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Theorem xmulcom 11234
Description: Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmulcom  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  =  ( B xe A ) )

Proof of Theorem xmulcom
StepHypRef Expression
1 xmullem 11232 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  ( A  =  0  \/  B  =  0
) )  /\  -.  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )  /\  -.  (
( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) )  ->  A  e.  RR )
21recnd 9417 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  ( A  =  0  \/  B  =  0
) )  /\  -.  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )  /\  -.  (
( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) )  ->  A  e.  CC )
3 ancom 450 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  <->  ( B  e.  RR*  /\  A  e. 
RR* ) )
4 orcom 387 . . . . . . . . . . . . . 14  |-  ( ( A  =  0  \/  B  =  0 )  <-> 
( B  =  0  \/  A  =  0 ) )
54notbii 296 . . . . . . . . . . . . 13  |-  ( -.  ( A  =  0  \/  B  =  0 )  <->  -.  ( B  =  0  \/  A  =  0 ) )
63, 5anbi12i 697 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  <->  ( ( B  e.  RR*  /\  A  e.  RR* )  /\  -.  ( B  =  0  \/  A  =  0
) ) )
7 orcom 387 . . . . . . . . . . . . 13  |-  ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  <-> 
( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) )
87notbii 296 . . . . . . . . . . . 12  |-  ( -.  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  <->  -.  ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) )
96, 8anbi12i 697 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  /\  -.  (
( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )  <->  ( ( ( B  e.  RR*  /\  A  e.  RR* )  /\  -.  ( B  =  0  \/  A  =  0
) )  /\  -.  ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) ) )
10 orcom 387 . . . . . . . . . . . 12  |-  ( ( ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) )  <-> 
( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) )
1110notbii 296 . . . . . . . . . . 11  |-  ( -.  ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) )  <->  -.  ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) )
12 xmullem 11232 . . . . . . . . . . 11  |-  ( ( ( ( ( B  e.  RR*  /\  A  e. 
RR* )  /\  -.  ( B  =  0  \/  A  =  0
) )  /\  -.  ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) )  /\  -.  (
( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) )  ->  B  e.  RR )
139, 11, 12syl2anb 479 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  ( A  =  0  \/  B  =  0
) )  /\  -.  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )  /\  -.  (
( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) )  ->  B  e.  RR )
1413recnd 9417 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  ( A  =  0  \/  B  =  0
) )  /\  -.  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )  /\  -.  (
( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) )  ->  B  e.  CC )
152, 14mulcomd 9412 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  ( A  =  0  \/  B  =  0
) )  /\  -.  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )  /\  -.  (
( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) )  ->  ( A  x.  B )  =  ( B  x.  A ) )
1615ifeq2da 3825 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  /\  -.  (
( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )  ->  if (
( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) )  =  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( B  x.  A ) ) )
1710a1i 11 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  /\  -.  (
( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )  ->  ( (
( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) )  <-> 
( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) ) )
1817ifbid 3816 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  /\  -.  (
( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )  ->  if (
( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( B  x.  A ) )  =  if ( ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) , -oo ,  ( B  x.  A ) ) )
1916, 18eqtrd 2475 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  /\  -.  (
( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )  ->  if (
( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) )  =  if ( ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) , -oo ,  ( B  x.  A ) ) )
2019ifeq2da 3825 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  ->  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  (
( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) , -oo ,  ( B  x.  A ) ) ) )
217a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  <-> 
( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) ) )
2221ifbid 3816 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  ->  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  (
( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) , -oo ,  ( B  x.  A ) ) )  =  if ( ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  (
( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  (
( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) , -oo ,  ( B  x.  A ) ) ) )
2320, 22eqtrd 2475 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  ->  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  if ( ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  (
( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  (
( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) , -oo ,  ( B  x.  A ) ) ) )
2423ifeq2da 3825 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  (
( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) , -oo ,  ( B  x.  A ) ) ) ) )
254a1i 11 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  =  0  \/  B  =  0 )  <->  ( B  =  0  \/  A  =  0 ) ) )
2625ifbid 3816 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  (
( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) , -oo ,  ( B  x.  A ) ) ) )  =  if ( ( B  =  0  \/  A  =  0 ) ,  0 ,  if ( ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  (
( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) , -oo ,  ( B  x.  A ) ) ) ) )
2724, 26eqtrd 2475 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )  =  if ( ( B  =  0  \/  A  =  0 ) ,  0 ,  if ( ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  (
( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) , -oo ,  ( B  x.  A ) ) ) ) )
28 xmulval 11200 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) ) )
29 xmulval 11200 . . 3  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B xe A )  =  if ( ( B  =  0  \/  A  =  0 ) ,  0 ,  if ( ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  (
( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  (
( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) , -oo ,  ( B  x.  A ) ) ) ) )
3029ancoms 453 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B xe A )  =  if ( ( B  =  0  \/  A  =  0 ) ,  0 ,  if ( ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  (
( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  (
( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) , -oo ,  ( B  x.  A ) ) ) ) )
3127, 28, 303eqtr4d 2485 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  =  ( B xe A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   ifcif 3796   class class class wbr 4297  (class class class)co 6096   RRcr 9286   0cc0 9287    x. cmul 9292   +oocpnf 9420   -oocmnf 9421   RR*cxr 9422    < clt 9423   xecxmu 11093
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-i2m1 9355  ax-1ne0 9356  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-xmul 11096
This theorem is referenced by:  xmul02  11236  xmulneg2  11238  xmulpnf2  11243  xmulmnf2  11245  xmulid2  11248  xlemul2a  11257  xlemul2  11259  xltmul2  11261  xadddir  11264  xadddi2r  11266  xrsmcmn  17844  xmulcand  26101  xdivrec  26107  xrge0adddi  26161  xrmulc1cn  26365  esummulc2  26536
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