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Theorem nmods 18778
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 957 . . 3  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  F  e.  ( S NGHom  T ) )
2 nghmrcl1 18766 . . . . 5  |-  ( F  e.  ( S NGHom  T
)  ->  S  e. NrmGrp )
3 ngpgrp 18646 . . . . 5  |-  ( S  e. NrmGrp  ->  S  e.  Grp )
42, 3syl 16 . . . 4  |-  ( F  e.  ( S NGHom  T
)  ->  S  e.  Grp )
5 nmods.v . . . . 5  |-  V  =  ( Base `  S
)
6 eqid 2436 . . . . 5  |-  ( -g `  S )  =  (
-g `  S )
75, 6grpsubcl 14869 . . . 4  |-  ( ( S  e.  Grp  /\  A  e.  V  /\  B  e.  V )  ->  ( A ( -g `  S ) B )  e.  V )
84, 7syl3an1 1217 . . 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 2436 . . . 4  |-  ( norm `  S )  =  (
norm `  S )
11 eqid 2436 . . . 4  |-  ( norm `  T )  =  (
norm `  T )
129, 5, 10, 11nmoi 18762 . . 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 643 . 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 18767 . . . . 5  |-  ( F  e.  ( S NGHom  T
)  ->  T  e. NrmGrp )
15143ad2ant1 978 . . . 4  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  T  e. NrmGrp )
16 nghmghm 18768 . . . . . . 7  |-  ( F  e.  ( S NGHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
17163ad2ant1 978 . . . . . 6  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  F  e.  ( S  GrpHom  T ) )
18 eqid 2436 . . . . . . 7  |-  ( Base `  T )  =  (
Base `  T )
195, 18ghmf 15010 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  F : V
--> ( Base `  T
) )
2017, 19syl 16 . . . . 5  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  F : V --> ( Base `  T
) )
21 simp2 958 . . . . 5  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  A  e.  V )
2220, 21ffvelrnd 5871 . . . 4  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  ( F `  A )  e.  ( Base `  T
) )
23 simp3 959 . . . . 5  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  B  e.  V )
2420, 23ffvelrnd 5871 . . . 4  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  ( F `  B )  e.  ( Base `  T
) )
25 eqid 2436 . . . . 5  |-  ( -g `  T )  =  (
-g `  T )
26 nmods.d . . . . 5  |-  D  =  ( dist `  T
)
2711, 18, 25, 26ngpds 18650 . . . 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
) ) ) )
2815, 22, 24, 27syl3anc 1184 . . 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 ) ) ) )
295, 6, 25ghmsub 15014 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  A  e.  V  /\  B  e.  V )  ->  ( F `  ( A
( -g `  S ) B ) )  =  ( ( F `  A ) ( -g `  T ) ( F `
 B ) ) )
3016, 29syl3an1 1217 . . . 4  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  ( F `  ( A
( -g `  S ) B ) )  =  ( ( F `  A ) ( -g `  T ) ( F `
 B ) ) )
3130fveq2d 5732 . . 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 ) ) ) )
3228, 31eqtr4d 2471 . 2  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  (
( F `  A
) D ( F `
 B ) )  =  ( ( norm `  T ) `  ( F `  ( A
( -g `  S ) B ) ) ) )
33 nmods.c . . . . 5  |-  C  =  ( dist `  S
)
3410, 5, 6, 33ngpds 18650 . . . 4  |-  ( ( S  e. NrmGrp  /\  A  e.  V  /\  B  e.  V )  ->  ( A C B )  =  ( ( norm `  S
) `  ( A
( -g `  S ) B ) ) )
352, 34syl3an1 1217 . . 3  |-  ( ( F  e.  ( S NGHom 
T )  /\  A  e.  V  /\  B  e.  V )  ->  ( A C B )  =  ( ( norm `  S
) `  ( A
( -g `  S ) B ) ) )
3635oveq2d 6097 . 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 ) ) ) )
3713, 32, 363brtr4d 4242 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 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   -->wf 5450   ` cfv 5454  (class class class)co 6081    x. cmul 8995    <_ cle 9121   Basecbs 13469   distcds 13538   Grpcgrp 14685   -gcsg 14688    GrpHom cghm 15003   normcnm 18624  NrmGrpcngp 18625   normOpcnmo 18739   NGHom cnghm 18740
This theorem is referenced by:  nghmcn  18779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ico 10922  df-topgen 13667  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-ghm 15004  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-xms 18350  df-ms 18351  df-nm 18630  df-ngp 18631  df-nmo 18742  df-nghm 18743
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