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Theorem axmulass 8795
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulass 8819. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
Assertion
Ref Expression
axmulass  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( A  x.  ( B  x.  C
) ) )

Proof of Theorem axmulass
Dummy variables  x  y  z  w  v  u  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfcnqs 8780 . 2  |-  CC  =  ( ( R.  X.  R. ) /. `'  _E  )
2 mulcnsrec 8782 . 2  |-  ( ( ( x  e.  R.  /\  y  e.  R. )  /\  ( z  e.  R.  /\  w  e.  R. )
)  ->  ( [ <. x ,  y >. ] `'  _E  x.  [ <. z ,  w >. ] `'  _E  )  =  [ <. ( ( x  .R  z )  +R  ( -1R  .R  (
y  .R  w )
) ) ,  ( ( y  .R  z
)  +R  ( x  .R  w ) )
>. ] `'  _E  )
3 mulcnsrec 8782 . 2  |-  ( ( ( z  e.  R.  /\  w  e.  R. )  /\  ( v  e.  R.  /\  u  e.  R. )
)  ->  ( [ <. z ,  w >. ] `'  _E  x.  [ <. v ,  u >. ] `'  _E  )  =  [ <. ( ( z  .R  v )  +R  ( -1R  .R  ( w  .R  u ) ) ) ,  ( ( w  .R  v )  +R  ( z  .R  u
) ) >. ] `'  _E  )
4 mulcnsrec 8782 . 2  |-  ( ( ( ( ( x  .R  z )  +R  ( -1R  .R  (
y  .R  w )
) )  e.  R.  /\  ( ( y  .R  z )  +R  (
x  .R  w )
)  e.  R. )  /\  ( v  e.  R.  /\  u  e.  R. )
)  ->  ( [ <. ( ( x  .R  z )  +R  ( -1R  .R  ( y  .R  w ) ) ) ,  ( ( y  .R  z )  +R  ( x  .R  w
) ) >. ] `'  _E  x.  [ <. v ,  u >. ] `'  _E  )  =  [ <. (
( ( ( x  .R  z )  +R  ( -1R  .R  (
y  .R  w )
) )  .R  v
)  +R  ( -1R 
.R  ( ( ( y  .R  z )  +R  ( x  .R  w ) )  .R  u ) ) ) ,  ( ( ( ( y  .R  z
)  +R  ( x  .R  w ) )  .R  v )  +R  ( ( ( x  .R  z )  +R  ( -1R  .R  (
y  .R  w )
) )  .R  u
) ) >. ] `'  _E  )
5 mulcnsrec 8782 . 2  |-  ( ( ( x  e.  R.  /\  y  e.  R. )  /\  ( ( ( z  .R  v )  +R  ( -1R  .R  (
w  .R  u )
) )  e.  R.  /\  ( ( w  .R  v )  +R  (
z  .R  u )
)  e.  R. )
)  ->  ( [ <. x ,  y >. ] `'  _E  x.  [ <. ( ( z  .R  v )  +R  ( -1R  .R  (
w  .R  u )
) ) ,  ( ( w  .R  v
)  +R  ( z  .R  u ) )
>. ] `'  _E  )  =  [ <. ( ( x  .R  ( ( z  .R  v )  +R  ( -1R  .R  (
w  .R  u )
) ) )  +R  ( -1R  .R  (
y  .R  ( (
w  .R  v )  +R  ( z  .R  u
) ) ) ) ) ,  ( ( y  .R  ( ( z  .R  v )  +R  ( -1R  .R  ( w  .R  u
) ) ) )  +R  ( x  .R  ( ( w  .R  v )  +R  (
z  .R  u )
) ) ) >. ] `'  _E  )
6 mulclsr 8722 . . . . 5  |-  ( ( x  e.  R.  /\  z  e.  R. )  ->  ( x  .R  z
)  e.  R. )
7 m1r 8720 . . . . . 6  |-  -1R  e.  R.
8 mulclsr 8722 . . . . . 6  |-  ( ( y  e.  R.  /\  w  e.  R. )  ->  ( y  .R  w
)  e.  R. )
9 mulclsr 8722 . . . . . 6  |-  ( ( -1R  e.  R.  /\  ( y  .R  w
)  e.  R. )  ->  ( -1R  .R  (
y  .R  w )
)  e.  R. )
107, 8, 9sylancr 644 . . . . 5  |-  ( ( y  e.  R.  /\  w  e.  R. )  ->  ( -1R  .R  (
y  .R  w )
)  e.  R. )
11 addclsr 8721 . . . . 5  |-  ( ( ( x  .R  z
)  e.  R.  /\  ( -1R  .R  ( y  .R  w ) )  e.  R. )  -> 
( ( x  .R  z )  +R  ( -1R  .R  ( y  .R  w ) ) )  e.  R. )
126, 10, 11syl2an 463 . . . 4  |-  ( ( ( x  e.  R.  /\  z  e.  R. )  /\  ( y  e.  R.  /\  w  e.  R. )
)  ->  ( (
x  .R  z )  +R  ( -1R  .R  (
y  .R  w )
) )  e.  R. )
1312an4s 799 . . 3  |-  ( ( ( x  e.  R.  /\  y  e.  R. )  /\  ( z  e.  R.  /\  w  e.  R. )
)  ->  ( (
x  .R  z )  +R  ( -1R  .R  (
y  .R  w )
) )  e.  R. )
14 mulclsr 8722 . . . . 5  |-  ( ( y  e.  R.  /\  z  e.  R. )  ->  ( y  .R  z
)  e.  R. )
15 mulclsr 8722 . . . . 5  |-  ( ( x  e.  R.  /\  w  e.  R. )  ->  ( x  .R  w
)  e.  R. )
16 addclsr 8721 . . . . 5  |-  ( ( ( y  .R  z
)  e.  R.  /\  ( x  .R  w
)  e.  R. )  ->  ( ( y  .R  z )  +R  (
x  .R  w )
)  e.  R. )
1714, 15, 16syl2anr 464 . . . 4  |-  ( ( ( x  e.  R.  /\  w  e.  R. )  /\  ( y  e.  R.  /\  z  e.  R. )
)  ->  ( (
y  .R  z )  +R  ( x  .R  w
) )  e.  R. )
1817an42s 800 . . 3  |-  ( ( ( x  e.  R.  /\  y  e.  R. )  /\  ( z  e.  R.  /\  w  e.  R. )
)  ->  ( (
y  .R  z )  +R  ( x  .R  w
) )  e.  R. )
1913, 18jca 518 . 2  |-  ( ( ( x  e.  R.  /\  y  e.  R. )  /\  ( z  e.  R.  /\  w  e.  R. )
)  ->  ( (
( x  .R  z
)  +R  ( -1R 
.R  ( y  .R  w ) ) )  e.  R.  /\  (
( y  .R  z
)  +R  ( x  .R  w ) )  e.  R. ) )
20 mulclsr 8722 . . . . 5  |-  ( ( z  e.  R.  /\  v  e.  R. )  ->  ( z  .R  v
)  e.  R. )
21 mulclsr 8722 . . . . . 6  |-  ( ( w  e.  R.  /\  u  e.  R. )  ->  ( w  .R  u
)  e.  R. )
22 mulclsr 8722 . . . . . 6  |-  ( ( -1R  e.  R.  /\  ( w  .R  u
)  e.  R. )  ->  ( -1R  .R  (
w  .R  u )
)  e.  R. )
237, 21, 22sylancr 644 . . . . 5  |-  ( ( w  e.  R.  /\  u  e.  R. )  ->  ( -1R  .R  (
w  .R  u )
)  e.  R. )
24 addclsr 8721 . . . . 5  |-  ( ( ( z  .R  v
)  e.  R.  /\  ( -1R  .R  ( w  .R  u ) )  e.  R. )  -> 
( ( z  .R  v )  +R  ( -1R  .R  ( w  .R  u ) ) )  e.  R. )
2520, 23, 24syl2an 463 . . . 4  |-  ( ( ( z  e.  R.  /\  v  e.  R. )  /\  ( w  e.  R.  /\  u  e.  R. )
)  ->  ( (
z  .R  v )  +R  ( -1R  .R  (
w  .R  u )
) )  e.  R. )
2625an4s 799 . . 3  |-  ( ( ( z  e.  R.  /\  w  e.  R. )  /\  ( v  e.  R.  /\  u  e.  R. )
)  ->  ( (
z  .R  v )  +R  ( -1R  .R  (
w  .R  u )
) )  e.  R. )
27 mulclsr 8722 . . . . 5  |-  ( ( w  e.  R.  /\  v  e.  R. )  ->  ( w  .R  v
)  e.  R. )
28 mulclsr 8722 . . . . 5  |-  ( ( z  e.  R.  /\  u  e.  R. )  ->  ( z  .R  u
)  e.  R. )
29 addclsr 8721 . . . . 5  |-  ( ( ( w  .R  v
)  e.  R.  /\  ( z  .R  u
)  e.  R. )  ->  ( ( w  .R  v )  +R  (
z  .R  u )
)  e.  R. )
3027, 28, 29syl2anr 464 . . . 4  |-  ( ( ( z  e.  R.  /\  u  e.  R. )  /\  ( w  e.  R.  /\  v  e.  R. )
)  ->  ( (
w  .R  v )  +R  ( z  .R  u
) )  e.  R. )
3130an42s 800 . . 3  |-  ( ( ( z  e.  R.  /\  w  e.  R. )  /\  ( v  e.  R.  /\  u  e.  R. )
)  ->  ( (
w  .R  v )  +R  ( z  .R  u
) )  e.  R. )
3226, 31jca 518 . 2  |-  ( ( ( z  e.  R.  /\  w  e.  R. )  /\  ( v  e.  R.  /\  u  e.  R. )
)  ->  ( (
( z  .R  v
)  +R  ( -1R 
.R  ( w  .R  u ) ) )  e.  R.  /\  (
( w  .R  v
)  +R  ( z  .R  u ) )  e.  R. ) )
33 ovex 5899 . . . 4  |-  ( x  .R  ( z  .R  v ) )  e. 
_V
34 ovex 5899 . . . 4  |-  ( x  .R  ( -1R  .R  ( w  .R  u
) ) )  e. 
_V
35 ovex 5899 . . . 4  |-  ( -1R 
.R  ( y  .R  ( w  .R  v
) ) )  e. 
_V
36 addcomsr 8725 . . . 4  |-  ( f  +R  g )  =  ( g  +R  f
)
37 addasssr 8726 . . . 4  |-  ( ( f  +R  g )  +R  h )  =  ( f  +R  (
g  +R  h ) )
38 ovex 5899 . . . 4  |-  ( -1R 
.R  ( y  .R  ( z  .R  u
) ) )  e. 
_V
3933, 34, 35, 36, 37, 38caov42 6069 . . 3  |-  ( ( ( x  .R  (
z  .R  v )
)  +R  ( x  .R  ( -1R  .R  ( w  .R  u
) ) ) )  +R  ( ( -1R 
.R  ( y  .R  ( w  .R  v
) ) )  +R  ( -1R  .R  (
y  .R  ( z  .R  u ) ) ) ) )  =  ( ( ( x  .R  ( z  .R  v
) )  +R  ( -1R  .R  ( y  .R  ( w  .R  v
) ) ) )  +R  ( ( -1R 
.R  ( y  .R  ( z  .R  u
) ) )  +R  ( x  .R  ( -1R  .R  ( w  .R  u ) ) ) ) )
40 distrsr 8729 . . . 4  |-  ( x  .R  ( ( z  .R  v )  +R  ( -1R  .R  (
w  .R  u )
) ) )  =  ( ( x  .R  ( z  .R  v
) )  +R  (
x  .R  ( -1R  .R  ( w  .R  u
) ) ) )
41 distrsr 8729 . . . . . 6  |-  ( y  .R  ( ( w  .R  v )  +R  ( z  .R  u
) ) )  =  ( ( y  .R  ( w  .R  v
) )  +R  (
y  .R  ( z  .R  u ) ) )
4241oveq2i 5885 . . . . 5  |-  ( -1R 
.R  ( y  .R  ( ( w  .R  v )  +R  (
z  .R  u )
) ) )  =  ( -1R  .R  (
( y  .R  (
w  .R  v )
)  +R  ( y  .R  ( z  .R  u ) ) ) )
43 distrsr 8729 . . . . 5  |-  ( -1R 
.R  ( ( y  .R  ( w  .R  v ) )  +R  ( y  .R  (
z  .R  u )
) ) )  =  ( ( -1R  .R  ( y  .R  (
w  .R  v )
) )  +R  ( -1R  .R  ( y  .R  ( z  .R  u
) ) ) )
4442, 43eqtri 2316 . . . 4  |-  ( -1R 
.R  ( y  .R  ( ( w  .R  v )  +R  (
z  .R  u )
) ) )  =  ( ( -1R  .R  ( y  .R  (
w  .R  v )
) )  +R  ( -1R  .R  ( y  .R  ( z  .R  u
) ) ) )
4540, 44oveq12i 5886 . . 3  |-  ( ( x  .R  ( ( z  .R  v )  +R  ( -1R  .R  ( w  .R  u
) ) ) )  +R  ( -1R  .R  ( y  .R  (
( w  .R  v
)  +R  ( z  .R  u ) ) ) ) )  =  ( ( ( x  .R  ( z  .R  v ) )  +R  ( x  .R  ( -1R  .R  ( w  .R  u ) ) ) )  +R  ( ( -1R  .R  ( y  .R  ( w  .R  v ) ) )  +R  ( -1R  .R  ( y  .R  (
z  .R  u )
) ) ) )
46 vex 2804 . . . . . 6  |-  x  e. 
_V
477elexi 2810 . . . . . 6  |-  -1R  e.  _V
48 vex 2804 . . . . . 6  |-  z  e. 
_V
49 mulcomsr 8727 . . . . . 6  |-  ( f  .R  g )  =  ( g  .R  f
)
50 distrsr 8729 . . . . . 6  |-  ( f  .R  ( g  +R  h ) )  =  ( ( f  .R  g )  +R  (
f  .R  h )
)
51 ovex 5899 . . . . . 6  |-  ( y  .R  w )  e. 
_V
52 vex 2804 . . . . . 6  |-  v  e. 
_V
53 mulasssr 8728 . . . . . 6  |-  ( ( f  .R  g )  .R  h )  =  ( f  .R  (
g  .R  h )
)
5446, 47, 48, 49, 50, 51, 52, 53caovdilem 6071 . . . . 5  |-  ( ( ( x  .R  z
)  +R  ( -1R 
.R  ( y  .R  w ) ) )  .R  v )  =  ( ( x  .R  ( z  .R  v
) )  +R  ( -1R  .R  ( ( y  .R  w )  .R  v ) ) )
55 mulasssr 8728 . . . . . . 7  |-  ( ( y  .R  w )  .R  v )  =  ( y  .R  (
w  .R  v )
)
5655oveq2i 5885 . . . . . 6  |-  ( -1R 
.R  ( ( y  .R  w )  .R  v ) )  =  ( -1R  .R  (
y  .R  ( w  .R  v ) ) )
5756oveq2i 5885 . . . . 5  |-  ( ( x  .R  ( z  .R  v ) )  +R  ( -1R  .R  ( ( y  .R  w )  .R  v
) ) )  =  ( ( x  .R  ( z  .R  v
) )  +R  ( -1R  .R  ( y  .R  ( w  .R  v
) ) ) )
5854, 57eqtri 2316 . . . 4  |-  ( ( ( x  .R  z
)  +R  ( -1R 
.R  ( y  .R  w ) ) )  .R  v )  =  ( ( x  .R  ( z  .R  v
) )  +R  ( -1R  .R  ( y  .R  ( w  .R  v
) ) ) )
59 vex 2804 . . . . . . 7  |-  y  e. 
_V
60 vex 2804 . . . . . . 7  |-  w  e. 
_V
61 vex 2804 . . . . . . 7  |-  u  e. 
_V
6259, 46, 48, 49, 50, 60, 61, 53caovdilem 6071 . . . . . 6  |-  ( ( ( y  .R  z
)  +R  ( x  .R  w ) )  .R  u )  =  ( ( y  .R  ( z  .R  u
) )  +R  (
x  .R  ( w  .R  u ) ) )
6362oveq2i 5885 . . . . 5  |-  ( -1R 
.R  ( ( ( y  .R  z )  +R  ( x  .R  w ) )  .R  u ) )  =  ( -1R  .R  (
( y  .R  (
z  .R  u )
)  +R  ( x  .R  ( w  .R  u ) ) ) )
64 distrsr 8729 . . . . . 6  |-  ( -1R 
.R  ( ( y  .R  ( z  .R  u ) )  +R  ( x  .R  (
w  .R  u )
) ) )  =  ( ( -1R  .R  ( y  .R  (
z  .R  u )
) )  +R  ( -1R  .R  ( x  .R  ( w  .R  u
) ) ) )
65 ovex 5899 . . . . . . . 8  |-  ( w  .R  u )  e. 
_V
6647, 46, 65, 49, 53caov12 6064 . . . . . . 7  |-  ( -1R 
.R  ( x  .R  ( w  .R  u
) ) )  =  ( x  .R  ( -1R  .R  ( w  .R  u ) ) )
6766oveq2i 5885 . . . . . 6  |-  ( ( -1R  .R  ( y  .R  ( z  .R  u ) ) )  +R  ( -1R  .R  ( x  .R  (
w  .R  u )
) ) )  =  ( ( -1R  .R  ( y  .R  (
z  .R  u )
) )  +R  (
x  .R  ( -1R  .R  ( w  .R  u
) ) ) )
6864, 67eqtri 2316 . . . . 5  |-  ( -1R 
.R  ( ( y  .R  ( z  .R  u ) )  +R  ( x  .R  (
w  .R  u )
) ) )  =  ( ( -1R  .R  ( y  .R  (
z  .R  u )
) )  +R  (
x  .R  ( -1R  .R  ( w  .R  u
) ) ) )
6963, 68eqtri 2316 . . . 4  |-  ( -1R 
.R  ( ( ( y  .R  z )  +R  ( x  .R  w ) )  .R  u ) )  =  ( ( -1R  .R  ( y  .R  (
z  .R  u )
) )  +R  (
x  .R  ( -1R  .R  ( w  .R  u
) ) ) )
7058, 69oveq12i 5886 . . 3  |-  ( ( ( ( x  .R  z )  +R  ( -1R  .R  ( y  .R  w ) ) )  .R  v )  +R  ( -1R  .R  (
( ( y  .R  z )  +R  (
x  .R  w )
)  .R  u )
) )  =  ( ( ( x  .R  ( z  .R  v
) )  +R  ( -1R  .R  ( y  .R  ( w  .R  v
) ) ) )  +R  ( ( -1R 
.R  ( y  .R  ( z  .R  u
) ) )  +R  ( x  .R  ( -1R  .R  ( w  .R  u ) ) ) ) )
7139, 45, 703eqtr4ri 2327 . 2  |-  ( ( ( ( x  .R  z )  +R  ( -1R  .R  ( y  .R  w ) ) )  .R  v )  +R  ( -1R  .R  (
( ( y  .R  z )  +R  (
x  .R  w )
)  .R  u )
) )  =  ( ( x  .R  (
( z  .R  v
)  +R  ( -1R 
.R  ( w  .R  u ) ) ) )  +R  ( -1R 
.R  ( y  .R  ( ( w  .R  v )  +R  (
z  .R  u )
) ) ) )
72 ovex 5899 . . . 4  |-  ( y  .R  ( z  .R  v ) )  e. 
_V
73 ovex 5899 . . . 4  |-  ( y  .R  ( -1R  .R  ( w  .R  u
) ) )  e. 
_V
74 ovex 5899 . . . 4  |-  ( x  .R  ( w  .R  v ) )  e. 
_V
75 ovex 5899 . . . 4  |-  ( x  .R  ( z  .R  u ) )  e. 
_V
7672, 73, 74, 36, 37, 75caov42 6069 . . 3  |-  ( ( ( y  .R  (
z  .R  v )
)  +R  ( y  .R  ( -1R  .R  ( w  .R  u
) ) ) )  +R  ( ( x  .R  ( w  .R  v ) )  +R  ( x  .R  (
z  .R  u )
) ) )  =  ( ( ( y  .R  ( z  .R  v ) )  +R  ( x  .R  (
w  .R  v )
) )  +R  (
( x  .R  (
z  .R  u )
)  +R  ( y  .R  ( -1R  .R  ( w  .R  u
) ) ) ) )
77 distrsr 8729 . . . 4  |-  ( y  .R  ( ( z  .R  v )  +R  ( -1R  .R  (
w  .R  u )
) ) )  =  ( ( y  .R  ( z  .R  v
) )  +R  (
y  .R  ( -1R  .R  ( w  .R  u
) ) ) )
78 distrsr 8729 . . . 4  |-  ( x  .R  ( ( w  .R  v )  +R  ( z  .R  u
) ) )  =  ( ( x  .R  ( w  .R  v
) )  +R  (
x  .R  ( z  .R  u ) ) )
7977, 78oveq12i 5886 . . 3  |-  ( ( y  .R  ( ( z  .R  v )  +R  ( -1R  .R  ( w  .R  u
) ) ) )  +R  ( x  .R  ( ( w  .R  v )  +R  (
z  .R  u )
) ) )  =  ( ( ( y  .R  ( z  .R  v ) )  +R  ( y  .R  ( -1R  .R  ( w  .R  u ) ) ) )  +R  ( ( x  .R  ( w  .R  v ) )  +R  ( x  .R  ( z  .R  u
) ) ) )
8059, 46, 48, 49, 50, 60, 52, 53caovdilem 6071 . . . 4  |-  ( ( ( y  .R  z
)  +R  ( x  .R  w ) )  .R  v )  =  ( ( y  .R  ( z  .R  v
) )  +R  (
x  .R  ( w  .R  v ) ) )
8146, 47, 48, 49, 50, 51, 61, 53caovdilem 6071 . . . . 5  |-  ( ( ( x  .R  z
)  +R  ( -1R 
.R  ( y  .R  w ) ) )  .R  u )  =  ( ( x  .R  ( z  .R  u
) )  +R  ( -1R  .R  ( ( y  .R  w )  .R  u ) ) )
82 mulasssr 8728 . . . . . . . 8  |-  ( ( y  .R  w )  .R  u )  =  ( y  .R  (
w  .R  u )
)
8382oveq2i 5885 . . . . . . 7  |-  ( -1R 
.R  ( ( y  .R  w )  .R  u ) )  =  ( -1R  .R  (
y  .R  ( w  .R  u ) ) )
8447, 59, 65, 49, 53caov12 6064 . . . . . . 7  |-  ( -1R 
.R  ( y  .R  ( w  .R  u
) ) )  =  ( y  .R  ( -1R  .R  ( w  .R  u ) ) )
8583, 84eqtri 2316 . . . . . 6  |-  ( -1R 
.R  ( ( y  .R  w )  .R  u ) )  =  ( y  .R  ( -1R  .R  ( w  .R  u ) ) )
8685oveq2i 5885 . . . . 5  |-  ( ( x  .R  ( z  .R  u ) )  +R  ( -1R  .R  ( ( y  .R  w )  .R  u
) ) )  =  ( ( x  .R  ( z  .R  u
) )  +R  (
y  .R  ( -1R  .R  ( w  .R  u
) ) ) )
8781, 86eqtri 2316 . . . 4  |-  ( ( ( x  .R  z
)  +R  ( -1R 
.R  ( y  .R  w ) ) )  .R  u )  =  ( ( x  .R  ( z  .R  u
) )  +R  (
y  .R  ( -1R  .R  ( w  .R  u
) ) ) )
8880, 87oveq12i 5886 . . 3  |-  ( ( ( ( y  .R  z )  +R  (
x  .R  w )
)  .R  v )  +R  ( ( ( x  .R  z )  +R  ( -1R  .R  (
y  .R  w )
) )  .R  u
) )  =  ( ( ( y  .R  ( z  .R  v
) )  +R  (
x  .R  ( w  .R  v ) ) )  +R  ( ( x  .R  ( z  .R  u ) )  +R  ( y  .R  ( -1R  .R  ( w  .R  u ) ) ) ) )
8976, 79, 883eqtr4ri 2327 . 2  |-  ( ( ( ( y  .R  z )  +R  (
x  .R  w )
)  .R  v )  +R  ( ( ( x  .R  z )  +R  ( -1R  .R  (
y  .R  w )
) )  .R  u
) )  =  ( ( y  .R  (
( z  .R  v
)  +R  ( -1R 
.R  ( w  .R  u ) ) ) )  +R  ( x  .R  ( ( w  .R  v )  +R  ( z  .R  u
) ) ) )
901, 2, 3, 4, 5, 19, 32, 71, 89ecovass 6786 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( A  x.  ( B  x.  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    _E cep 4319   `'ccnv 4704  (class class class)co 5874   R.cnr 8505   -1Rcm1r 8508    +R cplr 8509    .R cmr 8510   CCcc 8751    x. cmul 8758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-1p 8622  df-plp 8623  df-mp 8624  df-ltp 8625  df-plpr 8695  df-mpr 8696  df-enr 8697  df-nr 8698  df-plr 8699  df-mr 8700  df-m1r 8704  df-c 8759  df-mul 8765
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