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Theorem tngnm 18167
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 5575 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  =  ( x  e.  X  |->  ( N `  x ) ) )
3 tngnm.x . . . . . . . 8  |-  X  =  ( Base `  G
)
4 eqid 2283 . . . . . . . 8  |-  ( -g `  G )  =  (
-g `  G )
53, 4grpsubf 14545 . . . . . . 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 2283 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
93, 8grpidcl 14510 . . . . . . . 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 4721 . . . . . . 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 5596 . . . . . 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 5861 . . . . 5  |-  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G
) )  =  ( ( N  o.  ( -g `  G ) ) `
 <. x ,  ( 0g `  G )
>. )
16 df-ov 5861 . . . . . 6  |-  ( x ( -g `  G
) ( 0g `  G ) )  =  ( ( -g `  G
) `  <. x ,  ( 0g `  G
) >. )
1716fveq2i 5528 . . . . 5  |-  ( N `
 ( x (
-g `  G )
( 0g `  G
) ) )  =  ( N `  (
( -g `  G ) `
 <. x ,  ( 0g `  G )
>. ) )
1814, 15, 173eqtr4g 2340 . . . 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 14551 . . . . . 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 5529 . . . 4  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( N `  ( x ( -g `  G ) ( 0g
`  G ) ) )  =  ( N `
 x ) )
2218, 21eqtr2d 2316 . . 3  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( N `  x )  =  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) ) )
2322mpteq2dva 4106 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x  e.  X  |->  ( N `  x ) )  =  ( x  e.  X  |->  ( x ( N  o.  ( -g `  G
) ) ( 0g
`  G ) ) ) )
24 fvex 5539 . . . . . . . 8  |-  ( Base `  G )  e.  _V
253, 24eqeltri 2353 . . . . . . 7  |-  X  e. 
_V
26 tngnm.a . . . . . . 7  |-  A  e. 
_V
27 fex2 5401 . . . . . . 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 18157 . . . . 5  |-  ( N  e.  _V  ->  X  =  ( Base `  T
) )
3229, 31syl 15 . . . 4  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  X  =  (
Base `  T )
)
3330, 4tngds 18164 . . . . . 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 2284 . . . . 5  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  x  =  x )
3630, 8tng0 18159 . . . . . 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 5879 . . . 4  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) )  =  ( x ( dist `  T
) ( 0g `  T ) ) )
3932, 38mpteq12dv 4098 . . 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 2283 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
41 eqid 2283 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
42 eqid 2283 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
43 eqid 2283 . . . 4  |-  ( dist `  T )  =  (
dist `  T )
4440, 41, 42, 43nmfval 18111 . . 3  |-  ( norm `  T )  =  ( x  e.  ( Base `  T )  |->  ( x ( dist `  T
) ( 0g `  T ) ) )
4539, 44syl6eqr 2333 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x  e.  X  |->  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) ) )  =  (
norm `  T )
)
462, 23, 453eqtrd 2319 1  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  =  (
norm `  T )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    e. cmpt 4077    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   distcds 13217   0gc0g 13400   Grpcgrp 14362   -gcsg 14365   normcnm 18099   toNrmGrp ctng 18101
This theorem is referenced by:  tngngp2  18168  tngngp  18170  tngnrg  18185  tchnmfval  18659  tchcph  18667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-tset 13227  df-ds 13230  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-nm 18105  df-tng 18107
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