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Theorem isassad 16063
Description: Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.)
Hypotheses
Ref Expression
isassad.v  |-  ( ph  ->  V  =  ( Base `  W ) )
isassad.f  |-  ( ph  ->  F  =  (Scalar `  W ) )
isassad.b  |-  ( ph  ->  B  =  ( Base `  F ) )
isassad.s  |-  ( ph  ->  .x.  =  ( .s
`  W ) )
isassad.t  |-  ( ph  ->  .X.  =  ( .r
`  W ) )
isassad.1  |-  ( ph  ->  W  e.  LMod )
isassad.2  |-  ( ph  ->  W  e.  Ring )
isassad.3  |-  ( ph  ->  F  e.  CRing )
isassad.4  |-  ( (
ph  /\  ( r  e.  B  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( r  .x.  x )  .X.  y
)  =  ( r 
.x.  ( x  .X.  y ) ) )
isassad.5  |-  ( (
ph  /\  ( r  e.  B  /\  x  e.  V  /\  y  e.  V ) )  -> 
( x  .X.  (
r  .x.  y )
)  =  ( r 
.x.  ( x  .X.  y ) ) )
Assertion
Ref Expression
isassad  |-  ( ph  ->  W  e. AssAlg )
Distinct variable groups:    x, r,
y, B    ph, r, x, y    x, V, y    W, r, x, y
Allowed substitution hints:    .x. ( x, y, r)    .X. ( x, y, r)    F( x, y, r)    V( r)

Proof of Theorem isassad
StepHypRef Expression
1 isassad.1 . . 3  |-  ( ph  ->  W  e.  LMod )
2 isassad.2 . . 3  |-  ( ph  ->  W  e.  Ring )
3 isassad.f . . . 4  |-  ( ph  ->  F  =  (Scalar `  W ) )
4 isassad.3 . . . 4  |-  ( ph  ->  F  e.  CRing )
53, 4eqeltrrd 2358 . . 3  |-  ( ph  ->  (Scalar `  W )  e.  CRing )
61, 2, 53jca 1132 . 2  |-  ( ph  ->  ( W  e.  LMod  /\  W  e.  Ring  /\  (Scalar `  W )  e.  CRing ) )
7 isassad.4 . . . . 5  |-  ( (
ph  /\  ( r  e.  B  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( r  .x.  x )  .X.  y
)  =  ( r 
.x.  ( x  .X.  y ) ) )
8 isassad.5 . . . . 5  |-  ( (
ph  /\  ( r  e.  B  /\  x  e.  V  /\  y  e.  V ) )  -> 
( x  .X.  (
r  .x.  y )
)  =  ( r 
.x.  ( x  .X.  y ) ) )
97, 8jca 518 . . . 4  |-  ( (
ph  /\  ( r  e.  B  /\  x  e.  V  /\  y  e.  V ) )  -> 
( ( ( r 
.x.  x )  .X.  y )  =  ( r  .x.  ( x 
.X.  y ) )  /\  ( x  .X.  ( r  .x.  y
) )  =  ( r  .x.  ( x 
.X.  y ) ) ) )
109ralrimivvva 2636 . . 3  |-  ( ph  ->  A. r  e.  B  A. x  e.  V  A. y  e.  V  ( ( ( r 
.x.  x )  .X.  y )  =  ( r  .x.  ( x 
.X.  y ) )  /\  ( x  .X.  ( r  .x.  y
) )  =  ( r  .x.  ( x 
.X.  y ) ) ) )
11 isassad.b . . . . 5  |-  ( ph  ->  B  =  ( Base `  F ) )
123fveq2d 5529 . . . . 5  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (Scalar `  W )
) )
1311, 12eqtrd 2315 . . . 4  |-  ( ph  ->  B  =  ( Base `  (Scalar `  W )
) )
14 isassad.v . . . . 5  |-  ( ph  ->  V  =  ( Base `  W ) )
15 isassad.t . . . . . . . . 9  |-  ( ph  ->  .X.  =  ( .r
`  W ) )
16 isassad.s . . . . . . . . . 10  |-  ( ph  ->  .x.  =  ( .s
`  W ) )
1716oveqd 5875 . . . . . . . . 9  |-  ( ph  ->  ( r  .x.  x
)  =  ( r ( .s `  W
) x ) )
18 eqidd 2284 . . . . . . . . 9  |-  ( ph  ->  y  =  y )
1915, 17, 18oveq123d 5879 . . . . . . . 8  |-  ( ph  ->  ( ( r  .x.  x )  .X.  y
)  =  ( ( r ( .s `  W ) x ) ( .r `  W
) y ) )
20 eqidd 2284 . . . . . . . . 9  |-  ( ph  ->  r  =  r )
2115oveqd 5875 . . . . . . . . 9  |-  ( ph  ->  ( x  .X.  y
)  =  ( x ( .r `  W
) y ) )
2216, 20, 21oveq123d 5879 . . . . . . . 8  |-  ( ph  ->  ( r  .x.  (
x  .X.  y )
)  =  ( r ( .s `  W
) ( x ( .r `  W ) y ) ) )
2319, 22eqeq12d 2297 . . . . . . 7  |-  ( ph  ->  ( ( ( r 
.x.  x )  .X.  y )  =  ( r  .x.  ( x 
.X.  y ) )  <-> 
( ( r ( .s `  W ) x ) ( .r
`  W ) y )  =  ( r ( .s `  W
) ( x ( .r `  W ) y ) ) ) )
24 eqidd 2284 . . . . . . . . 9  |-  ( ph  ->  x  =  x )
2516oveqd 5875 . . . . . . . . 9  |-  ( ph  ->  ( r  .x.  y
)  =  ( r ( .s `  W
) y ) )
2615, 24, 25oveq123d 5879 . . . . . . . 8  |-  ( ph  ->  ( x  .X.  (
r  .x.  y )
)  =  ( x ( .r `  W
) ( r ( .s `  W ) y ) ) )
2726, 22eqeq12d 2297 . . . . . . 7  |-  ( ph  ->  ( ( x  .X.  ( r  .x.  y
) )  =  ( r  .x.  ( x 
.X.  y ) )  <-> 
( x ( .r
`  W ) ( r ( .s `  W ) y ) )  =  ( r ( .s `  W
) ( x ( .r `  W ) y ) ) ) )
2823, 27anbi12d 691 . . . . . 6  |-  ( ph  ->  ( ( ( ( r  .x.  x ) 
.X.  y )  =  ( r  .x.  (
x  .X.  y )
)  /\  ( x  .X.  ( r  .x.  y
) )  =  ( r  .x.  ( x 
.X.  y ) ) )  <->  ( ( ( r ( .s `  W ) x ) ( .r `  W
) y )  =  ( r ( .s
`  W ) ( x ( .r `  W ) y ) )  /\  ( x ( .r `  W
) ( r ( .s `  W ) y ) )  =  ( r ( .s
`  W ) ( x ( .r `  W ) y ) ) ) ) )
2914, 28raleqbidv 2748 . . . . 5  |-  ( ph  ->  ( A. y  e.  V  ( ( ( r  .x.  x ) 
.X.  y )  =  ( r  .x.  (
x  .X.  y )
)  /\  ( x  .X.  ( r  .x.  y
) )  =  ( r  .x.  ( x 
.X.  y ) ) )  <->  A. y  e.  (
Base `  W )
( ( ( r ( .s `  W
) x ) ( .r `  W ) y )  =  ( r ( .s `  W ) ( x ( .r `  W
) y ) )  /\  ( x ( .r `  W ) ( r ( .s
`  W ) y ) )  =  ( r ( .s `  W ) ( x ( .r `  W
) y ) ) ) ) )
3014, 29raleqbidv 2748 . . . 4  |-  ( ph  ->  ( A. x  e.  V  A. y  e.  V  ( ( ( r  .x.  x ) 
.X.  y )  =  ( r  .x.  (
x  .X.  y )
)  /\  ( x  .X.  ( r  .x.  y
) )  =  ( r  .x.  ( x 
.X.  y ) ) )  <->  A. x  e.  (
Base `  W ) A. y  e.  ( Base `  W ) ( ( ( r ( .s `  W ) x ) ( .r
`  W ) y )  =  ( r ( .s `  W
) ( x ( .r `  W ) y ) )  /\  ( x ( .r
`  W ) ( r ( .s `  W ) y ) )  =  ( r ( .s `  W
) ( x ( .r `  W ) y ) ) ) ) )
3113, 30raleqbidv 2748 . . 3  |-  ( ph  ->  ( A. r  e.  B  A. x  e.  V  A. y  e.  V  ( ( ( r  .x.  x ) 
.X.  y )  =  ( r  .x.  (
x  .X.  y )
)  /\  ( x  .X.  ( r  .x.  y
) )  =  ( r  .x.  ( x 
.X.  y ) ) )  <->  A. r  e.  (
Base `  (Scalar `  W
) ) A. x  e.  ( Base `  W
) A. y  e.  ( Base `  W
) ( ( ( r ( .s `  W ) x ) ( .r `  W
) y )  =  ( r ( .s
`  W ) ( x ( .r `  W ) y ) )  /\  ( x ( .r `  W
) ( r ( .s `  W ) y ) )  =  ( r ( .s
`  W ) ( x ( .r `  W ) y ) ) ) ) )
3210, 31mpbid 201 . 2  |-  ( ph  ->  A. r  e.  (
Base `  (Scalar `  W
) ) A. x  e.  ( Base `  W
) A. y  e.  ( Base `  W
) ( ( ( r ( .s `  W ) x ) ( .r `  W
) y )  =  ( r ( .s
`  W ) ( x ( .r `  W ) y ) )  /\  ( x ( .r `  W
) ( r ( .s `  W ) y ) )  =  ( r ( .s
`  W ) ( x ( .r `  W ) y ) ) ) )
33 eqid 2283 . . 3  |-  ( Base `  W )  =  (
Base `  W )
34 eqid 2283 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
35 eqid 2283 . . 3  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
36 eqid 2283 . . 3  |-  ( .s
`  W )  =  ( .s `  W
)
37 eqid 2283 . . 3  |-  ( .r
`  W )  =  ( .r `  W
)
3833, 34, 35, 36, 37isassa 16056 . 2  |-  ( W  e. AssAlg 
<->  ( ( W  e. 
LMod  /\  W  e.  Ring  /\  (Scalar `  W )  e.  CRing )  /\  A. r  e.  ( Base `  (Scalar `  W )
) A. x  e.  ( Base `  W
) A. y  e.  ( Base `  W
) ( ( ( r ( .s `  W ) x ) ( .r `  W
) y )  =  ( r ( .s
`  W ) ( x ( .r `  W ) y ) )  /\  ( x ( .r `  W
) ( r ( .s `  W ) y ) )  =  ( r ( .s
`  W ) ( x ( .r `  W ) y ) ) ) ) )
396, 32, 38sylanbrc 645 1  |-  ( ph  ->  W  e. AssAlg )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   Ringcrg 15337   CRingccrg 15338   LModclmod 15627  AssAlgcasa 16050
This theorem is referenced by:  issubassa  16064  sraassa  16065  psrassa  16158  zlmassa  16478  matassa  27481  mendassa  27502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-assa 16053
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