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Theorem relcmpcmet 23315
 Description: If 𝐷 is a metric space such that all the balls of some fixed size are relatively compact, then 𝐷 is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
relcmpcmet.1 𝐽 = (MetOpen‘𝐷)
relcmpcmet.2 (𝜑𝐷 ∈ (Met‘𝑋))
relcmpcmet.3 (𝜑𝑅 ∈ ℝ+)
relcmpcmet.4 ((𝜑𝑥𝑋) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
Assertion
Ref Expression
relcmpcmet (𝜑𝐷 ∈ (CMet‘𝑋))
Distinct variable groups:   𝑥,𝐷   𝑥,𝐽   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋

Proof of Theorem relcmpcmet
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 relcmpcmet.2 . 2 (𝜑𝐷 ∈ (Met‘𝑋))
2 metxmet 22340 . . . . . . 7 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
31, 2syl 17 . . . . . 6 (𝜑𝐷 ∈ (∞Met‘𝑋))
43adantr 472 . . . . 5 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → 𝐷 ∈ (∞Met‘𝑋))
5 simpr 479 . . . . 5 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (CauFil‘𝐷))
6 relcmpcmet.3 . . . . . 6 (𝜑𝑅 ∈ ℝ+)
76adantr 472 . . . . 5 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → 𝑅 ∈ ℝ+)
8 cfil3i 23267 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
94, 5, 7, 8syl3anc 1477 . . . 4 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → ∃𝑥𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
103ad2antrr 764 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐷 ∈ (∞Met‘𝑋))
11 relcmpcmet.1 . . . . . . . . 9 𝐽 = (MetOpen‘𝐷)
1211mopntopon 22445 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
1310, 12syl 17 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
14 cfilfil 23265 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
153, 14sylan 489 . . . . . . . 8 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
1615adantr 472 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
17 simprr 813 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
18 topontop 20920 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1913, 18syl 17 . . . . . . . . . 10 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ Top)
20 simprl 811 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑥𝑋)
216rpxrd 12066 . . . . . . . . . . . . 13 (𝜑𝑅 ∈ ℝ*)
2221ad2antrr 764 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑅 ∈ ℝ*)
23 blssm 22424 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑅 ∈ ℝ*) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
2410, 20, 22, 23syl3anc 1477 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
25 toponuni 20921 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2613, 25syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑋 = 𝐽)
2724, 26sseqtrd 3782 . . . . . . . . . 10 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝐽)
28 eqid 2760 . . . . . . . . . . 11 𝐽 = 𝐽
2928clsss3 21065 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ 𝐽) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝐽)
3019, 27, 29syl2anc 696 . . . . . . . . 9 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝐽)
3130, 26sseqtr4d 3783 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋)
3228sscls 21062 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ 𝐽) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
3319, 27, 32syl2anc 696 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
34 filss 21858 . . . . . . . 8 ((𝑓 ∈ (Fil‘𝑋) ∧ ((𝑥(ball‘𝐷)𝑅) ∈ 𝑓 ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
3516, 17, 31, 33, 34syl13anc 1479 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
36 fclsrest 22029 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
3713, 16, 35, 36syl3anc 1477 . . . . . 6 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
38 inss1 3976 . . . . . . 7 ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fClus 𝑓)
39 eqid 2760 . . . . . . . . 9 dom dom 𝐷 = dom dom 𝐷
4011, 39cfilfcls 23272 . . . . . . . 8 (𝑓 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
4140ad2antlr 765 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
4238, 41syl5sseq 3794 . . . . . 6 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fLim 𝑓))
4337, 42eqsstrd 3780 . . . . 5 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓))
44 relcmpcmet.4 . . . . . . 7 ((𝜑𝑥𝑋) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
4544ad2ant2r 800 . . . . . 6 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
46 filfbas 21853 . . . . . . . . . 10 (𝑓 ∈ (Fil‘𝑋) → 𝑓 ∈ (fBas‘𝑋))
4716, 46syl 17 . . . . . . . . 9 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (fBas‘𝑋))
48 fbncp 21844 . . . . . . . . 9 ((𝑓 ∈ (fBas‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
4947, 35, 48syl2anc 696 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
50 trfil3 21893 . . . . . . . . 9 ((𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → ((𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
5116, 31, 50syl2anc 696 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
5249, 51mpbird 247 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
53 resttopon 21167 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
5413, 31, 53syl2anc 696 . . . . . . . . 9 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
55 toponuni 20921 . . . . . . . . 9 ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
5654, 55syl 17 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
5756fveq2d 6356 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = (Fil‘ (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
5852, 57eleqtrd 2841 . . . . . 6 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘ (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
59 eqid 2760 . . . . . . 7 (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
6059fclscmpi 22034 . . . . . 6 (((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp ∧ (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘ (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
6145, 58, 60syl2anc 696 . . . . 5 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
62 ssn0 4119 . . . . 5 ((((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓) ∧ ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅) → (𝐽 fLim 𝑓) ≠ ∅)
6343, 61, 62syl2anc 696 . . . 4 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fLim 𝑓) ≠ ∅)
649, 63rexlimddv 3173 . . 3 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝑓) ≠ ∅)
6564ralrimiva 3104 . 2 (𝜑 → ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)
6611iscmet 23282 . 2 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
671, 65, 66sylanbrc 701 1 (𝜑𝐷 ∈ (CMet‘𝑋))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  ∀wral 3050  ∃wrex 3051   ∖ cdif 3712   ∩ cin 3714   ⊆ wss 3715  ∅c0 4058  ∪ cuni 4588  dom cdm 5266  ‘cfv 6049  (class class class)co 6813  ℝ*cxr 10265  ℝ+crp 12025   ↾t crest 16283  ∞Metcxmt 19933  Metcme 19934  ballcbl 19935  fBascfbas 19936  MetOpencmopn 19938  Topctop 20900  TopOnctopon 20917  clsccl 21024  Compccmp 21391  Filcfil 21850   fLim cflim 21939   fClus cfcls 21941  CauFilccfil 23250  CMetcms 23252 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-rep 4923  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  ax-pre-sup 10206 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-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  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-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  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-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fi 8482  df-sup 8513  df-inf 8514  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-nn 11213  df-2 11271  df-n0 11485  df-z 11570  df-uz 11880  df-q 11982  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-ico 12374  df-rest 16285  df-topgen 16306  df-psmet 19940  df-xmet 19941  df-met 19942  df-bl 19943  df-mopn 19944  df-fbas 19945  df-fg 19946  df-top 20901  df-topon 20918  df-bases 20952  df-cld 21025  df-ntr 21026  df-cls 21027  df-nei 21104  df-cmp 21392  df-fil 21851  df-flim 21944  df-fcls 21946  df-cfil 23253  df-cmet 23255 This theorem is referenced by:  cmpcmet  23316  cncmet  23319
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