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Theorem isnlm 20256
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 967 . . . 4  |-  ( ( W  e. NrmGrp  /\  W  e. 
LMod  /\  F  e. NrmRing )  <->  ( ( W  e. NrmGrp  /\  W  e.  LMod )  /\  F  e. NrmRing ) )
3 elin 3539 . . . . 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 5701 . . . . 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 5691 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
11 isnlm.f . . . . . . . 8  |-  F  =  (Scalar `  W )
1210, 11syl6eqr 2493 . . . . . . 7  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
139, 12sylan9eqr 2497 . . . . . 6  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  f  =  F )
1413eleq1d 2509 . . . . 5  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
f  e. NrmRing  <->  F  e. NrmRing ) )
1513fveq2d 5695 . . . . . . 7  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  ( Base `  F
) )
16 isnlm.k . . . . . . 7  |-  K  =  ( Base `  F
)
1715, 16syl6eqr 2493 . . . . . 6  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  K )
18 simpl 457 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  w  =  W )
1918fveq2d 5695 . . . . . . . 8  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  w )  =  ( Base `  W
) )
20 isnlm.v . . . . . . . 8  |-  V  =  ( Base `  W
)
2119, 20syl6eqr 2493 . . . . . . 7  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  w )  =  V )
2218fveq2d 5695 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( norm `  w )  =  ( norm `  W
) )
23 isnlm.n . . . . . . . . . 10  |-  N  =  ( norm `  W
)
2422, 23syl6eqr 2493 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( norm `  w )  =  N )
2518fveq2d 5695 . . . . . . . . . . 11  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( .s `  w )  =  ( .s `  W
) )
26 isnlm.s . . . . . . . . . . 11  |-  .x.  =  ( .s `  W )
2725, 26syl6eqr 2493 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( .s `  w )  = 
.x.  )
2827oveqd 6108 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
x ( .s `  w ) y )  =  ( x  .x.  y ) )
2924, 28fveq12d 5697 . . . . . . . 8  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( norm `  w ) `  ( x ( .s
`  w ) y ) )  =  ( N `  ( x 
.x.  y ) ) )
3013fveq2d 5695 . . . . . . . . . . 11  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( norm `  f )  =  ( norm `  F
) )
31 isnlm.a . . . . . . . . . . 11  |-  A  =  ( norm `  F
)
3230, 31syl6eqr 2493 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( norm `  f )  =  A )
3332fveq1d 5693 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( norm `  f ) `  x )  =  ( A `  x ) )
3424fveq1d 5693 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( norm `  w ) `  y )  =  ( N `  y ) )
3533, 34oveq12d 6109 . . . . . . . 8  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( ( norm `  f
) `  x )  x.  ( ( norm `  w
) `  y )
)  =  ( ( A `  x )  x.  ( N `  y ) ) )
3629, 35eqeq12d 2457 . . . . . . 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 2931 . . . . . 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 2931 . . . . 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 3223 . . 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 20179 . . 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 3119 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972   [.wsbc 3186    i^i cin 3327   ` cfv 5418  (class class class)co 6091    x. cmul 9287   Basecbs 14174  Scalarcsca 14241   .scvsca 14242   LModclmod 16948   normcnm 20169  NrmGrpcngp 20170  NrmRingcnrg 20172  NrmModcnlm 20173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-ov 6094  df-nlm 20179
This theorem is referenced by:  nmvs  20257  nlmngp  20258  nlmlmod  20259  nlmnrg  20260  sranlm  20265  lssnlm  20281  tchcph  20752  cnzh  26399  rezh  26400
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