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Theorem nmods 18305
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 18293 . . . . 5  |-  ( F  e.  ( S NGHom  T
)  ->  S  e. NrmGrp )
3 ngpgrp 18173 . . . . 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 2316 . . . . 5  |-  ( -g `  S )  =  (
-g `  S )
75, 6grpsubcl 14595 . . . 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 2316 . . . 4  |-  ( norm `  S )  =  (
norm `  S )
11 eqid 2316 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
129, 5, 10, 11nmoi 18289 . . 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 18294 . . . . 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 18295 . . . . . . 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 2316 . . . . . . 7  |-  ( Base `  T )  =  (
Base `  T )
195, 18ghmf 14736 . . . . . 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 5701 . . . . 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 5701 . . . . 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 2316 . . . . 5  |-  ( -g `  T )  =  (
-g `  T )
28 nmods.d . . . . 5  |-  D  =  ( dist `  T
)
2911, 18, 27, 28ngpds 18177 . . . 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 14740 . . . . 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 5567 . . 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 2351 . 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 18177 . . . 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 5916 . 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 4090 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 1633    e. wcel 1701   class class class wbr 4060   -->wf 5288   ` cfv 5292  (class class class)co 5900    x. cmul 8787    <_ cle 8913   Basecbs 13195   distcds 13264   Grpcgrp 14411   -gcsg 14414    GrpHom cghm 14729   normcnm 18151  NrmGrpcngp 18152   normOpcnmo 18266   NGHom cnghm 18267
This theorem is referenced by:  nghmcn  18306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-n0 10013  df-z 10072  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ico 10709  df-topgen 13393  df-0g 13453  df-mnd 14416  df-grp 14538  df-minusg 14539  df-sbg 14540  df-ghm 14730  df-xmet 16425  df-met 16426  df-bl 16427  df-mopn 16428  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-xms 17937  df-ms 17938  df-nm 18157  df-ngp 18158  df-nmo 18269  df-nghm 18270
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