . . 9 𝑦((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+)
73 nfv 1911 . . . . . . . . 9 𝑦(𝐷 “ (0[,)𝑎)) ⊆ 𝑣
7472, 73nfan 1896 . . . . . . . 8 𝑦(((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
7554ad4antr 732 . . . . . . . . . . . 12 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ∧ 𝑦𝐶) → 𝐶 ∈ (fBas‘𝑋))
76 fbelss 23856 . . . . . . . . . . . 12 ((𝐶 ∈ (fBas‘𝑋) ∧ 𝑦𝐶) → 𝑦𝑋)
7775, 76sylancom 588 . . . . . . . . . . 11 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ∧ 𝑦𝐶) → 𝑦𝑋)
78 xpss12 5703 . . . . . . . . . . 11 ((𝑦𝑋𝑦𝑋) → (𝑦 × 𝑦) ⊆ (𝑋 × 𝑋))
7977, 77, 78syl2anc 584 . . . . . . . . . 10 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ∧ 𝑦𝐶) → (𝑦 × 𝑦) ⊆ (𝑋 × 𝑋))
80 simp-6r 788 . . . . . . . . . . 11 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ∧ 𝑦𝐶) → 𝐷 ∈ (PsMet‘𝑋))
8180, 6, 143syl 18 . . . . . . . . . 10 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ∧ 𝑦𝐶) → dom 𝐷 = (𝑋 × 𝑋))
8279, 81sseqtrrd 4036 . . . . . . . . 9 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ∧ 𝑦𝐶) → (𝑦 × 𝑦) ⊆ dom 𝐷)
8382ex 412 . . . . . . . 8 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → (𝑦𝐶 → (𝑦 × 𝑦) ⊆ dom 𝐷))
8474, 83ralrimi 3254 . . . . . . 7 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∀𝑦𝐶 (𝑦 × 𝑦) ⊆ dom 𝐷)
85 r19.29r 3113 . . . . . . . 8 ((∃𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ ∀𝑦𝐶 (𝑦 × 𝑦) ⊆ dom 𝐷) → ∃𝑦𝐶 ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷))
86 sseqin2 4230 . . . . . . . . . . . . 13 ((𝑦 × 𝑦) ⊆ dom 𝐷 ↔ (dom 𝐷 ∩ (𝑦 × 𝑦)) = (𝑦 × 𝑦))
8786biimpi 216 . . . . . . . . . . . 12 ((𝑦 × 𝑦) ⊆ dom 𝐷 → (dom 𝐷 ∩ (𝑦 × 𝑦)) = (𝑦 × 𝑦))
8887adantl 481 . . . . . . . . . . 11 (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (dom 𝐷 ∩ (𝑦 × 𝑦)) = (𝑦 × 𝑦))
89 dminss 6174 . . . . . . . . . . 11 (dom 𝐷 ∩ (𝑦 × 𝑦)) ⊆ (𝐷 “ (𝐷 “ (𝑦 × 𝑦)))
9088, 89eqsstrrdi 4050 . . . . . . . . . 10 (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (𝑦 × 𝑦) ⊆ (𝐷 “ (𝐷 “ (𝑦 × 𝑦))))
91 imass2 6122 . . . . . . . . . . 11 ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) → (𝐷 “ (𝐷 “ (𝑦 × 𝑦))) ⊆ (𝐷 “ (0[,)𝑎)))
9291adantr 480 . . . . . . . . . 10 (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (𝐷 “ (𝐷 “ (𝑦 × 𝑦))) ⊆ (𝐷 “ (0[,)𝑎)))
9390, 92sstrd 4005 . . . . . . . . 9 (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)))
9493reximi 3081 . . . . . . . 8 (∃𝑦𝐶 ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → ∃𝑦𝐶 (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)))
9585, 94syl 17 . . . . . . 7 ((∃𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ ∀𝑦𝐶 (𝑦 × 𝑦) ⊆ dom 𝐷) → ∃𝑦𝐶 (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)))
9661, 84, 95syl2anc 584 . . . . . 6 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∃𝑦𝐶 (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)))
97 r19.41v 3186 . . . . . . 7 (∃𝑦𝐶 ((𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ↔ (∃𝑦𝐶 (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣))
98 sstr 4003 . . . . . . . 8 (((𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → (𝑦 × 𝑦) ⊆ 𝑣)
9998reximi 3081 . . . . . . 7 (∃𝑦𝐶 ((𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∃𝑦𝐶 (𝑦 × 𝑦) ⊆ 𝑣)
10097, 99sylbir 235 . . . . . 6 ((∃𝑦𝐶 (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∃𝑦𝐶 (𝑦 × 𝑦) ⊆ 𝑣)
10196, 100sylancom 588 . . . . 5 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑎 ∈ ℝ+) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∃𝑦𝐶 (𝑦 × 𝑦) ⊆ 𝑣)
102 simp-5r 786 . . . . . . . 8 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤𝐹) ∧ 𝑤𝑣) → 𝐷 ∈ (PsMet‘𝑋))
103 simplr 769 . . . . . . . 8 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤𝐹) ∧ 𝑤𝑣) → 𝑤𝐹)
1041metustel 24578 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → (𝑤𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑎))))
105104biimpa 476 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑤𝐹) → ∃𝑎 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑎)))
106102, 103, 105syl2anc 584 . . . . . . 7 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤𝐹) ∧ 𝑤𝑣) → ∃𝑎 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑎)))
107 r19.41v 3186 . . . . . . . 8 (∃𝑎 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑎)) ∧ 𝑤𝑣) ↔ (∃𝑎 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑎)) ∧ 𝑤𝑣))
108 sseq1 4020 . . . . . . . . . 10 (𝑤 = (𝐷 “ (0[,)𝑎)) → (𝑤𝑣 ↔ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣))
109108biimpa 476 . . . . . . . . 9 ((𝑤 = (𝐷 “ (0[,)𝑎)) ∧ 𝑤𝑣) → (𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
110109reximi 3081 . . . . . . . 8 (∃𝑎 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑎)) ∧ 𝑤𝑣) → ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
111107, 110sylbir 235 . . . . . . 7 ((∃𝑎 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑎)) ∧ 𝑤𝑣) → ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
112106, 111sylancom 588 . . . . . 6 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤𝐹) ∧ 𝑤𝑣) → ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
11311ad2antrr 726 . . . . . . . 8 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
114 elfg 23894 . . . . . . . . 9 (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → (𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤𝐹 𝑤𝑣)))
115114biimpa 476 . . . . . . . 8 ((𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤𝐹 𝑤𝑣))
116113, 115sylancom 588 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤𝐹 𝑤𝑣))
117116simprd 495 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑤𝐹 𝑤𝑣)
118112, 117r19.29a 3159 . . . . 5 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
119101, 118r19.29a 3159 . . . 4 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) → ∃𝑦𝐶 (𝑦 × 𝑦) ⊆ 𝑣)
120119ralrimiva 3143 . . 3 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → ∀𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)∃𝑦𝐶 (𝑦 × 𝑦) ⊆ 𝑣)
1212adantr 480 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
122 iscfilu 24312 . . . 4 (((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋) → (𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)∃𝑦𝐶 (𝑦 × 𝑦) ⊆ 𝑣)))
123121, 122syl 17 . . 3 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → (𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)∃𝑦𝐶 (𝑦 × 𝑦) ⊆ 𝑣)))
12454, 120, 123mpbir2and 713 . 2 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹)))
12553, 124impbida 801 1 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  wrex 3067  cin 3961  wss 3962  c0 4338   class class class wbr 5147  cmpt 5230   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689  cima 5691  Fun wfun 6556  wf 6558  cfv 6562  (class class class)co 7430  0cc0 11152  *cxr 11291  cle 11293   / cdiv 11917  2c2 12318  +crp 13031  [,)cico 13385  PsMetcpsmet 21365  fBascfbas 21369  filGencfg 21370  UnifOncust 24223  CauFiluccfilu 24310
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-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-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ico 13389  df-psmet 21373  df-fbas 21378  df-fg 21379  df-fil 23869  df-ust 24224  df-cfilu 24311
This theorem is referenced by:  cfilucfil2  24589
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