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Mirrors > Home > MPE Home > Th. List > tngdsOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of tngds 24027 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 17272 | . . . 4 β’ dist = Slot (distβndx) | |
2 | 9re 12257 | . . . . . 6 β’ 9 β β | |
3 | 1nn 12169 | . . . . . . 7 β’ 1 β β | |
4 | 2nn0 12435 | . . . . . . 7 β’ 2 β β0 | |
5 | 9nn0 12442 | . . . . . . 7 β’ 9 β β0 | |
6 | 9lt10 12754 | . . . . . . 7 β’ 9 < ;10 | |
7 | 3, 4, 5, 6 | declti 12661 | . . . . . 6 β’ 9 < ;12 |
8 | 2, 7 | gtneii 11272 | . . . . 5 β’ ;12 β 9 |
9 | dsndx 17271 | . . . . . 6 β’ (distβndx) = ;12 | |
10 | tsetndx 17238 | . . . . . 6 β’ (TopSetβndx) = 9 | |
11 | 9, 10 | neeq12i 3007 | . . . . 5 β’ ((distβndx) β (TopSetβndx) β ;12 β 9) |
12 | 8, 11 | mpbir 230 | . . . 4 β’ (distβndx) β (TopSetβndx) |
13 | 1, 12 | setsnid 17086 | . . 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 6857 | . . . . 5 β’ β β V |
16 | coexg 7867 | . . . . 5 β’ ((π β π β§ β β V) β (π β β ) β V) | |
17 | 15, 16 | mpan2 690 | . . . 4 β’ (π β π β (π β β ) β V) |
18 | 1 | setsid 17085 | . . . 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 594 | . . 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 2733 | . . . . 5 β’ (π β β ) = (π β β ) | |
22 | eqid 2733 | . . . . 5 β’ (MetOpenβ(π β β )) = (MetOpenβ(π β β )) | |
23 | 20, 14, 21, 22 | tngval 24011 | . . . 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 6847 | . . 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 2799 | . 2 β’ ((πΊ β V β§ π β π) β (π β β ) = (distβπ)) |
26 | co02 6213 | . . . . 5 β’ (π β β ) = β | |
27 | df-ds 17160 | . . . . . 6 β’ dist = Slot ;12 | |
28 | 27 | str0 17066 | . . . . 5 β’ β = (distββ ) |
29 | 26, 28 | eqtri 2761 | . . . 4 β’ (π β β ) = (distββ ) |
30 | fvprc 6835 | . . . . . 6 β’ (Β¬ πΊ β V β (-gβπΊ) = β ) | |
31 | 14, 30 | eqtrid 2785 | . . . . 5 β’ (Β¬ πΊ β V β β = β ) |
32 | 31 | coeq2d 5819 | . . . 4 β’ (Β¬ πΊ β V β (π β β ) = (π β β )) |
33 | reldmtng 24010 | . . . . . . 7 β’ Rel dom toNrmGrp | |
34 | 33 | ovprc1 7397 | . . . . . 6 β’ (Β¬ πΊ β V β (πΊ toNrmGrp π) = β ) |
35 | 20, 34 | eqtrid 2785 | . . . . 5 β’ (Β¬ πΊ β V β π = β ) |
36 | 35 | fveq2d 6847 | . . . 4 β’ (Β¬ πΊ β V β (distβπ) = (distββ )) |
37 | 29, 32, 36 | 3eqtr4a 2799 | . . 3 β’ (Β¬ πΊ β V β (π β β ) = (distβπ)) |
38 | 37 | adantr 482 | . 2 β’ ((Β¬ πΊ β V β§ π β π) β (π β β ) = (distβπ)) |
39 | 25, 38 | 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 397 = wceq 1542 β wcel 2107 β wne 2940 Vcvv 3444 β c0 4283 β¨cop 4593 β ccom 5638 βcfv 6497 (class class class)co 7358 1c1 11057 2c2 12213 9c9 12220 ;cdc 12623 sSet csts 17040 ndxcnx 17070 TopSetcts 17144 distcds 17147 -gcsg 18755 MetOpencmopn 20802 toNrmGrp ctng 23950 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-sets 17041 df-slot 17059 df-ndx 17071 df-tset 17157 df-ds 17160 df-tng 23956 |
This theorem is referenced by: (None) |
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