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Theorem ressnm 27873
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 14774 . . . 4  |-  ( A 
C_  B  ->  A  =  ( Base `  H
) )
433ad2ant3 1017 . . 3  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  A  =  ( Base `  H
) )
5 fvex 5858 . . . . . . . 8  |-  ( Base `  G )  e.  _V
62, 5eqeltri 2538 . . . . . . 7  |-  B  e. 
_V
76ssex 4581 . . . . . 6  |-  ( A 
C_  B  ->  A  e.  _V )
8 eqid 2454 . . . . . . 7  |-  ( dist `  G )  =  (
dist `  G )
91, 8ressds 14902 . . . . . 6  |-  ( A  e.  _V  ->  ( dist `  G )  =  ( dist `  H
) )
107, 9syl 16 . . . . 5  |-  ( A 
C_  B  ->  ( dist `  G )  =  ( dist `  H
) )
11103ad2ant3 1017 . . . 4  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  ( dist `  G )  =  ( dist `  H
) )
12 eqidd 2455 . . . 4  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  x  =  x )
13 ressnm.3 . . . . 5  |-  .0.  =  ( 0g `  G )
141, 2, 13ress0g 16148 . . . 4  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  .0.  =  ( 0g `  H ) )
1511, 12, 14oveq123d 6291 . . 3  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  (
x ( dist `  G
)  .0.  )  =  ( x ( dist `  H ) ( 0g
`  H ) ) )
164, 15mpteq12dv 4517 . 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 21275 . . . . 5  |-  N  =  ( x  e.  B  |->  ( x ( dist `  G )  .0.  )
)
1918reseq1i 5258 . . . 4  |-  ( N  |`  A )  =  ( ( x  e.  B  |->  ( x ( dist `  G )  .0.  )
)  |`  A )
20 resmpt 5311 . . . 4  |-  ( A 
C_  B  ->  (
( x  e.  B  |->  ( x ( dist `  G )  .0.  )
)  |`  A )  =  ( x  e.  A  |->  ( x ( dist `  G )  .0.  )
) )
2119, 20syl5eq 2507 . . 3  |-  ( A 
C_  B  ->  ( N  |`  A )  =  ( x  e.  A  |->  ( x ( dist `  G )  .0.  )
) )
22213ad2ant3 1017 . 2  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  ( N  |`  A )  =  ( x  e.  A  |->  ( x ( dist `  G )  .0.  )
) )
23 eqid 2454 . . . 4  |-  ( norm `  H )  =  (
norm `  H )
24 eqid 2454 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
25 eqid 2454 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
26 eqid 2454 . . . 4  |-  ( dist `  H )  =  (
dist `  H )
2723, 24, 25, 26nmfval 21275 . . 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 2505 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 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461    |-> cmpt 4497    |` cres 4990   ` cfv 5570  (class class class)co 6270   Basecbs 14716   ↾s cress 14717   distcds 14793   0gc0g 14929   Mndcmnd 16118   normcnm 21263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-ds 14806  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-nm 21269
This theorem is referenced by:  zringnm  28175  rezh  28186
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