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Theorem cfilucfil 23915
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 24629. (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 23914 . . . 4 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
3 cfilufbas 23641 . . . 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 774 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋))
6 psmetf 23659 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
7 ffun 6671 . . . . . 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 724 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
10 simplr 767 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹)))
111metustfbas 23913 . . . . . . . 8 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
1211ad2antrr 724 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
13 cnvimass 6033 . . . . . . . 8 (𝐷 “ (0[,)𝑥)) ⊆ dom 𝐷
14 fdm 6677 . . . . . . . . 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 3996 . . . . . . 7 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → (𝐷 “ (0[,)𝑥)) ⊆ (𝑋 × 𝑋))
17 simpr 485 . . . . . . . . . . 11 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
1817rphalfcld 12969 . . . . . . . . . 10 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → (𝑥 / 2) ∈ ℝ+)
19 eqidd 2737 . . . . . . . . . 10 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → (𝐷 “ (0[,)(𝑥 / 2))) = (𝐷 “ (0[,)(𝑥 / 2))))
20 oveq2 7365 . . . . . . . . . . . 12 (𝑎 = (𝑥 / 2) → (0[,)𝑎) = (0[,)(𝑥 / 2)))
2120imaeq2d 6013 . . . . . . . . . . 11 (𝑎 = (𝑥 / 2) → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)(𝑥 / 2))))
2221rspceeqv 3595 . . . . . . . . . 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 23906 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → ((𝐷 “ (0[,)(𝑥 / 2))) ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)(𝑥 / 2))) = (𝐷 “ (0[,)𝑎))))
2524biimpar 478 . . . . . . . . 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 11202 . . . . . . . . . . 11 0 ∈ ℝ*
2827a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → 0 ∈ ℝ*)
29 rpxr 12924 . . . . . . . . . 10 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
30 0le0 12254 . . . . . . . . . . 11 0 ≤ 0
3130a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → 0 ≤ 0)
32 rpre 12923 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
3332rehalfcld 12400 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ)
34 rphalflt 12944 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑥 / 2) < 𝑥)
3533, 32, 34ltled 11303 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (𝑥 / 2) ≤ 𝑥)
36 icossico 13334 . . . . . . . . . 10 (((0 ∈ ℝ*𝑥 ∈ ℝ*) ∧ (0 ≤ 0 ∧ (𝑥 / 2) ≤ 𝑥)) → (0[,)(𝑥 / 2)) ⊆ (0[,)𝑥))
3728, 29, 31, 35, 36syl22anc 837 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (0[,)(𝑥 / 2)) ⊆ (0[,)𝑥))
38 imass2 6054 . . . . . . . . 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 3969 . . . . . . . . 9 (𝑤 = (𝐷 “ (0[,)(𝑥 / 2))) → (𝑤 ⊆ (𝐷 “ (0[,)𝑥)) ↔ (𝐷 “ (0[,)(𝑥 / 2))) ⊆ (𝐷 “ (0[,)𝑥))))
4140rspcev 3581 . . . . . . . 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 23222 . . . . . . . 8 (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → ((𝐷 “ (0[,)𝑥)) ∈ ((𝑋 × 𝑋)filGen𝐹) ↔ ((𝐷 “ (0[,)𝑥)) ⊆ (𝑋 × 𝑋) ∧ ∃𝑤𝐹 𝑤 ⊆ (𝐷 “ (0[,)𝑥)))))
4443biimpar 478 . . . . . . 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 835 . . . . . 6 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → (𝐷 “ (0[,)𝑥)) ∈ ((𝑋 × 𝑋)filGen𝐹))
46 cfiluexsm 23642 . . . . . 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 1371 . . . . 5 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → ∃𝑦𝐶 (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑥)))
48 funimass2 6584 . . . . . . 7 ((Fun 𝐷 ∧ (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑥))) → (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
4948ex 413 . . . . . 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 512 . 2 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) → (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
54 simprl 769 . . 3 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → 𝐶 ∈ (fBas‘𝑋))
55 oveq2 7365 . . . . . . . . . 10 (𝑥 = 𝑎 → (0[,)𝑥) = (0[,)𝑎))
5655sseq2d 3976 . . . . . . . . 9 (𝑥 = 𝑎 → ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎)))
5756rexbidv 3175 . . . . . . . 8 (𝑥 = 𝑎 → (∃𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∃𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎)))
58 simp-4r 782 . . . . . . . . 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 496 . . . . . . . 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 767 . . . . . . . 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 3582 . . . . . . 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 1917 . . . . . . . . . . . 12 𝑦(𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋))
63 nfv 1917 . . . . . . . . . . . . 13 𝑦 𝐶 ∈ (fBas‘𝑋)
64 nfcv 2907 . . . . . . . . . . . . . 14 𝑦+
65 nfre1 3268 . . . . . . . . . . . . . 14 𝑦𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)
6664, 65nfralw 3294 . . . . . . . . . . . . 13 𝑦𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)
6763, 66nfan 1902 . . . . . . . . . . . 12 𝑦(𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
6862, 67nfan 1902 . . . . . . . . . . 11 𝑦((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
69 nfv 1917 . . . . . . . . . . 11 𝑦 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)
7068, 69nfan 1902 . . . . . . . . . 10 𝑦(((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
71 nfv 1917 . . . . . . . . . 10 𝑦 𝑎 ∈ ℝ+
7270, 71nfan 1902 . . . . . . . . 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 1917 . . . . . . . . 9 𝑦(𝐷 “ (0[,)𝑎)) ⊆ 𝑣
7472, 73nfan 1902 . . . . . . . 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 730 . . . . . . . . . . . 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 23184 . . . . . . . . . . . 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 5648 . . . . . . . . . . 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 786 . . . . . . . . . . 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 3985 . . . . . . . . 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 413 . . . . . . . 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 3240 . . . . . . 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 3119 . . . . . . . 8 ((∃𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ ∀𝑦𝐶 (𝑦 × 𝑦) ⊆ dom 𝐷) → ∃𝑦𝐶 ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷))
86 sseqin2 4175 . . . . . . . . . . . . 13 ((𝑦 × 𝑦) ⊆ dom 𝐷 ↔ (dom 𝐷 ∩ (𝑦 × 𝑦)) = (𝑦 × 𝑦))
8786biimpi 215 . . . . . . . . . . . 12 ((𝑦 × 𝑦) ⊆ dom 𝐷 → (dom 𝐷 ∩ (𝑦 × 𝑦)) = (𝑦 × 𝑦))
8887adantl 482 . . . . . . . . . . 11 (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (dom 𝐷 ∩ (𝑦 × 𝑦)) = (𝑦 × 𝑦))
89 dminss 6105 . . . . . . . . . . 11 (dom 𝐷 ∩ (𝑦 × 𝑦)) ⊆ (𝐷 “ (𝐷 “ (𝑦 × 𝑦)))
9088, 89eqsstrrdi 3999 . . . . . . . . . 10 (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (𝑦 × 𝑦) ⊆ (𝐷 “ (𝐷 “ (𝑦 × 𝑦))))
91 imass2 6054 . . . . . . . . . . 11 ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) → (𝐷 “ (𝐷 “ (𝑦 × 𝑦))) ⊆ (𝐷 “ (0[,)𝑎)))
9291adantr 481 . . . . . . . . . 10 (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (𝐷 “ (𝐷 “ (𝑦 × 𝑦))) ⊆ (𝐷 “ (0[,)𝑎)))
9390, 92sstrd 3954 . . . . . . . . 9 (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)))
9493reximi 3087 . . . . . . . 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 3185 . . . . . . 7 (∃𝑦𝐶 ((𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ↔ (∃𝑦𝐶 (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣))
98 sstr 3952 . . . . . . . 8 (((𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → (𝑦 × 𝑦) ⊆ 𝑣)
9998reximi 3087 . . . . . . 7 (∃𝑦𝐶 ((𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → ∃𝑦𝐶 (𝑦 × 𝑦) ⊆ 𝑣)
10097, 99sylbir 234 . . . . . 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 784 . . . . . . . 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 767 . . . . . . . 8 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)) ∧ 𝑤𝐹) ∧ 𝑤𝑣) → 𝑤𝐹)
1041metustel 23906 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → (𝑤𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑎))))
105104biimpa 477 . . . . . . . 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 3185 . . . . . . . 8 (∃𝑎 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑎)) ∧ 𝑤𝑣) ↔ (∃𝑎 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑎)) ∧ 𝑤𝑣))
108 sseq1 3969 . . . . . . . . . 10 (𝑤 = (𝐷 “ (0[,)𝑎)) → (𝑤𝑣 ↔ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣))
109108biimpa 477 . . . . . . . . 9 ((𝑤 = (𝐷 “ (0[,)𝑎)) ∧ 𝑤𝑣) → (𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
110109reximi 3087 . . . . . . . 8 (∃𝑎 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑎)) ∧ 𝑤𝑣) → ∃𝑎 ∈ ℝ+ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
111107, 110sylbir 234 . . . . . . 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 724 . . . . . . . 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 23222 . . . . . . . . 9 (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) → (𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹) ↔ (𝑣 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤𝐹 𝑤𝑣)))
115114biimpa 477 . . . . . . . 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 496 . . . . . 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 481 . . . 4 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
122 iscfilu 23640 . . . 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 711 . 2 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) → 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹)))
12553, 124impbida 799 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 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  cin 3909  wss 3910  c0 4282   class class class wbr 5105  cmpt 5188   × cxp 5631  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636  Fun wfun 6490  wf 6492  cfv 6496  (class class class)co 7357  0cc0 11051  *cxr 11188  cle 11190   / cdiv 11812  2c2 12208  +crp 12915  [,)cico 13266  PsMetcpsmet 20780  fBascfbas 20784  filGencfg 20785  UnifOncust 23551  CauFiluccfilu 23638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-2 12216  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ico 13270  df-psmet 20788  df-fbas 20793  df-fg 20794  df-fil 23197  df-ust 23552  df-cfilu 23639
This theorem is referenced by:  cfilucfil2  23917
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