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Theorem tngnm 20196
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 458 . . 3  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N : X --> A )
21feqmptd 5741 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  =  ( x  e.  X  |->  ( N `  x ) ) )
3 tngnm.x . . . . . . . 8  |-  X  =  ( Base `  G
)
4 eqid 2441 . . . . . . . 8  |-  ( -g `  G )  =  (
-g `  G )
53, 4grpsubf 15598 . . . . . . 7  |-  ( G  e.  Grp  ->  ( -g `  G ) : ( X  X.  X
) --> X )
65ad2antrr 720 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( -g `  G ) : ( X  X.  X ) --> X )
7 simpr 458 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  x  e.  X )
8 eqid 2441 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
93, 8grpidcl 15559 . . . . . . . 8  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  X )
109ad2antrr 720 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( 0g `  G )  e.  X
)
11 opelxpi 4867 . . . . . . 7  |-  ( ( x  e.  X  /\  ( 0g `  G )  e.  X )  ->  <. x ,  ( 0g
`  G ) >.  e.  ( X  X.  X
) )
127, 10, 11syl2anc 656 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  <. x ,  ( 0g `  G
) >.  e.  ( X  X.  X ) )
13 fvco3 5765 . . . . . 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 656 . . . . 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 6093 . . . . 5  |-  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G
) )  =  ( ( N  o.  ( -g `  G ) ) `
 <. x ,  ( 0g `  G )
>. )
16 df-ov 6093 . . . . . 6  |-  ( x ( -g `  G
) ( 0g `  G ) )  =  ( ( -g `  G
) `  <. x ,  ( 0g `  G
) >. )
1716fveq2i 5691 . . . . 5  |-  ( N `
 ( x (
-g `  G )
( 0g `  G
) ) )  =  ( N `  (
( -g `  G ) `
 <. x ,  ( 0g `  G )
>. ) )
1814, 15, 173eqtr4g 2498 . . . 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 15604 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( x ( -g `  G ) ( 0g
`  G ) )  =  x )
2019adantlr 709 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( x
( -g `  G ) ( 0g `  G
) )  =  x )
2120fveq2d 5692 . . . 4  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( N `  ( x ( -g `  G ) ( 0g
`  G ) ) )  =  ( N `
 x ) )
2218, 21eqtr2d 2474 . . 3  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( N `  x )  =  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) ) )
2322mpteq2dva 4375 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x  e.  X  |->  ( N `  x ) )  =  ( x  e.  X  |->  ( x ( N  o.  ( -g `  G
) ) ( 0g
`  G ) ) ) )
24 fvex 5698 . . . . . . . 8  |-  ( Base `  G )  e.  _V
253, 24eqeltri 2511 . . . . . . 7  |-  X  e. 
_V
26 tngnm.a . . . . . . 7  |-  A  e. 
_V
27 fex2 6531 . . . . . . 7  |-  ( ( N : X --> A  /\  X  e.  _V  /\  A  e.  _V )  ->  N  e.  _V )
2825, 26, 27mp3an23 1301 . . . . . 6  |-  ( N : X --> A  ->  N  e.  _V )
2928adantl 463 . . . . 5  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  e.  _V )
30 tngnm.t . . . . . 6  |-  T  =  ( G toNrmGrp  N )
3130, 3tngbas 20186 . . . . 5  |-  ( N  e.  _V  ->  X  =  ( Base `  T
) )
3229, 31syl 16 . . . 4  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  X  =  (
Base `  T )
)
3330, 4tngds 20193 . . . . . 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 2442 . . . . 5  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  x  =  x )
3630, 8tng0 20188 . . . . . 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 6111 . . . 4  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) )  =  ( x ( dist `  T
) ( 0g `  T ) ) )
3932, 38mpteq12dv 4367 . . 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 2441 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
41 eqid 2441 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
42 eqid 2441 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
43 eqid 2441 . . . 4  |-  ( dist `  T )  =  (
dist `  T )
4440, 41, 42, 43nmfval 20140 . . 3  |-  ( norm `  T )  =  ( x  e.  ( Base `  T )  |->  ( x ( dist `  T
) ( 0g `  T ) ) )
4539, 44syl6eqr 2491 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x  e.  X  |->  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) ) )  =  (
norm `  T )
)
462, 23, 453eqtrd 2477 1  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  =  (
norm `  T )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970   <.cop 3880    e. cmpt 4347    X. cxp 4834    o. ccom 4840   -->wf 5411   ` cfv 5415  (class class class)co 6090   Basecbs 14170   distcds 14243   0gc0g 14374   Grpcgrp 15406   -gcsg 15409   normcnm 20128   toNrmGrp ctng 20130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-tset 14253  df-ds 14256  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-sbg 15540  df-nm 20134  df-tng 20136
This theorem is referenced by:  tngngp2  20197  tngngp  20199  tngnrg  20214  tchnmfval  20702  tchcph  20711
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