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Mirrors > Home > MPE Home > Th. List > tngtset | Structured version Visualization version GIF version |
Description: The topology generated by a normed structure. (Contributed by Mario Carneiro, 3-Oct-2015.) |
Ref | Expression |
---|---|
tngbas.t | β’ π = (πΊ toNrmGrp π) |
tngtset.2 | β’ π· = (distβπ) |
tngtset.3 | β’ π½ = (MetOpenβπ·) |
Ref | Expression |
---|---|
tngtset | β’ ((πΊ β π β§ π β π) β π½ = (TopSetβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7394 | . . 3 β’ (πΊ sSet β¨(distβndx), (π β (-gβπΊ))β©) β V | |
2 | fvex 6859 | . . 3 β’ (MetOpenβ(π β (-gβπΊ))) β V | |
3 | tsetid 17242 | . . . 4 β’ TopSet = Slot (TopSetβndx) | |
4 | 3 | setsid 17088 | . . 3 β’ (((πΊ sSet β¨(distβndx), (π β (-gβπΊ))β©) β V β§ (MetOpenβ(π β (-gβπΊ))) β V) β (MetOpenβ(π β (-gβπΊ))) = (TopSetβ((πΊ sSet β¨(distβndx), (π β (-gβπΊ))β©) sSet β¨(TopSetβndx), (MetOpenβ(π β (-gβπΊ)))β©))) |
5 | 1, 2, 4 | mp2an 691 | . 2 β’ (MetOpenβ(π β (-gβπΊ))) = (TopSetβ((πΊ sSet β¨(distβndx), (π β (-gβπΊ))β©) sSet β¨(TopSetβndx), (MetOpenβ(π β (-gβπΊ)))β©)) |
6 | tngtset.3 | . . 3 β’ π½ = (MetOpenβπ·) | |
7 | tngtset.2 | . . . . . 6 β’ π· = (distβπ) | |
8 | tngbas.t | . . . . . . 7 β’ π = (πΊ toNrmGrp π) | |
9 | eqid 2733 | . . . . . . 7 β’ (-gβπΊ) = (-gβπΊ) | |
10 | 8, 9 | tngds 24034 | . . . . . 6 β’ (π β π β (π β (-gβπΊ)) = (distβπ)) |
11 | 7, 10 | eqtr4id 2792 | . . . . 5 β’ (π β π β π· = (π β (-gβπΊ))) |
12 | 11 | adantl 483 | . . . 4 β’ ((πΊ β π β§ π β π) β π· = (π β (-gβπΊ))) |
13 | 12 | fveq2d 6850 | . . 3 β’ ((πΊ β π β§ π β π) β (MetOpenβπ·) = (MetOpenβ(π β (-gβπΊ)))) |
14 | 6, 13 | eqtrid 2785 | . 2 β’ ((πΊ β π β§ π β π) β π½ = (MetOpenβ(π β (-gβπΊ)))) |
15 | eqid 2733 | . . . 4 β’ (π β (-gβπΊ)) = (π β (-gβπΊ)) | |
16 | eqid 2733 | . . . 4 β’ (MetOpenβ(π β (-gβπΊ))) = (MetOpenβ(π β (-gβπΊ))) | |
17 | 8, 9, 15, 16 | tngval 24018 | . . 3 β’ ((πΊ β π β§ π β π) β π = ((πΊ sSet β¨(distβndx), (π β (-gβπΊ))β©) sSet β¨(TopSetβndx), (MetOpenβ(π β (-gβπΊ)))β©)) |
18 | 17 | fveq2d 6850 | . 2 β’ ((πΊ β π β§ π β π) β (TopSetβπ) = (TopSetβ((πΊ sSet β¨(distβndx), (π β (-gβπΊ))β©) sSet β¨(TopSetβndx), (MetOpenβ(π β (-gβπΊ)))β©))) |
19 | 5, 14, 18 | 3eqtr4a 2799 | 1 β’ ((πΊ β π β§ π β π) β π½ = (TopSetβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3447 β¨cop 4596 β ccom 5641 βcfv 6500 (class class class)co 7361 sSet csts 17043 ndxcnx 17073 TopSetcts 17147 distcds 17150 -gcsg 18758 MetOpencmopn 20809 toNrmGrp ctng 23957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-sets 17044 df-slot 17062 df-ndx 17074 df-tset 17160 df-ds 17163 df-tng 23963 |
This theorem is referenced by: tngtopn 24037 |
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