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Mirrors > Home > MPE Home > Th. List > tngds | Structured version Visualization version GIF version |
Description: The metric function of a structure augmented with a norm. (Contributed by Mario Carneiro, 3-Oct-2015.) (Proof shortened by AV, 29-Oct-2024.) |
Ref | Expression |
---|---|
tngbas.t | β’ π = (πΊ toNrmGrp π) |
tngds.2 | β’ β = (-gβπΊ) |
Ref | Expression |
---|---|
tngds | β’ (π β π β (π β β ) = (distβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dsid 17366 | . . . 4 β’ dist = Slot (distβndx) | |
2 | dsndxntsetndx 17373 | . . . 4 β’ (distβndx) β (TopSetβndx) | |
3 | 1, 2 | setsnid 17177 | . . 3 β’ (distβ(πΊ sSet β¨(distβndx), (π β β )β©)) = (distβ((πΊ sSet β¨(distβndx), (π β β )β©) sSet β¨(TopSetβndx), (MetOpenβ(π β β ))β©)) |
4 | tngds.2 | . . . . . 6 β’ β = (-gβπΊ) | |
5 | 4 | fvexi 6911 | . . . . 5 β’ β β V |
6 | coexg 7937 | . . . . 5 β’ ((π β π β§ β β V) β (π β β ) β V) | |
7 | 5, 6 | mpan2 690 | . . . 4 β’ (π β π β (π β β ) β V) |
8 | 1 | setsid 17176 | . . . 4 β’ ((πΊ β V β§ (π β β ) β V) β (π β β ) = (distβ(πΊ sSet β¨(distβndx), (π β β )β©))) |
9 | 7, 8 | sylan2 592 | . . 3 β’ ((πΊ β V β§ π β π) β (π β β ) = (distβ(πΊ sSet β¨(distβndx), (π β β )β©))) |
10 | tngbas.t | . . . . 5 β’ π = (πΊ toNrmGrp π) | |
11 | eqid 2728 | . . . . 5 β’ (π β β ) = (π β β ) | |
12 | eqid 2728 | . . . . 5 β’ (MetOpenβ(π β β )) = (MetOpenβ(π β β )) | |
13 | 10, 4, 11, 12 | tngval 24547 | . . . 4 β’ ((πΊ β V β§ π β π) β π = ((πΊ sSet β¨(distβndx), (π β β )β©) sSet β¨(TopSetβndx), (MetOpenβ(π β β ))β©)) |
14 | 13 | fveq2d 6901 | . . 3 β’ ((πΊ β V β§ π β π) β (distβπ) = (distβ((πΊ sSet β¨(distβndx), (π β β )β©) sSet β¨(TopSetβndx), (MetOpenβ(π β β ))β©))) |
15 | 3, 9, 14 | 3eqtr4a 2794 | . 2 β’ ((πΊ β V β§ π β π) β (π β β ) = (distβπ)) |
16 | co02 6264 | . . . . 5 β’ (π β β ) = β | |
17 | 1 | str0 17157 | . . . . 5 β’ β = (distββ ) |
18 | 16, 17 | eqtri 2756 | . . . 4 β’ (π β β ) = (distββ ) |
19 | fvprc 6889 | . . . . . 6 β’ (Β¬ πΊ β V β (-gβπΊ) = β ) | |
20 | 4, 19 | eqtrid 2780 | . . . . 5 β’ (Β¬ πΊ β V β β = β ) |
21 | 20 | coeq2d 5865 | . . . 4 β’ (Β¬ πΊ β V β (π β β ) = (π β β )) |
22 | reldmtng 24546 | . . . . . . 7 β’ Rel dom toNrmGrp | |
23 | 22 | ovprc1 7459 | . . . . . 6 β’ (Β¬ πΊ β V β (πΊ toNrmGrp π) = β ) |
24 | 10, 23 | eqtrid 2780 | . . . . 5 β’ (Β¬ πΊ β V β π = β ) |
25 | 24 | fveq2d 6901 | . . . 4 β’ (Β¬ πΊ β V β (distβπ) = (distββ )) |
26 | 18, 21, 25 | 3eqtr4a 2794 | . . 3 β’ (Β¬ πΊ β V β (π β β ) = (distβπ)) |
27 | 26 | adantr 480 | . 2 β’ ((Β¬ πΊ β V β§ π β π) β (π β β ) = (distβπ)) |
28 | 15, 27 | pm2.61ian 811 | 1 β’ (π β π β (π β β ) = (distβπ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 Vcvv 3471 β c0 4323 β¨cop 4635 β ccom 5682 βcfv 6548 (class class class)co 7420 sSet csts 17131 ndxcnx 17161 TopSetcts 17238 distcds 17241 -gcsg 18891 MetOpencmopn 21268 toNrmGrp ctng 24486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-sets 17132 df-slot 17150 df-ndx 17162 df-tset 17251 df-ds 17254 df-tng 24492 |
This theorem is referenced by: tngtset 24565 tngtopn 24566 tngnm 24567 tngngp2 24568 tngngpd 24569 nrmtngdist 24573 tngnrg 24590 cnindmet 25089 tcphds 25158 |
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