![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 17327 | . . . 4 β’ dist = Slot (distβndx) | |
2 | dsndxntsetndx 17334 | . . . 4 β’ (distβndx) β (TopSetβndx) | |
3 | 1, 2 | setsnid 17138 | . . 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 6902 | . . . . 5 β’ β β V |
6 | coexg 7916 | . . . . 5 β’ ((π β π β§ β β V) β (π β β ) β V) | |
7 | 5, 6 | mpan2 689 | . . . 4 β’ (π β π β (π β β ) β V) |
8 | 1 | setsid 17137 | . . . 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 593 | . . 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 2732 | . . . . 5 β’ (π β β ) = (π β β ) | |
12 | eqid 2732 | . . . . 5 β’ (MetOpenβ(π β β )) = (MetOpenβ(π β β )) | |
13 | 10, 4, 11, 12 | tngval 24139 | . . . 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 6892 | . . 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 2798 | . 2 β’ ((πΊ β V β§ π β π) β (π β β ) = (distβπ)) |
16 | co02 6256 | . . . . 5 β’ (π β β ) = β | |
17 | 1 | str0 17118 | . . . . 5 β’ β = (distββ ) |
18 | 16, 17 | eqtri 2760 | . . . 4 β’ (π β β ) = (distββ ) |
19 | fvprc 6880 | . . . . . 6 β’ (Β¬ πΊ β V β (-gβπΊ) = β ) | |
20 | 4, 19 | eqtrid 2784 | . . . . 5 β’ (Β¬ πΊ β V β β = β ) |
21 | 20 | coeq2d 5860 | . . . 4 β’ (Β¬ πΊ β V β (π β β ) = (π β β )) |
22 | reldmtng 24138 | . . . . . . 7 β’ Rel dom toNrmGrp | |
23 | 22 | ovprc1 7444 | . . . . . 6 β’ (Β¬ πΊ β V β (πΊ toNrmGrp π) = β ) |
24 | 10, 23 | eqtrid 2784 | . . . . 5 β’ (Β¬ πΊ β V β π = β ) |
25 | 24 | fveq2d 6892 | . . . 4 β’ (Β¬ πΊ β V β (distβπ) = (distββ )) |
26 | 18, 21, 25 | 3eqtr4a 2798 | . . 3 β’ (Β¬ πΊ β V β (π β β ) = (distβπ)) |
27 | 26 | adantr 481 | . 2 β’ ((Β¬ πΊ β V β§ π β π) β (π β β ) = (distβπ)) |
28 | 15, 27 | pm2.61ian 810 | 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 396 = wceq 1541 β wcel 2106 Vcvv 3474 β c0 4321 β¨cop 4633 β ccom 5679 βcfv 6540 (class class class)co 7405 sSet csts 17092 ndxcnx 17122 TopSetcts 17199 distcds 17202 -gcsg 18817 MetOpencmopn 20926 toNrmGrp ctng 24078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-sets 17093 df-slot 17111 df-ndx 17123 df-tset 17212 df-ds 17215 df-tng 24084 |
This theorem is referenced by: tngtset 24157 tngtopn 24158 tngnm 24159 tngngp2 24160 tngngpd 24161 nrmtngdist 24165 tngnrg 24182 cnindmet 24670 tcphds 24739 |
Copyright terms: Public domain | W3C validator |