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Theorem ressnm 26029
Description: The norm in a restricted structure. (Contributed by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
ressnm.1  |-  H  =  ( Gs  A )
ressnm.2  |-  B  =  ( Base `  G
)
ressnm.3  |-  .0.  =  ( 0g `  G )
ressnm.4  |-  N  =  ( norm `  G
)
Assertion
Ref Expression
ressnm  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  ( N  |`  A )  =  ( norm `  H
) )

Proof of Theorem ressnm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ressnm.1 . . . . 5  |-  H  =  ( Gs  A )
2 ressnm.2 . . . . 5  |-  B  =  ( Base `  G
)
31, 2ressbas2 14225 . . . 4  |-  ( A 
C_  B  ->  A  =  ( Base `  H
) )
433ad2ant3 1006 . . 3  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  A  =  ( Base `  H
) )
5 fvex 5698 . . . . . . . 8  |-  ( Base `  G )  e.  _V
62, 5eqeltri 2511 . . . . . . 7  |-  B  e. 
_V
76ssex 4433 . . . . . 6  |-  ( A 
C_  B  ->  A  e.  _V )
8 eqid 2441 . . . . . . 7  |-  ( dist `  G )  =  (
dist `  G )
91, 8ressds 14348 . . . . . 6  |-  ( A  e.  _V  ->  ( dist `  G )  =  ( dist `  H
) )
107, 9syl 16 . . . . 5  |-  ( A 
C_  B  ->  ( dist `  G )  =  ( dist `  H
) )
11103ad2ant3 1006 . . . 4  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  ( dist `  G )  =  ( dist `  H
) )
12 eqidd 2442 . . . 4  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  x  =  x )
13 ressnm.3 . . . . 5  |-  .0.  =  ( 0g `  G )
141, 2, 13ress0g 15446 . . . 4  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  .0.  =  ( 0g `  H ) )
1511, 12, 14oveq123d 6111 . . 3  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  (
x ( dist `  G
)  .0.  )  =  ( x ( dist `  H ) ( 0g
`  H ) ) )
164, 15mpteq12dv 4367 . 2  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  (
x  e.  A  |->  ( x ( dist `  G
)  .0.  ) )  =  ( x  e.  ( Base `  H
)  |->  ( x (
dist `  H )
( 0g `  H
) ) ) )
17 ressnm.4 . . . . . 6  |-  N  =  ( norm `  G
)
1817, 2, 13, 8nmfval 20081 . . . . 5  |-  N  =  ( x  e.  B  |->  ( x ( dist `  G )  .0.  )
)
1918reseq1i 5102 . . . 4  |-  ( N  |`  A )  =  ( ( x  e.  B  |->  ( x ( dist `  G )  .0.  )
)  |`  A )
20 resmpt 5153 . . . 4  |-  ( A 
C_  B  ->  (
( x  e.  B  |->  ( x ( dist `  G )  .0.  )
)  |`  A )  =  ( x  e.  A  |->  ( x ( dist `  G )  .0.  )
) )
2119, 20syl5eq 2485 . . 3  |-  ( A 
C_  B  ->  ( N  |`  A )  =  ( x  e.  A  |->  ( x ( dist `  G )  .0.  )
) )
22213ad2ant3 1006 . 2  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  ( N  |`  A )  =  ( x  e.  A  |->  ( x ( dist `  G )  .0.  )
) )
23 eqid 2441 . . . 4  |-  ( norm `  H )  =  (
norm `  H )
24 eqid 2441 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
25 eqid 2441 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
26 eqid 2441 . . . 4  |-  ( dist `  H )  =  (
dist `  H )
2723, 24, 25, 26nmfval 20081 . . 3  |-  ( norm `  H )  =  ( x  e.  ( Base `  H )  |->  ( x ( dist `  H
) ( 0g `  H ) ) )
2827a1i 11 . 2  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  ( norm `  H )  =  ( x  e.  (
Base `  H )  |->  ( x ( dist `  H ) ( 0g
`  H ) ) ) )
2916, 22, 283eqtr4d 2483 1  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  ( N  |`  A )  =  ( norm `  H
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 960    = wceq 1364    e. wcel 1761   _Vcvv 2970    C_ wss 3325    e. cmpt 4347    |` cres 4838   ` cfv 5415  (class class class)co 6090   Basecbs 14170   ↾s cress 14171   distcds 14243   0gc0g 14374   Mndcmnd 15405   normcnm 20069
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-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 2263  df-mo 2264  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-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-ress 14177  df-plusg 14247  df-ds 14256  df-0g 14376  df-mnd 15411  df-nm 20075
This theorem is referenced by:  zringnm  26308  zzsnmOLD  26310  rezh  26320
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