Metamath Proof Explorer


Theorem cfilucfil

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 . (Contributed by Thierry Arnoux, 29-Nov-2017) (Revised by Thierry Arnoux, 11-Feb-2018)

Ref Expression
Hypothesis metust.1 F=rana+D-10a
Assertion cfilucfil XDPsMetXCCauFilUX×XfilGenFCfBasXx+yCDy×y0x

Proof

Step Hyp Ref Expression
1 metust.1 F=rana+D-10a
2 1 metust XDPsMetXX×XfilGenFUnifOnX
3 cfilufbas X×XfilGenFUnifOnXCCauFilUX×XfilGenFCfBasX
4 2 3 sylan XDPsMetXCCauFilUX×XfilGenFCfBasX
5 simpllr XDPsMetXCCauFilUX×XfilGenFx+DPsMetX
6 psmetf DPsMetXD:X×X*
7 ffun D:X×X*FunD
8 5 6 7 3syl XDPsMetXCCauFilUX×XfilGenFx+FunD
9 2 ad2antrr XDPsMetXCCauFilUX×XfilGenFx+X×XfilGenFUnifOnX
10 simplr XDPsMetXCCauFilUX×XfilGenFx+CCauFilUX×XfilGenF
11 1 metustfbas XDPsMetXFfBasX×X
12 11 ad2antrr XDPsMetXCCauFilUX×XfilGenFx+FfBasX×X
13 cnvimass D-10xdomD
14 fdm D:X×X*domD=X×X
15 5 6 14 3syl XDPsMetXCCauFilUX×XfilGenFx+domD=X×X
16 13 15 sseqtrid XDPsMetXCCauFilUX×XfilGenFx+D-10xX×X
17 simpr XDPsMetXCCauFilUX×XfilGenFx+x+
18 17 rphalfcld XDPsMetXCCauFilUX×XfilGenFx+x2+
19 eqidd XDPsMetXCCauFilUX×XfilGenFx+D-10x2=D-10x2
20 oveq2 a=x20a=0x2
21 20 imaeq2d a=x2D-10a=D-10x2
22 21 rspceeqv x2+D-10x2=D-10x2a+D-10x2=D-10a
23 18 19 22 syl2anc XDPsMetXCCauFilUX×XfilGenFx+a+D-10x2=D-10a
24 1 metustel DPsMetXD-10x2Fa+D-10x2=D-10a
25 24 biimpar DPsMetXa+D-10x2=D-10aD-10x2F
26 5 23 25 syl2anc XDPsMetXCCauFilUX×XfilGenFx+D-10x2F
27 0xr 0*
28 27 a1i x+0*
29 rpxr x+x*
30 0le0 00
31 30 a1i x+00
32 rpre x+x
33 32 rehalfcld x+x2
34 rphalflt x+x2<x
35 33 32 34 ltled x+x2x
36 icossico 0*x*00x2x0x20x
37 28 29 31 35 36 syl22anc x+0x20x
38 imass2 0x20xD-10x2D-10x
39 17 37 38 3syl XDPsMetXCCauFilUX×XfilGenFx+D-10x2D-10x
40 sseq1 w=D-10x2wD-10xD-10x2D-10x
41 40 rspcev D-10x2FD-10x2D-10xwFwD-10x
42 26 39 41 syl2anc XDPsMetXCCauFilUX×XfilGenFx+wFwD-10x
43 elfg FfBasX×XD-10xX×XfilGenFD-10xX×XwFwD-10x
44 43 biimpar FfBasX×XD-10xX×XwFwD-10xD-10xX×XfilGenF
45 12 16 42 44 syl12anc XDPsMetXCCauFilUX×XfilGenFx+D-10xX×XfilGenF
46 cfiluexsm X×XfilGenFUnifOnXCCauFilUX×XfilGenFD-10xX×XfilGenFyCy×yD-10x
47 9 10 45 46 syl3anc XDPsMetXCCauFilUX×XfilGenFx+yCy×yD-10x
48 funimass2 FunDy×yD-10xDy×y0x
49 48 ex FunDy×yD-10xDy×y0x
50 49 reximdv FunDyCy×yD-10xyCDy×y0x
51 8 47 50 sylc XDPsMetXCCauFilUX×XfilGenFx+yCDy×y0x
52 51 ralrimiva XDPsMetXCCauFilUX×XfilGenFx+yCDy×y0x
53 4 52 jca XDPsMetXCCauFilUX×XfilGenFCfBasXx+yCDy×y0x
54 simprl XDPsMetXCfBasXx+yCDy×y0xCfBasX
55 oveq2 x=a0x=0a
56 55 sseq2d x=aDy×y0xDy×y0a
57 56 rexbidv x=ayCDy×y0xyCDy×y0a
58 simp-4r XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10avCfBasXx+yCDy×y0x
59 58 simprd XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10avx+yCDy×y0x
60 simplr XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10ava+
61 57 59 60 rspcdva XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10avyCDy×y0a
62 nfv yXDPsMetX
63 nfv yCfBasX
64 nfcv _y+
65 nfre1 yyCDy×y0x
66 64 65 nfralw yx+yCDy×y0x
67 63 66 nfan yCfBasXx+yCDy×y0x
68 62 67 nfan yXDPsMetXCfBasXx+yCDy×y0x
69 nfv yvX×XfilGenF
70 68 69 nfan yXDPsMetXCfBasXx+yCDy×y0xvX×XfilGenF
71 nfv ya+
72 70 71 nfan yXDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+
73 nfv yD-10av
74 72 73 nfan yXDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10av
75 54 ad4antr XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10avyCCfBasX
76 fbelss CfBasXyCyX
77 75 76 sylancom XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10avyCyX
78 xpss12 yXyXy×yX×X
79 77 77 78 syl2anc XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10avyCy×yX×X
80 simp-6r XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10avyCDPsMetX
81 80 6 14 3syl XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10avyCdomD=X×X
82 79 81 sseqtrrd XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10avyCy×ydomD
83 82 ex XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10avyCy×ydomD
84 74 83 ralrimi XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10avyCy×ydomD
85 r19.29r yCDy×y0ayCy×ydomDyCDy×y0ay×ydomD
86 sseqin2 y×ydomDdomDy×y=y×y
87 86 biimpi y×ydomDdomDy×y=y×y
88 87 adantl Dy×y0ay×ydomDdomDy×y=y×y
89 dminss domDy×yD-1Dy×y
90 88 89 eqsstrrdi Dy×y0ay×ydomDy×yD-1Dy×y
91 imass2 Dy×y0aD-1Dy×yD-10a
92 91 adantr Dy×y0ay×ydomDD-1Dy×yD-10a
93 90 92 sstrd Dy×y0ay×ydomDy×yD-10a
94 93 reximi yCDy×y0ay×ydomDyCy×yD-10a
95 85 94 syl yCDy×y0ayCy×ydomDyCy×yD-10a
96 61 84 95 syl2anc XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10avyCy×yD-10a
97 r19.41v yCy×yD-10aD-10avyCy×yD-10aD-10av
98 sstr y×yD-10aD-10avy×yv
99 98 reximi yCy×yD-10aD-10avyCy×yv
100 97 99 sylbir yCy×yD-10aD-10avyCy×yv
101 96 100 sylancom XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10avyCy×yv
102 simp-5r XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFwFwvDPsMetX
103 simplr XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFwFwvwF
104 1 metustel DPsMetXwFa+w=D-10a
105 104 biimpa DPsMetXwFa+w=D-10a
106 102 103 105 syl2anc XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFwFwva+w=D-10a
107 r19.41v a+w=D-10awva+w=D-10awv
108 sseq1 w=D-10awvD-10av
109 108 biimpa w=D-10awvD-10av
110 109 reximi a+w=D-10awva+D-10av
111 107 110 sylbir a+w=D-10awva+D-10av
112 106 111 sylancom XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFwFwva+D-10av
113 11 ad2antrr XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFFfBasX×X
114 elfg FfBasX×XvX×XfilGenFvX×XwFwv
115 114 biimpa FfBasX×XvX×XfilGenFvX×XwFwv
116 113 115 sylancom XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFvX×XwFwv
117 116 simprd XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFwFwv
118 112 117 r19.29a XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFa+D-10av
119 101 118 r19.29a XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFyCy×yv
120 119 ralrimiva XDPsMetXCfBasXx+yCDy×y0xvX×XfilGenFyCy×yv
121 2 adantr XDPsMetXCfBasXx+yCDy×y0xX×XfilGenFUnifOnX
122 iscfilu X×XfilGenFUnifOnXCCauFilUX×XfilGenFCfBasXvX×XfilGenFyCy×yv
123 121 122 syl XDPsMetXCfBasXx+yCDy×y0xCCauFilUX×XfilGenFCfBasXvX×XfilGenFyCy×yv
124 54 120 123 mpbir2and XDPsMetXCfBasXx+yCDy×y0xCCauFilUX×XfilGenF
125 53 124 impbida XDPsMetXCCauFilUX×XfilGenFCfBasXx+yCDy×y0x