MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xmullem Structured version   Unicode version

Theorem xmullem 11509
Description: Lemma for rexmul 11516. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmullem  |-  ( ( ( ( ( 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 )

Proof of Theorem xmullem
StepHypRef Expression
1 ioran 488 . . . 4  |-  ( -.  ( A  =  0  \/  B  =  0 )  <->  ( -.  A  =  0  /\  -.  B  =  0 ) )
21anbi2i 692 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( -.  A  =  0  /\  -.  B  =  0 ) ) )
3 ioran 488 . . . . 5  |-  ( -.  ( ( ( 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 ) ) ) )
4 ioran 488 . . . . . 6  |-  ( -.  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  <->  ( -.  ( 0  <  B  /\  A  = +oo )  /\  -.  ( B  <  0  /\  A  = -oo ) ) )
5 ioran 488 . . . . . 6  |-  ( -.  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  <->  ( -.  ( 0  <  A  /\  B  = +oo )  /\  -.  ( A  <  0  /\  B  = -oo ) ) )
64, 5anbi12i 695 . . . . 5  |-  ( ( -.  ( ( 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 ) ) ) )
73, 6bitri 249 . . . 4  |-  ( -.  ( ( ( 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 ) ) ) )
8 ioran 488 . . . . 5  |-  ( -.  ( ( ( 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 ) ) ) )
9 ioran 488 . . . . . 6  |-  ( -.  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  <->  ( -.  ( 0  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) ) )
10 ioran 488 . . . . . 6  |-  ( -.  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  <->  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) )
119, 10anbi12i 695 . . . . 5  |-  ( ( -.  ( ( 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 ) ) ) )
128, 11bitri 249 . . . 4  |-  ( -.  ( ( ( 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 ) ) ) )
137, 12anbi12i 695 . . 3  |-  ( ( -.  ( ( ( 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  < 
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 ) ) ) ) )
14 simplll 760 . . . . 5  |-  ( ( ( ( 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* )
15 elxr 11378 . . . . 5  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
1614, 15sylib 196 . . . 4  |-  ( ( ( ( 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  \/  A  = +oo  \/  A  = -oo ) )
17 idd 24 . . . . 5  |-  ( ( ( ( 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  ->  A  e.  RR ) )
18 simprlr 765 . . . . . . . . 9  |-  ( ( ( ( -.  (
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  <  0  /\  A  = +oo ) )
1918adantl 464 . . . . . . . 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 ) ) ) ) )  ->  -.  ( B  <  0  /\  A  = +oo ) )
2019pm2.21d 106 . . . . . . 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 ) ) )  /\  ( ( -.  (
0  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  ->  ( ( B  <  0  /\  A  = +oo )  ->  A  e.  RR ) )
2120expdimp 435 . . . . . 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  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  /\  B  <  0 )  ->  ( A  = +oo  ->  A  e.  RR ) )
22 simplrr 763 . . . . . . . 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 ) ) ) ) )  ->  -.  B  =  0 )
2322pm2.21d 106 . . . . . . 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 ) ) )  /\  ( ( -.  (
0  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  ->  ( B  =  0  ->  ( A  = +oo  ->  A  e.  RR ) ) )
2423imp 427 . . . . . 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  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  /\  B  =  0 )  ->  ( A  = +oo  ->  A  e.  RR ) )
25 simplll 760 . . . . . . . . 9  |-  ( ( ( ( -.  (
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  <  B  /\  A  = +oo ) )
2625adantl 464 . . . . . . . 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  <  B  /\  A  = +oo )
)
2726pm2.21d 106 . . . . . . 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 ) ) )  /\  ( ( -.  (
0  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  ->  ( (
0  <  B  /\  A  = +oo )  ->  A  e.  RR ) )
2827expdimp 435 . . . . . 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  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  /\  0  < 
B )  ->  ( A  = +oo  ->  A  e.  RR ) )
29 simpllr 761 . . . . . . 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 ) ) )  /\  ( ( -.  (
0  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  ->  B  e.  RR* )
30 0xr 9670 . . . . . . 7  |-  0  e.  RR*
31 xrltso 11400 . . . . . . . 8  |-  <  Or  RR*
32 solin 4767 . . . . . . . 8  |-  ( (  <  Or  RR*  /\  ( B  e.  RR*  /\  0  e.  RR* ) )  -> 
( B  <  0  \/  B  =  0  \/  0  <  B ) )
3331, 32mpan 668 . . . . . . 7  |-  ( ( B  e.  RR*  /\  0  e.  RR* )  ->  ( B  <  0  \/  B  =  0  \/  0  <  B ) )
3429, 30, 33sylancl 660 . . . . . 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  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  ->  ( B  <  0  \/  B  =  0  \/  0  < 
B ) )
3521, 24, 28, 34mpjao3dan 1297 . . . . 5  |-  ( ( ( ( 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  = +oo  ->  A  e.  RR ) )
36 simpllr 761 . . . . . . . . 9  |-  ( ( ( ( -.  (
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  <  0  /\  A  = -oo ) )
3736adantl 464 . . . . . . . 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 ) ) ) ) )  ->  -.  ( B  <  0  /\  A  = -oo ) )
3837pm2.21d 106 . . . . . . 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 ) ) )  /\  ( ( -.  (
0  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  ->  ( ( B  <  0  /\  A  = -oo )  ->  A  e.  RR ) )
3938expdimp 435 . . . . . 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  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  /\  B  <  0 )  ->  ( A  = -oo  ->  A  e.  RR ) )
4022pm2.21d 106 . . . . . . 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 ) ) )  /\  ( ( -.  (
0  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  ->  ( B  =  0  ->  ( A  = -oo  ->  A  e.  RR ) ) )
4140imp 427 . . . . . 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  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  /\  B  =  0 )  ->  ( A  = -oo  ->  A  e.  RR ) )
42 simprll 764 . . . . . . . . 9  |-  ( ( ( ( -.  (
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  <  B  /\  A  = -oo ) )
4342adantl 464 . . . . . . . 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  <  B  /\  A  = -oo )
)
4443pm2.21d 106 . . . . . . 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 ) ) )  /\  ( ( -.  (
0  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  ->  ( (
0  <  B  /\  A  = -oo )  ->  A  e.  RR ) )
4544expdimp 435 . . . . . 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  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) )  /\  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) ) )  /\  0  < 
B )  ->  ( A  = -oo  ->  A  e.  RR ) )
4639, 41, 45, 34mpjao3dan 1297 . . . . 5  |-  ( ( ( ( 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  = -oo  ->  A  e.  RR ) )
4717, 35, 463jaod 1294 . . . 4  |-  ( ( ( ( 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  \/  A  = +oo  \/  A  = -oo )  ->  A  e.  RR ) )
4816, 47mpd 15 . . 3  |-  ( ( ( ( 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 )
492, 13, 48syl2anb 477 . 2  |-  ( ( ( ( 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 )
5049anassrs 646 1  |-  ( ( ( ( ( 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 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    \/ w3o 973    = wceq 1405    e. wcel 1842   class class class wbr 4395    Or wor 4743   RRcr 9521   0cc0 9522   +oocpnf 9655   -oocmnf 9656   RR*cxr 9657    < clt 9658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-i2m1 9590  ax-1ne0 9591  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663
This theorem is referenced by:  xmulcom  11511  xmulneg1  11514  xmulf  11517
  Copyright terms: Public domain W3C validator