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Theorem nmods 18253
Description: Upper bound for the distance between the values of a bounded linear operator. (Contributed by Mario Carneiro, 22-Oct-2015.)
Hypotheses
Ref Expression
nmods.n  |-  N  =  ( S normOp T )
nmods.v  |-  V  =  ( Base `  S
)
nmods.c  |-  C  =  ( dist `  S
)
nmods.d  |-  D  =  ( dist `  T
)
Assertion
Ref Expression
nmods  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  (
( F `  A
) D ( F `
 B ) )  <_  ( ( N `
 F )  x.  ( A C B ) ) )

Proof of Theorem nmods
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  F  e.  ( S NGHom  T ) )
2 nghmrcl1 18241 . . . . 5  |-  ( F  e.  ( S NGHom  T
)  ->  S  e. NrmGrp )
3 ngpgrp 18121 . . . . 5  |-  ( S  e. NrmGrp  ->  S  e.  Grp )
42, 3syl 15 . . . 4  |-  ( F  e.  ( S NGHom  T
)  ->  S  e.  Grp )
5 nmods.v . . . . 5  |-  V  =  ( Base `  S
)
6 eqid 2283 . . . . 5  |-  ( -g `  S )  =  (
-g `  S )
75, 6grpsubcl 14546 . . . 4  |-  ( ( S  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( A ( -g `  S ) B )  e.  V )
84, 7syl3an1 1215 . . 3  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  ( A ( -g `  S
) B )  e.  V )
9 nmods.n . . . 4  |-  N  =  ( S normOp T )
10 eqid 2283 . . . 4  |-  ( norm `  S )  =  (
norm `  S )
11 eqid 2283 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
129, 5, 10, 11nmoi 18237 . . 3  |-  ( ( F  e.  ( S NGHom 
T )  /\  ( A ( -g `  S
) B )  e.  V )  ->  (
( norm `  T ) `  ( F `  ( A ( -g `  S
) B ) ) )  <_  ( ( N `  F )  x.  ( ( norm `  S
) `  ( A
( -g `  S ) B ) ) ) )
131, 8, 12syl2anc 642 . 2  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  (
( norm `  T ) `  ( F `  ( A ( -g `  S
) B ) ) )  <_  ( ( N `  F )  x.  ( ( norm `  S
) `  ( A
( -g `  S ) B ) ) ) )
14 nghmrcl2 18242 . . . . 5  |-  ( F  e.  ( S NGHom  T
)  ->  T  e. NrmGrp )
15143ad2ant1 976 . . . 4  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  T  e. NrmGrp )
16 nghmghm 18243 . . . . . . 7  |-  ( F  e.  ( S NGHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
17163ad2ant1 976 . . . . . 6  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  F  e.  ( S  GrpHom  T ) )
18 eqid 2283 . . . . . . 7  |-  ( Base `  T )  =  (
Base `  T )
195, 18ghmf 14687 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  F : V
--> ( Base `  T
) )
2017, 19syl 15 . . . . 5  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  F : V --> ( Base `  T
) )
21 simp2 956 . . . . 5  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  A  e.  V )
22 ffvelrn 5663 . . . . 5  |-  ( ( F : V --> ( Base `  T )  /\  A  e.  V )  ->  ( F `  A )  e.  ( Base `  T
) )
2320, 21, 22syl2anc 642 . . . 4  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  ( F `  A )  e.  ( Base `  T
) )
24 simp3 957 . . . . 5  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  B  e.  V )
25 ffvelrn 5663 . . . . 5  |-  ( ( F : V --> ( Base `  T )  /\  B  e.  V )  ->  ( F `  B )  e.  ( Base `  T
) )
2620, 24, 25syl2anc 642 . . . 4  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  ( F `  B )  e.  ( Base `  T
) )
27 eqid 2283 . . . . 5  |-  ( -g `  T )  =  (
-g `  T )
28 nmods.d . . . . 5  |-  D  =  ( dist `  T
)
2911, 18, 27, 28ngpds 18125 . . . 4  |-  ( ( T  e. NrmGrp  /\  ( F `  A )  e.  ( Base `  T
)  /\  ( F `  B )  e.  (
Base `  T )
)  ->  ( ( F `  A ) D ( F `  B ) )  =  ( ( norm `  T
) `  ( ( F `  A )
( -g `  T ) ( F `  B
) ) ) )
3015, 23, 26, 29syl3anc 1182 . . 3  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  (
( F `  A
) D ( F `
 B ) )  =  ( ( norm `  T ) `  (
( F `  A
) ( -g `  T
) ( F `  B ) ) ) )
315, 6, 27ghmsub 14691 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  A  e.  V  /\  B  e.  V )  ->  ( F `  ( A
( -g `  S ) B ) )  =  ( ( F `  A ) ( -g `  T ) ( F `
 B ) ) )
3216, 31syl3an1 1215 . . . 4  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  ( F `  ( A
( -g `  S ) B ) )  =  ( ( F `  A ) ( -g `  T ) ( F `
 B ) ) )
3332fveq2d 5529 . . 3  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  (
( norm `  T ) `  ( F `  ( A ( -g `  S
) B ) ) )  =  ( (
norm `  T ) `  ( ( F `  A ) ( -g `  T ) ( F `
 B ) ) ) )
3430, 33eqtr4d 2318 . 2  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  (
( F `  A
) D ( F `
 B ) )  =  ( ( norm `  T ) `  ( F `  ( A
( -g `  S ) B ) ) ) )
35 nmods.c . . . . 5  |-  C  =  ( dist `  S
)
3610, 5, 6, 35ngpds 18125 . . . 4  |-  ( ( S  e. NrmGrp  /\  A  e.  V  /\  B  e.  V )  ->  ( A C B )  =  ( ( norm `  S
) `  ( A
( -g `  S ) B ) ) )
372, 36syl3an1 1215 . . 3  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  ( A C B )  =  ( ( norm `  S
) `  ( A
( -g `  S ) B ) ) )
3837oveq2d 5874 . 2  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  (
( N `  F
)  x.  ( A C B ) )  =  ( ( N `
 F )  x.  ( ( norm `  S
) `  ( A
( -g `  S ) B ) ) ) )
3913, 34, 383brtr4d 4053 1  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  (
( F `  A
) D ( F `
 B ) )  <_  ( ( N `
 F )  x.  ( A C B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858    x. cmul 8742    <_ cle 8868   Basecbs 13148   distcds 13217   Grpcgrp 14362   -gcsg 14365    GrpHom cghm 14680   normcnm 18099  NrmGrpcngp 18100   normOpcnmo 18214   NGHom cnghm 18215
This theorem is referenced by:  nghmcn  18254
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  ax-pre-sup 8815
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-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ico 10662  df-topgen 13344  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-ghm 14681  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-xms 17885  df-ms 17886  df-nm 18105  df-ngp 18106  df-nmo 18217  df-nghm 18218
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