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Theorem ressnm 26284
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 14352 . . . 4  |-  ( A 
C_  B  ->  A  =  ( Base `  H
) )
433ad2ant3 1011 . . 3  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  A  =  ( Base `  H
) )
5 fvex 5812 . . . . . . . 8  |-  ( Base `  G )  e.  _V
62, 5eqeltri 2538 . . . . . . 7  |-  B  e. 
_V
76ssex 4547 . . . . . 6  |-  ( A 
C_  B  ->  A  e.  _V )
8 eqid 2454 . . . . . . 7  |-  ( dist `  G )  =  (
dist `  G )
91, 8ressds 14475 . . . . . 6  |-  ( A  e.  _V  ->  ( dist `  G )  =  ( dist `  H
) )
107, 9syl 16 . . . . 5  |-  ( A 
C_  B  ->  ( dist `  G )  =  ( dist `  H
) )
11103ad2ant3 1011 . . . 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 15573 . . . 4  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  .0.  =  ( 0g `  H ) )
1511, 12, 14oveq123d 6224 . . 3  |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  (
x ( dist `  G
)  .0.  )  =  ( x ( dist `  H ) ( 0g
`  H ) ) )
164, 15mpteq12dv 4481 . 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 20323 . . . . 5  |-  N  =  ( x  e.  B  |->  ( x ( dist `  G )  .0.  )
)
1918reseq1i 5217 . . . 4  |-  ( N  |`  A )  =  ( ( x  e.  B  |->  ( x ( dist `  G )  .0.  )
)  |`  A )
20 resmpt 5267 . . . 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 1011 . 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 20323 . . 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 965    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3439    |-> cmpt 4461    |` cres 4953   ` cfv 5529  (class class class)co 6203   Basecbs 14296   ↾s cress 14297   distcds 14370   0gc0g 14501   Mndcmnd 15532   normcnm 20311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-ds 14383  df-0g 14503  df-mnd 15538  df-nm 20317
This theorem is referenced by:  zringnm  26556  zzsnmOLD  26558  rezh  26568
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