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Theorem tngnm 18645
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 5738 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  =  ( x  e.  X  |->  ( N `  x ) ) )
3 tngnm.x . . . . . . . 8  |-  X  =  ( Base `  G
)
4 eqid 2404 . . . . . . . 8  |-  ( -g `  G )  =  (
-g `  G )
53, 4grpsubf 14823 . . . . . . 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 2404 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
93, 8grpidcl 14788 . . . . . . . 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 4869 . . . . . . 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 5759 . . . . . 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 6043 . . . . 5  |-  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G
) )  =  ( ( N  o.  ( -g `  G ) ) `
 <. x ,  ( 0g `  G )
>. )
16 df-ov 6043 . . . . . 6  |-  ( x ( -g `  G
) ( 0g `  G ) )  =  ( ( -g `  G
) `  <. x ,  ( 0g `  G
) >. )
1716fveq2i 5690 . . . . 5  |-  ( N `
 ( x (
-g `  G )
( 0g `  G
) ) )  =  ( N `  (
( -g `  G ) `
 <. x ,  ( 0g `  G )
>. ) )
1814, 15, 173eqtr4g 2461 . . . 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 14829 . . . . . 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 5691 . . . 4  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( N `  ( x ( -g `  G ) ( 0g
`  G ) ) )  =  ( N `
 x ) )
2218, 21eqtr2d 2437 . . 3  |-  ( ( ( G  e.  Grp  /\  N : X --> A )  /\  x  e.  X
)  ->  ( N `  x )  =  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) ) )
2322mpteq2dva 4255 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x  e.  X  |->  ( N `  x ) )  =  ( x  e.  X  |->  ( x ( N  o.  ( -g `  G
) ) ( 0g
`  G ) ) ) )
24 fvex 5701 . . . . . . . 8  |-  ( Base `  G )  e.  _V
253, 24eqeltri 2474 . . . . . . 7  |-  X  e. 
_V
26 tngnm.a . . . . . . 7  |-  A  e. 
_V
27 fex2 5562 . . . . . . 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 18635 . . . . 5  |-  ( N  e.  _V  ->  X  =  ( Base `  T
) )
3229, 31syl 16 . . . 4  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  X  =  (
Base `  T )
)
3330, 4tngds 18642 . . . . . 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 2405 . . . . 5  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  x  =  x )
3630, 8tng0 18637 . . . . . 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 6061 . . . 4  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) )  =  ( x ( dist `  T
) ( 0g `  T ) ) )
3932, 38mpteq12dv 4247 . . 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 2404 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
41 eqid 2404 . . . 4  |-  ( Base `  T )  =  (
Base `  T )
42 eqid 2404 . . . 4  |-  ( 0g
`  T )  =  ( 0g `  T
)
43 eqid 2404 . . . 4  |-  ( dist `  T )  =  (
dist `  T )
4440, 41, 42, 43nmfval 18589 . . 3  |-  ( norm `  T )  =  ( x  e.  ( Base `  T )  |->  ( x ( dist `  T
) ( 0g `  T ) ) )
4539, 44syl6eqr 2454 . 2  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  ( x  e.  X  |->  ( x ( N  o.  ( -g `  G ) ) ( 0g `  G ) ) )  =  (
norm `  T )
)
462, 23, 453eqtrd 2440 1  |-  ( ( G  e.  Grp  /\  N : X --> A )  ->  N  =  (
norm `  T )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777    e. cmpt 4226    X. cxp 4835    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040   Basecbs 13424   distcds 13493   0gc0g 13678   Grpcgrp 14640   -gcsg 14643   normcnm 18577   toNrmGrp ctng 18579
This theorem is referenced by:  tngngp2  18646  tngngp  18648  tngnrg  18663  tchnmfval  19139  tchcph  19147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-plusg 13497  df-tset 13503  df-ds 13506  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-nm 18583  df-tng 18585
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