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Theorem isnlm 18703
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 631 . 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 938 . . . 4  |-  ( ( W  e. NrmGrp  /\  W  e. 
LMod  /\  F  e. NrmRing )  <->  ( ( W  e. NrmGrp  /\  W  e.  LMod )  /\  F  e. NrmRing ) )
3 elin 3522 . . . . 5  |-  ( W  e.  (NrmGrp  i^i  LMod ) 
<->  ( W  e. NrmGrp  /\  W  e.  LMod ) )
43anbi1i 677 . . . 4  |-  ( ( W  e.  (NrmGrp  i^i  LMod )  /\  F  e. NrmRing ) 
<->  ( ( W  e. NrmGrp  /\  W  e.  LMod )  /\  F  e. NrmRing )
)
52, 4bitr4i 244 . . 3  |-  ( ( W  e. NrmGrp  /\  W  e. 
LMod  /\  F  e. NrmRing )  <->  ( W  e.  (NrmGrp  i^i  LMod )  /\  F  e. NrmRing ) )
65anbi1i 677 . 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 5734 . . . . 5  |-  (Scalar `  w )  e.  _V
87a1i 11 . . . 4  |-  ( w  =  W  ->  (Scalar `  w )  e.  _V )
9 id 20 . . . . . . 7  |-  ( f  =  (Scalar `  w
)  ->  f  =  (Scalar `  w ) )
10 fveq2 5720 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
11 isnlm.f . . . . . . . 8  |-  F  =  (Scalar `  W )
1210, 11syl6eqr 2485 . . . . . . 7  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
139, 12sylan9eqr 2489 . . . . . 6  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  f  =  F )
1413eleq1d 2501 . . . . 5  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
f  e. NrmRing  <->  F  e. NrmRing ) )
1513fveq2d 5724 . . . . . . 7  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  ( Base `  F
) )
16 isnlm.k . . . . . . 7  |-  K  =  ( Base `  F
)
1715, 16syl6eqr 2485 . . . . . 6  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  =  K )
18 simpl 444 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  w  =  W )
1918fveq2d 5724 . . . . . . . 8  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  w )  =  ( Base `  W
) )
20 isnlm.v . . . . . . . 8  |-  V  =  ( Base `  W
)
2119, 20syl6eqr 2485 . . . . . . 7  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  w )  =  V )
2218fveq2d 5724 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( norm `  w )  =  ( norm `  W
) )
23 isnlm.n . . . . . . . . . 10  |-  N  =  ( norm `  W
)
2422, 23syl6eqr 2485 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( norm `  w )  =  N )
2518fveq2d 5724 . . . . . . . . . . 11  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( .s `  w )  =  ( .s `  W
) )
26 isnlm.s . . . . . . . . . . 11  |-  .x.  =  ( .s `  W )
2725, 26syl6eqr 2485 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( .s `  w )  = 
.x.  )
2827oveqd 6090 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
x ( .s `  w ) y )  =  ( x  .x.  y ) )
2924, 28fveq12d 5726 . . . . . . . 8  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( norm `  w ) `  ( x ( .s
`  w ) y ) )  =  ( N `  ( x 
.x.  y ) ) )
3013fveq2d 5724 . . . . . . . . . . 11  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( norm `  f )  =  ( norm `  F
) )
31 isnlm.a . . . . . . . . . . 11  |-  A  =  ( norm `  F
)
3230, 31syl6eqr 2485 . . . . . . . . . 10  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( norm `  f )  =  A )
3332fveq1d 5722 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( norm `  f ) `  x )  =  ( A `  x ) )
3424fveq1d 5722 . . . . . . . . 9  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( norm `  w ) `  y )  =  ( N `  y ) )
3533, 34oveq12d 6091 . . . . . . . 8  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  (
( ( norm `  f
) `  x )  x.  ( ( norm `  w
) `  y )
)  =  ( ( A `  x )  x.  ( N `  y ) ) )
3629, 35eqeq12d 2449 . . . . . . 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 2908 . . . . . 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 2908 . . . . 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 692 . . . 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 3189 . . 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 18626 . . 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 3086 . 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 270 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 set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   [.wsbc 3153    i^i cin 3311   ` cfv 5446  (class class class)co 6073    x. cmul 8987   Basecbs 13461  Scalarcsca 13524   .scvsca 13525   LModclmod 15942   normcnm 18616  NrmGrpcngp 18617  NrmRingcnrg 18619  NrmModcnlm 18620
This theorem is referenced by:  nmvs  18704  nlmngp  18705  nlmlmod  18706  nlmnrg  18707  sranlm  18712  lssnlm  18728  tchcph  19186  cnzh  24346  rezh  24347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-nlm 18626
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