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Theorem isnlm 21310
Description: A normed (left) module is a module which is also a normed group over a normed ring, such that the norm distributes over scalar multiplication. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
isnlm.v  |-  V  =  ( Base `  W
)
isnlm.n  |-  N  =  ( norm `  W
)
isnlm.s  |-  .x.  =  ( .s `  W )
isnlm.f  |-  F  =  (Scalar `  W )
isnlm.k  |-  K  =  ( Base `  F
)
isnlm.a  |-  A  =  ( norm `  F
)
Assertion
Ref Expression
isnlm  |-  ( W  e. NrmMod 
<->  ( ( W  e. NrmGrp  /\  W  e.  LMod  /\  F  e. NrmRing )  /\  A. x  e.  K  A. y  e.  V  ( N `  ( x  .x.  y ) )  =  ( ( A `  x )  x.  ( N `  y )
) ) )
Distinct variable groups:    x, y, A    x, N, y    x, V, y    x, K    x, W, y    x,  .x. , y
Allowed substitution hints:    F( x, y)    K( y)

Proof of Theorem isnlm
Dummy variables  w  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 anass 649 . 2  |-  ( ( ( W  e.  (NrmGrp 
i^i  LMod )  /\  F  e. NrmRing )  /\  A. x  e.  K  A. y  e.  V  ( N `  ( x  .x.  y
) )  =  ( ( A `  x
)  x.  ( N `
 y ) ) )  <->  ( W  e.  (NrmGrp  i^i  LMod )  /\  ( F  e. NrmRing  /\  A. x  e.  K  A. y  e.  V  ( N `  ( x  .x.  y ) )  =  ( ( A `  x )  x.  ( N `  y )
) ) ) )
2 df-3an 975 . . . 4  |-  ( ( W  e. NrmGrp  /\  W  e. 
LMod  /\  F  e. NrmRing )  <->  ( ( W  e. NrmGrp  /\  W  e.  LMod )  /\  F  e. NrmRing ) )
3 elin 3683 . . . . 5  |-  ( W  e.  (NrmGrp  i^i  LMod ) 
<->  ( W  e. NrmGrp  /\  W  e.  LMod ) )
43anbi1i 695 . . . 4  |-  ( ( W  e.  (NrmGrp  i^i  LMod )  /\  F  e. NrmRing ) 
<->  ( ( W  e. NrmGrp  /\  W  e.  LMod )  /\  F  e. NrmRing )
)
52, 4bitr4i 252 . . 3  |-  ( ( W  e. NrmGrp  /\  W  e. 
LMod  /\  F  e. NrmRing )  <->  ( W  e.  (NrmGrp  i^i  LMod )  /\  F  e. NrmRing ) )
65anbi1i 695 . 2  |-  ( ( ( W  e. NrmGrp  /\  W  e.  LMod  /\  F  e. NrmRing )  /\  A. x  e.  K  A. y  e.  V  ( N `  ( x  .x.  y ) )  =  ( ( A `  x )  x.  ( N `  y ) ) )  <-> 
( ( W  e.  (NrmGrp  i^i  LMod )  /\  F  e. NrmRing )  /\  A. x  e.  K  A. y  e.  V  ( N `  ( x  .x.  y ) )  =  ( ( A `  x )  x.  ( N `  y )
) ) )
7 fvex 5882 . . . . 5  |-  (Scalar `  w )  e.  _V
87a1i 11 . . . 4  |-  ( w  =  W  ->  (Scalar `  w )  e.  _V )
9 id 22 . . . . . . 7  |-  ( f  =  (Scalar `  w
)  ->  f  =  (Scalar `  w ) )
10 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
11 isnlm.f . . . . . . . 8  |-  F  =  (Scalar `  W )
1210, 11syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
139, 12sylan9eqr 2520 . . . . . 6  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  f  =  F )
1413eleq1d 2526 . . . . 5  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
f  e. NrmRing  <->  F  e. NrmRing ) )
1513fveq2d 5876 . . . . . . 7  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  ( Base `  F
) )
16 isnlm.k . . . . . . 7  |-  K  =  ( Base `  F
)
1715, 16syl6eqr 2516 . . . . . 6  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  K )
18 simpl 457 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  w  =  W )
1918fveq2d 5876 . . . . . . . 8  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  w )  =  ( Base `  W
) )
20 isnlm.v . . . . . . . 8  |-  V  =  ( Base `  W
)
2119, 20syl6eqr 2516 . . . . . . 7  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  w )  =  V )
2218fveq2d 5876 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( norm `  w )  =  ( norm `  W
) )
23 isnlm.n . . . . . . . . . 10  |-  N  =  ( norm `  W
)
2422, 23syl6eqr 2516 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( norm `  w )  =  N )
2518fveq2d 5876 . . . . . . . . . . 11  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( .s `  w )  =  ( .s `  W
) )
26 isnlm.s . . . . . . . . . . 11  |-  .x.  =  ( .s `  W )
2725, 26syl6eqr 2516 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( .s `  w )  = 
.x.  )
2827oveqd 6313 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
x ( .s `  w ) y )  =  ( x  .x.  y ) )
2924, 28fveq12d 5878 . . . . . . . 8  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( norm `  w ) `  ( x ( .s
`  w ) y ) )  =  ( N `  ( x 
.x.  y ) ) )
3013fveq2d 5876 . . . . . . . . . . 11  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( norm `  f )  =  ( norm `  F
) )
31 isnlm.a . . . . . . . . . . 11  |-  A  =  ( norm `  F
)
3230, 31syl6eqr 2516 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( norm `  f )  =  A )
3332fveq1d 5874 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( norm `  f ) `  x )  =  ( A `  x ) )
3424fveq1d 5874 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( norm `  w ) `  y )  =  ( N `  y ) )
3533, 34oveq12d 6314 . . . . . . . 8  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( ( norm `  f
) `  x )  x.  ( ( norm `  w
) `  y )
)  =  ( ( A `  x )  x.  ( N `  y ) ) )
3629, 35eqeq12d 2479 . . . . . . 7  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( ( norm `  w
) `  ( x
( .s `  w
) y ) )  =  ( ( (
norm `  f ) `  x )  x.  (
( norm `  w ) `  y ) )  <->  ( N `  ( x  .x.  y
) )  =  ( ( A `  x
)  x.  ( N `
 y ) ) ) )
3721, 36raleqbidv 3068 . . . . . 6  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( A. y  e.  ( Base `  w ) ( ( norm `  w
) `  ( x
( .s `  w
) y ) )  =  ( ( (
norm `  f ) `  x )  x.  (
( norm `  w ) `  y ) )  <->  A. y  e.  V  ( N `  ( x  .x.  y
) )  =  ( ( A `  x
)  x.  ( N `
 y ) ) ) )
3817, 37raleqbidv 3068 . . . . 5  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( A. x  e.  ( Base `  f ) A. y  e.  ( Base `  w ) ( (
norm `  w ) `  ( x ( .s
`  w ) y ) )  =  ( ( ( norm `  f
) `  x )  x.  ( ( norm `  w
) `  y )
)  <->  A. x  e.  K  A. y  e.  V  ( N `  ( x 
.x.  y ) )  =  ( ( A `
 x )  x.  ( N `  y
) ) ) )
3914, 38anbi12d 710 . . . 4  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( f  e. NrmRing  /\  A. x  e.  ( Base `  f ) A. y  e.  ( Base `  w
) ( ( norm `  w ) `  (
x ( .s `  w ) y ) )  =  ( ( ( norm `  f
) `  x )  x.  ( ( norm `  w
) `  y )
) )  <->  ( F  e. NrmRing  /\  A. x  e.  K  A. y  e.  V  ( N `  ( x  .x.  y ) )  =  ( ( A `  x )  x.  ( N `  y ) ) ) ) )
408, 39sbcied 3364 . . 3  |-  ( w  =  W  ->  ( [. (Scalar `  w )  /  f ]. (
f  e. NrmRing  /\  A. x  e.  ( Base `  f
) A. y  e.  ( Base `  w
) ( ( norm `  w ) `  (
x ( .s `  w ) y ) )  =  ( ( ( norm `  f
) `  x )  x.  ( ( norm `  w
) `  y )
) )  <->  ( F  e. NrmRing  /\  A. x  e.  K  A. y  e.  V  ( N `  ( x  .x.  y ) )  =  ( ( A `  x )  x.  ( N `  y ) ) ) ) )
41 df-nlm 21233 . . 3  |- NrmMod  =  {
w  e.  (NrmGrp  i^i  LMod )  |  [. (Scalar `  w )  /  f ]. ( f  e. NrmRing  /\  A. x  e.  ( Base `  f ) A. y  e.  ( Base `  w
) ( ( norm `  w ) `  (
x ( .s `  w ) y ) )  =  ( ( ( norm `  f
) `  x )  x.  ( ( norm `  w
) `  y )
) ) }
4240, 41elrab2 3259 . 2  |-  ( W  e. NrmMod 
<->  ( W  e.  (NrmGrp 
i^i  LMod )  /\  ( F  e. NrmRing  /\  A. x  e.  K  A. y  e.  V  ( N `  ( x  .x.  y
) )  =  ( ( A `  x
)  x.  ( N `
 y ) ) ) ) )
431, 6, 423bitr4ri 278 1  |-  ( W  e. NrmMod 
<->  ( ( W  e. NrmGrp  /\  W  e.  LMod  /\  F  e. NrmRing )  /\  A. x  e.  K  A. y  e.  V  ( N `  ( x  .x.  y ) )  =  ( ( A `  x )  x.  ( N `  y )
) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   [.wsbc 3327    i^i cin 3470   ` cfv 5594  (class class class)co 6296    x. cmul 9514   Basecbs 14644  Scalarcsca 14715   .scvsca 14716   LModclmod 17639   normcnm 21223  NrmGrpcngp 21224  NrmRingcnrg 21226  NrmModcnlm 21227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-nlm 21233
This theorem is referenced by:  nmvs  21311  nlmngp  21312  nlmlmod  21313  nlmnrg  21314  sranlm  21319  lssnlm  21335  tchcph  21806  cnzh  28112  rezh  28113
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