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Mirrors > Home > MPE Home > Th. List > tngdsOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of tngds 24519 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 17340 | . . . 4 β’ dist = Slot (distβndx) | |
2 | 9re 12315 | . . . . . 6 β’ 9 β β | |
3 | 1nn 12227 | . . . . . . 7 β’ 1 β β | |
4 | 2nn0 12493 | . . . . . . 7 β’ 2 β β0 | |
5 | 9nn0 12500 | . . . . . . 7 β’ 9 β β0 | |
6 | 9lt10 12812 | . . . . . . 7 β’ 9 < ;10 | |
7 | 3, 4, 5, 6 | declti 12719 | . . . . . 6 β’ 9 < ;12 |
8 | 2, 7 | gtneii 11330 | . . . . 5 β’ ;12 β 9 |
9 | dsndx 17339 | . . . . . 6 β’ (distβndx) = ;12 | |
10 | tsetndx 17306 | . . . . . 6 β’ (TopSetβndx) = 9 | |
11 | 9, 10 | neeq12i 3001 | . . . . 5 β’ ((distβndx) β (TopSetβndx) β ;12 β 9) |
12 | 8, 11 | mpbir 230 | . . . 4 β’ (distβndx) β (TopSetβndx) |
13 | 1, 12 | setsnid 17151 | . . 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 6899 | . . . . 5 β’ β β V |
16 | coexg 7919 | . . . . 5 β’ ((π β π β§ β β V) β (π β β ) β V) | |
17 | 15, 16 | mpan2 688 | . . . 4 β’ (π β π β (π β β ) β V) |
18 | 1 | setsid 17150 | . . . 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 592 | . . 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 2726 | . . . . 5 β’ (π β β ) = (π β β ) | |
22 | eqid 2726 | . . . . 5 β’ (MetOpenβ(π β β )) = (MetOpenβ(π β β )) | |
23 | 20, 14, 21, 22 | tngval 24503 | . . . 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 6889 | . . 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 2792 | . 2 β’ ((πΊ β V β§ π β π) β (π β β ) = (distβπ)) |
26 | co02 6253 | . . . . 5 β’ (π β β ) = β | |
27 | df-ds 17228 | . . . . . 6 β’ dist = Slot ;12 | |
28 | 27 | str0 17131 | . . . . 5 β’ β = (distββ ) |
29 | 26, 28 | eqtri 2754 | . . . 4 β’ (π β β ) = (distββ ) |
30 | fvprc 6877 | . . . . . 6 β’ (Β¬ πΊ β V β (-gβπΊ) = β ) | |
31 | 14, 30 | eqtrid 2778 | . . . . 5 β’ (Β¬ πΊ β V β β = β ) |
32 | 31 | coeq2d 5856 | . . . 4 β’ (Β¬ πΊ β V β (π β β ) = (π β β )) |
33 | reldmtng 24502 | . . . . . . 7 β’ Rel dom toNrmGrp | |
34 | 33 | ovprc1 7444 | . . . . . 6 β’ (Β¬ πΊ β V β (πΊ toNrmGrp π) = β ) |
35 | 20, 34 | eqtrid 2778 | . . . . 5 β’ (Β¬ πΊ β V β π = β ) |
36 | 35 | fveq2d 6889 | . . . 4 β’ (Β¬ πΊ β V β (distβπ) = (distββ )) |
37 | 29, 32, 36 | 3eqtr4a 2792 | . . 3 β’ (Β¬ πΊ β V β (π β β ) = (distβπ)) |
38 | 37 | adantr 480 | . 2 β’ ((Β¬ πΊ β V β§ π β π) β (π β β ) = (distβπ)) |
39 | 25, 38 | pm2.61ian 809 | 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 1533 β wcel 2098 β wne 2934 Vcvv 3468 β c0 4317 β¨cop 4629 β ccom 5673 βcfv 6537 (class class class)co 7405 1c1 11113 2c2 12271 9c9 12278 ;cdc 12681 sSet csts 17105 ndxcnx 17135 TopSetcts 17212 distcds 17215 -gcsg 18865 MetOpencmopn 21230 toNrmGrp ctng 24442 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-sets 17106 df-slot 17124 df-ndx 17136 df-tset 17225 df-ds 17228 df-tng 24448 |
This theorem is referenced by: (None) |
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