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

Theorem cfilucfil 24572
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 25299. (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 24571 . . . 4 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
3 cfilufbas 24298 . . . 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 24316 . . . . . 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 24570 . . . . . . . 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 6100 . . . . . . . 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 4026 . . . . . . 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 13089 . . . . . . . . . 10 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → (𝑥 / 2) ∈ ℝ+)
19 eqidd 2738 . . . . . . . . . 10 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → (𝐷 “ (0[,)(𝑥 / 2))) = (𝐷 “ (0[,)(𝑥 / 2))))
20 oveq2 7439 . . . . . . . . . . . 12 (𝑎 = (𝑥 / 2) → (0[,)𝑎) = (0[,)(𝑥 / 2)))
2120imaeq2d 6078 . . . . . . . . . . 11 (𝑎 = (𝑥 / 2) → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)(𝑥 / 2))))
2221rspceeqv 3645 . . . . . . . . . 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 24563 . . . . . . . . . 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 11308 . . . . . . . . . . 11 0 ∈ ℝ*
2827a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → 0 ∈ ℝ*)
29 rpxr 13044 . . . . . . . . . 10 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
30 0le0 12367 . . . . . . . . . . 11 0 ≤ 0
3130a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → 0 ≤ 0)
32 rpre 13043 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
3332rehalfcld 12513 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ)
34 rphalflt 13064 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑥 / 2) < 𝑥)
3533, 32, 34ltled 11409 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (𝑥 / 2) ≤ 𝑥)
36 icossico 13457 . . . . . . . . . 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 6120 . . . . . . . . 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 4009 . . . . . . . . 9 (𝑤 = (𝐷 “ (0[,)(𝑥 / 2))) → (𝑤 ⊆ (𝐷 “ (0[,)𝑥)) ↔ (𝐷 “ (0[,)(𝑥 / 2))) ⊆ (𝐷 “ (0[,)𝑥))))
4140rspcev 3622 . . . . . . . 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 23879 . . . . . . . 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 24299 . . . . . 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 1373 . . . . 5 ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹))) ∧ 𝑥 ∈ ℝ+) → ∃𝑦𝐶 (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑥)))
48 funimass2 6649 . . . . . . 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 3170 . . . . 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 3146 . . 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 7439 . . . . . . . . . 10 (𝑥 = 𝑎 → (0[,)𝑥) = (0[,)𝑎))
5655sseq2d 4016 . . . . . . . . 9 (𝑥 = 𝑎 → ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎)))
5756rexbidv 3179 . . . . . . . 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 3623 . . . . . . 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 1914 . . . . . . . . . . . 12 𝑦(𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋))
63 nfv 1914 . . . . . . . . . . . . 13 𝑦 𝐶 ∈ (fBas‘𝑋)
64 nfcv 2905 . . . . . . . . . . . . . 14 𝑦+
65 nfre1 3285 . . . . . . . . . . . . . 14 𝑦𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)
6664, 65nfralw 3311 . . . . . . . . . . . . 13 𝑦𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)
6763, 66nfan 1899 . . . . . . . . . . . 12 𝑦(𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
6862, 67nfan 1899 . . . . . . . . . . 11 𝑦((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
69 nfv 1914 . . . . . . . . . . 11 𝑦 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹)
7068, 69nfan 1899 . . . . . . . . . 10 𝑦(((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) ∧ 𝑣 ∈ ((𝑋 × 𝑋)filGen𝐹))
71 nfv 1914 . . . . . . . . . 10 𝑦 𝑎 ∈ ℝ+
7270, 71nfan 1899 . . . . . . . . 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 1914 . . . . . . . . 9 𝑦(𝐷 “ (0[,)𝑎)) ⊆ 𝑣
7472, 73nfan 1899 . . . . . . . 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 23841 . . . . . . . . . . . 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 5700 . . . . . . . . . . 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 4021 . . . . . . . . 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 3257 . . . . . . 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 3116 . . . . . . . 8 ((∃𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ ∀𝑦𝐶 (𝑦 × 𝑦) ⊆ dom 𝐷) → ∃𝑦𝐶 ((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷))
86 sseqin2 4223 . . . . . . . . . . . . 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 6173 . . . . . . . . . . 11 (dom 𝐷 ∩ (𝑦 × 𝑦)) ⊆ (𝐷 “ (𝐷 “ (𝑦 × 𝑦)))
9088, 89eqsstrrdi 4029 . . . . . . . . . 10 (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (𝑦 × 𝑦) ⊆ (𝐷 “ (𝐷 “ (𝑦 × 𝑦))))
91 imass2 6120 . . . . . . . . . . 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 3994 . . . . . . . . 9 (((𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑎) ∧ (𝑦 × 𝑦) ⊆ dom 𝐷) → (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)))
9493reximi 3084 . . . . . . . 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 3189 . . . . . . 7 (∃𝑦𝐶 ((𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) ↔ (∃𝑦𝐶 (𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣))
98 sstr 3992 . . . . . . . 8 (((𝑦 × 𝑦) ⊆ (𝐷 “ (0[,)𝑎)) ∧ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣) → (𝑦 × 𝑦) ⊆ 𝑣)
9998reximi 3084 . . . . . . 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 24563 . . . . . . . . 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 3189 . . . . . . . 8 (∃𝑎 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑎)) ∧ 𝑤𝑣) ↔ (∃𝑎 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑎)) ∧ 𝑤𝑣))
108 sseq1 4009 . . . . . . . . . 10 (𝑤 = (𝐷 “ (0[,)𝑎)) → (𝑤𝑣 ↔ (𝐷 “ (0[,)𝑎)) ⊆ 𝑣))
109108biimpa 476 . . . . . . . . 9 ((𝑤 = (𝐷 “ (0[,)𝑎)) ∧ 𝑤𝑣) → (𝐷 “ (0[,)𝑎)) ⊆ 𝑣)
110109reximi 3084 . . . . . . . 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 23879 . . . . . . . . 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 3162 . . . . 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 3162 . . . 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 3146 . . 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 24297 . . . 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 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cin 3950  wss 3951  c0 4333   class class class wbr 5143  cmpt 5225   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431  0cc0 11155  *cxr 11294  cle 11296   / cdiv 11920  2c2 12321  +crp 13034  [,)cico 13389  PsMetcpsmet 21348  fBascfbas 21352  filGencfg 21353  UnifOncust 24208  CauFiluccfilu 24295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-2 12329  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ico 13393  df-psmet 21356  df-fbas 21361  df-fg 21362  df-fil 23854  df-ust 24209  df-cfilu 24296
This theorem is referenced by:  cfilucfil2  24574
  Copyright terms: Public domain W3C validator