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Theorem tngnm 18692
Description: The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
tngnm.t  |-  T  =  ( G toNrmGrp  N )
tngnm.x  |-  X  =  ( Base `  G
)
tngnm.a  |-  A  e. 
_V
Assertion
Ref Expression
tngnm  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  =  (
norm `  T )
)

Proof of Theorem tngnm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr 448 . . 3  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N : X --> A )
21feqmptd 5779 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  =  ( x  e.  X  |->  ( N `  x ) ) )
3 tngnm.x . . . . . . . 8  |-  X  =  ( Base `  G
)
4 eqid 2436 . . . . . . . 8  |-  ( -g `  G )  =  (
-g `  G )
53, 4grpsubf 14868 . . . . . . 7  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( X  X.  X
) --> X )
65ad2antrr 707 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( -g `  G ) : ( X  X.  X ) --> X )
7 simpr 448 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  x  e.  X )
8 eqid 2436 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
93, 8grpidcl 14833 . . . . . . . 8  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  X )
109ad2antrr 707 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( 0g `  G )  e.  X
)
11 opelxpi 4910 . . . . . . 7  |-  ( ( x  e.  X  /\  ( 0g `  G )  e.  X )  ->  <. x ,  ( 0g
`  G ) >.  e.  ( X  X.  X
) )
127, 10, 11syl2anc 643 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  <. x ,  ( 0g `  G
) >.  e.  ( X  X.  X ) )
13 fvco3 5800 . . . . . 6  |-  ( ( ( -g `  G
) : ( X  X.  X ) --> X  /\  <. x ,  ( 0g `  G )
>.  e.  ( X  X.  X ) )  -> 
( ( N  o.  ( -g `  G ) ) `  <. x ,  ( 0g `  G ) >. )  =  ( N `  ( ( -g `  G
) `  <. x ,  ( 0g `  G
) >. ) ) )
146, 12, 13syl2anc 643 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( ( N  o.  ( -g `  G ) ) `  <. x ,  ( 0g
`  G ) >.
)  =  ( N `
 ( ( -g `  G ) `  <. x ,  ( 0g `  G ) >. )
) )
15 df-ov 6084 . . . . 5  |-  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G
) )  =  ( ( N  o.  ( -g `  G ) ) `
 <. x ,  ( 0g `  G )
>. )
16 df-ov 6084 . . . . . 6  |-  ( x ( -g `  G
) ( 0g `  G ) )  =  ( ( -g `  G
) `  <. x ,  ( 0g `  G
) >. )
1716fveq2i 5731 . . . . 5  |-  ( N `
 ( x (
-g `  G )
( 0g `  G
) ) )  =  ( N `  (
( -g `  G ) `
 <. x ,  ( 0g `  G )
>. ) )
1814, 15, 173eqtr4g 2493 . . . 4  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( x
( N  o.  ( -g `  G ) ) ( 0g `  G
) )  =  ( N `  ( x ( -g `  G
) ( 0g `  G ) ) ) )
193, 8, 4grpsubid1 14874 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( x ( -g `  G ) ( 0g
`  G ) )  =  x )
2019adantlr 696 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( x
( -g `  G ) ( 0g `  G
) )  =  x )
2120fveq2d 5732 . . . 4  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( N `  ( x ( -g `  G ) ( 0g
`  G ) ) )  =  ( N `
 x ) )
2218, 21eqtr2d 2469 . . 3  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( N `  x )  =  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) ) )
2322mpteq2dva 4295 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x  e.  X  |->  ( N `  x ) )  =  ( x  e.  X  |->  ( x ( N  o.  ( -g `  G
) ) ( 0g
`  G ) ) ) )
24 fvex 5742 . . . . . . . 8  |-  ( Base `  G )  e.  _V
253, 24eqeltri 2506 . . . . . . 7  |-  X  e. 
_V
26 tngnm.a . . . . . . 7  |-  A  e. 
_V
27 fex2 5603 . . . . . . 7  |-  ( ( N : X --> A  /\  X  e.  _V  /\  A  e.  _V )  ->  N  e.  _V )
2825, 26, 27mp3an23 1271 . . . . . 6  |-  ( N : X --> A  ->  N  e.  _V )
2928adantl 453 . . . . 5  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  e.  _V )
30 tngnm.t . . . . . 6  |-  T  =  ( G toNrmGrp  N )
3130, 3tngbas 18682 . . . . 5  |-  ( N  e.  _V  ->  X  =  ( Base `  T
) )
3229, 31syl 16 . . . 4  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  X  =  (
Base `  T )
)
3330, 4tngds 18689 . . . . . 6  |-  ( N  e.  _V  ->  ( N  o.  ( -g `  G ) )  =  ( dist `  T
) )
3429, 33syl 16 . . . . 5  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( N  o.  ( -g `  G ) )  =  ( dist `  T ) )
35 eqidd 2437 . . . . 5  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  x  =  x )
3630, 8tng0 18684 . . . . . 6  |-  ( N  e.  _V  ->  ( 0g `  G )  =  ( 0g `  T
) )
3729, 36syl 16 . . . . 5  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( 0g `  G )  =  ( 0g `  T ) )
3834, 35, 37oveq123d 6102 . . . 4  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) )  =  ( x ( dist `  T
) ( 0g `  T ) ) )
3932, 38mpteq12dv 4287 . . 3  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x  e.  X  |->  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) ) )  =  ( x  e.  ( Base `  T )  |->  ( x ( dist `  T
) ( 0g `  T ) ) ) )
40 eqid 2436 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
41 eqid 2436 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
42 eqid 2436 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
43 eqid 2436 . . . 4  |-  ( dist `  T )  =  (
dist `  T )
4440, 41, 42, 43nmfval 18636 . . 3  |-  ( norm `  T )  =  ( x  e.  ( Base `  T )  |->  ( x ( dist `  T
) ( 0g `  T ) ) )
4539, 44syl6eqr 2486 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x  e.  X  |->  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) ) )  =  (
norm `  T )
)
462, 23, 453eqtrd 2472 1  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  =  (
norm `  T )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817    e. cmpt 4266    X. cxp 4876    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   distcds 13538   0gc0g 13723   Grpcgrp 14685   -gcsg 14688   normcnm 18624   toNrmGrp ctng 18626
This theorem is referenced by:  tngngp2  18693  tngngp  18695  tngnrg  18710  tchnmfval  19186  tchcph  19194
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-tset 13548  df-ds 13551  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-nm 18630  df-tng 18632
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