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Theorem tngnm 18183
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 447 . . 3  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N : X --> A )
21feqmptd 5591 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  =  ( x  e.  X  |->  ( N `  x ) ) )
3 tngnm.x . . . . . . . 8  |-  X  =  ( Base `  G
)
4 eqid 2296 . . . . . . . 8  |-  ( -g `  G )  =  (
-g `  G )
53, 4grpsubf 14561 . . . . . . 7  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( X  X.  X
) --> X )
65ad2antrr 706 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( -g `  G ) : ( X  X.  X ) --> X )
7 simpr 447 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  x  e.  X )
8 eqid 2296 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
93, 8grpidcl 14526 . . . . . . . 8  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  X )
109ad2antrr 706 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( 0g `  G )  e.  X
)
11 opelxpi 4737 . . . . . . 7  |-  ( ( x  e.  X  /\  ( 0g `  G )  e.  X )  ->  <. x ,  ( 0g
`  G ) >.  e.  ( X  X.  X
) )
127, 10, 11syl2anc 642 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  <. x ,  ( 0g `  G
) >.  e.  ( X  X.  X ) )
13 fvco3 5612 . . . . . 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 642 . . . . 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 5877 . . . . 5  |-  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G
) )  =  ( ( N  o.  ( -g `  G ) ) `
 <. x ,  ( 0g `  G )
>. )
16 df-ov 5877 . . . . . 6  |-  ( x ( -g `  G
) ( 0g `  G ) )  =  ( ( -g `  G
) `  <. x ,  ( 0g `  G
) >. )
1716fveq2i 5544 . . . . 5  |-  ( N `
 ( x (
-g `  G )
( 0g `  G
) ) )  =  ( N `  (
( -g `  G ) `
 <. x ,  ( 0g `  G )
>. ) )
1814, 15, 173eqtr4g 2353 . . . 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 14567 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( x ( -g `  G ) ( 0g
`  G ) )  =  x )
2019adantlr 695 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( x
( -g `  G ) ( 0g `  G
) )  =  x )
2120fveq2d 5545 . . . 4  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( N `  ( x ( -g `  G ) ( 0g
`  G ) ) )  =  ( N `
 x ) )
2218, 21eqtr2d 2329 . . 3  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( N `  x )  =  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) ) )
2322mpteq2dva 4122 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x  e.  X  |->  ( N `  x ) )  =  ( x  e.  X  |->  ( x ( N  o.  ( -g `  G
) ) ( 0g
`  G ) ) ) )
24 fvex 5555 . . . . . . . 8  |-  ( Base `  G )  e.  _V
253, 24eqeltri 2366 . . . . . . 7  |-  X  e. 
_V
26 tngnm.a . . . . . . 7  |-  A  e. 
_V
27 fex2 5417 . . . . . . 7  |-  ( ( N : X --> A  /\  X  e.  _V  /\  A  e.  _V )  ->  N  e.  _V )
2825, 26, 27mp3an23 1269 . . . . . 6  |-  ( N : X --> A  ->  N  e.  _V )
2928adantl 452 . . . . 5  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  e.  _V )
30 tngnm.t . . . . . 6  |-  T  =  ( G toNrmGrp  N )
3130, 3tngbas 18173 . . . . 5  |-  ( N  e.  _V  ->  X  =  ( Base `  T
) )
3229, 31syl 15 . . . 4  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  X  =  (
Base `  T )
)
3330, 4tngds 18180 . . . . . 6  |-  ( N  e.  _V  ->  ( N  o.  ( -g `  G ) )  =  ( dist `  T
) )
3429, 33syl 15 . . . . 5  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( N  o.  ( -g `  G ) )  =  ( dist `  T ) )
35 eqidd 2297 . . . . 5  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  x  =  x )
3630, 8tng0 18175 . . . . . 6  |-  ( N  e.  _V  ->  ( 0g `  G )  =  ( 0g `  T
) )
3729, 36syl 15 . . . . 5  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( 0g `  G )  =  ( 0g `  T ) )
3834, 35, 37oveq123d 5895 . . . 4  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) )  =  ( x ( dist `  T
) ( 0g `  T ) ) )
3932, 38mpteq12dv 4114 . . 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 2296 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
41 eqid 2296 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
42 eqid 2296 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
43 eqid 2296 . . . 4  |-  ( dist `  T )  =  (
dist `  T )
4440, 41, 42, 43nmfval 18127 . . 3  |-  ( norm `  T )  =  ( x  e.  ( Base `  T )  |->  ( x ( dist `  T
) ( 0g `  T ) ) )
4539, 44syl6eqr 2346 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x  e.  X  |->  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) ) )  =  (
norm `  T )
)
462, 23, 453eqtrd 2332 1  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  =  (
norm `  T )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    e. cmpt 4093    X. cxp 4703    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   distcds 13233   0gc0g 13416   Grpcgrp 14378   -gcsg 14381   normcnm 18115   toNrmGrp ctng 18117
This theorem is referenced by:  tngngp2  18184  tngngp  18186  tngnrg  18201  tchnmfval  18675  tchcph  18683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-tset 13243  df-ds 13246  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-nm 18121  df-tng 18123
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