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Mirrors > Home > MPE Home > Th. List > tngdsOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of tngds 24592 as of 29-Oct-2024. The metric function of a structure augmented with a norm. (Contributed by Mario Carneiro, 3-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
tngbas.t | β’ π = (πΊ toNrmGrp π) |
tngds.2 | β’ β = (-gβπΊ) |
Ref | Expression |
---|---|
tngdsOLD | β’ (π β π β (π β β ) = (distβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dsid 17376 | . . . 4 β’ dist = Slot (distβndx) | |
2 | 9re 12351 | . . . . . 6 β’ 9 β β | |
3 | 1nn 12263 | . . . . . . 7 β’ 1 β β | |
4 | 2nn0 12529 | . . . . . . 7 β’ 2 β β0 | |
5 | 9nn0 12536 | . . . . . . 7 β’ 9 β β0 | |
6 | 9lt10 12848 | . . . . . . 7 β’ 9 < ;10 | |
7 | 3, 4, 5, 6 | declti 12755 | . . . . . 6 β’ 9 < ;12 |
8 | 2, 7 | gtneii 11366 | . . . . 5 β’ ;12 β 9 |
9 | dsndx 17375 | . . . . . 6 β’ (distβndx) = ;12 | |
10 | tsetndx 17342 | . . . . . 6 β’ (TopSetβndx) = 9 | |
11 | 9, 10 | neeq12i 3004 | . . . . 5 β’ ((distβndx) β (TopSetβndx) β ;12 β 9) |
12 | 8, 11 | mpbir 230 | . . . 4 β’ (distβndx) β (TopSetβndx) |
13 | 1, 12 | setsnid 17187 | . . 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©)) |
14 | tngds.2 | . . . . . 6 β’ β = (-gβπΊ) | |
15 | 14 | fvexi 6916 | . . . . 5 β’ β β V |
16 | coexg 7945 | . . . . 5 β’ ((π β π β§ β β V) β (π β β ) β V) | |
17 | 15, 16 | mpan2 689 | . . . 4 β’ (π β π β (π β β ) β V) |
18 | 1 | setsid 17186 | . . . 4 β’ ((πΊ β V β§ (π β β ) β V) β (π β β ) = (distβ(πΊ sSet β¨(distβndx), (π β β )β©))) |
19 | 17, 18 | sylan2 591 | . . 3 β’ ((πΊ β V β§ π β π) β (π β β ) = (distβ(πΊ sSet β¨(distβndx), (π β β )β©))) |
20 | tngbas.t | . . . . 5 β’ π = (πΊ toNrmGrp π) | |
21 | eqid 2728 | . . . . 5 β’ (π β β ) = (π β β ) | |
22 | eqid 2728 | . . . . 5 β’ (MetOpenβ(π β β )) = (MetOpenβ(π β β )) | |
23 | 20, 14, 21, 22 | tngval 24576 | . . . 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©)) |
24 | 23 | fveq2d 6906 | . . 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©))) |
25 | 13, 19, 24 | 3eqtr4a 2794 | . 2 β’ ((πΊ β V β§ π β π) β (π β β ) = (distβπ)) |
26 | co02 6269 | . . . . 5 β’ (π β β ) = β | |
27 | df-ds 17264 | . . . . . 6 β’ dist = Slot ;12 | |
28 | 27 | str0 17167 | . . . . 5 β’ β = (distββ ) |
29 | 26, 28 | eqtri 2756 | . . . 4 β’ (π β β ) = (distββ ) |
30 | fvprc 6894 | . . . . . 6 β’ (Β¬ πΊ β V β (-gβπΊ) = β ) | |
31 | 14, 30 | eqtrid 2780 | . . . . 5 β’ (Β¬ πΊ β V β β = β ) |
32 | 31 | coeq2d 5869 | . . . 4 β’ (Β¬ πΊ β V β (π β β ) = (π β β )) |
33 | reldmtng 24575 | . . . . . . 7 β’ Rel dom toNrmGrp | |
34 | 33 | ovprc1 7465 | . . . . . 6 β’ (Β¬ πΊ β V β (πΊ toNrmGrp π) = β ) |
35 | 20, 34 | eqtrid 2780 | . . . . 5 β’ (Β¬ πΊ β V β π = β ) |
36 | 35 | fveq2d 6906 | . . . 4 β’ (Β¬ πΊ β V β (distβπ) = (distββ )) |
37 | 29, 32, 36 | 3eqtr4a 2794 | . . 3 β’ (Β¬ πΊ β V β (π β β ) = (distβπ)) |
38 | 37 | adantr 479 | . 2 β’ ((Β¬ πΊ β V β§ π β π) β (π β β ) = (distβπ)) |
39 | 25, 38 | 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 394 = wceq 1533 β wcel 2098 β wne 2937 Vcvv 3473 β c0 4326 β¨cop 4638 β ccom 5686 βcfv 6553 (class class class)co 7426 1c1 11149 2c2 12307 9c9 12314 ;cdc 12717 sSet csts 17141 ndxcnx 17171 TopSetcts 17248 distcds 17251 -gcsg 18906 MetOpencmopn 21283 toNrmGrp ctng 24515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-sets 17142 df-slot 17160 df-ndx 17172 df-tset 17261 df-ds 17264 df-tng 24521 |
This theorem is referenced by: (None) |
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