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Theorem sgnmul 26923
Description: Signum of a product. (Contributed by Thierry Arnoux, 2-Oct-2018.)
Assertion
Ref Expression
sgnmul  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (sgn `  ( A  x.  B ) )  =  ( (sgn `  A
)  x.  (sgn `  B ) ) )

Proof of Theorem sgnmul
StepHypRef Expression
1 remulcl 9365 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
21rexrd 9431 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR* )
3 eqeq1 2447 . 2  |-  ( (sgn
`  ( A  x.  B ) )  =  0  ->  ( (sgn `  ( A  x.  B
) )  =  ( (sgn `  A )  x.  (sgn `  B )
)  <->  0  =  ( (sgn `  A )  x.  (sgn `  B )
) ) )
4 eqeq1 2447 . 2  |-  ( (sgn
`  ( A  x.  B ) )  =  1  ->  ( (sgn `  ( A  x.  B
) )  =  ( (sgn `  A )  x.  (sgn `  B )
)  <->  1  =  ( (sgn `  A )  x.  (sgn `  B )
) ) )
5 eqeq1 2447 . 2  |-  ( (sgn
`  ( A  x.  B ) )  = 
-u 1  ->  (
(sgn `  ( A  x.  B ) )  =  ( (sgn `  A
)  x.  (sgn `  B ) )  <->  -u 1  =  ( (sgn `  A
)  x.  (sgn `  B ) ) ) )
6 fveq2 5689 . . . . . . 7  |-  ( A  =  0  ->  (sgn `  A )  =  (sgn
`  0 ) )
7 sgn0 12576 . . . . . . 7  |-  (sgn ` 
0 )  =  0
86, 7syl6eq 2489 . . . . . 6  |-  ( A  =  0  ->  (sgn `  A )  =  0 )
98oveq1d 6104 . . . . 5  |-  ( A  =  0  ->  (
(sgn `  A )  x.  (sgn `  B )
)  =  ( 0  x.  (sgn `  B
) ) )
109adantl 466 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  /\  A  =  0 )  ->  ( (sgn `  A )  x.  (sgn `  B ) )  =  ( 0  x.  (sgn `  B ) ) )
11 sgnclre 26920 . . . . . . 7  |-  ( B  e.  RR  ->  (sgn `  B )  e.  RR )
1211recnd 9410 . . . . . 6  |-  ( B  e.  RR  ->  (sgn `  B )  e.  CC )
1312mul02d 9565 . . . . 5  |-  ( B  e.  RR  ->  (
0  x.  (sgn `  B ) )  =  0 )
1413ad3antlr 730 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  /\  A  =  0 )  ->  ( 0  x.  (sgn `  B )
)  =  0 )
1510, 14eqtr2d 2474 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  /\  A  =  0 )  ->  0  =  ( (sgn `  A )  x.  (sgn `  B )
) )
16 fveq2 5689 . . . . . . 7  |-  ( B  =  0  ->  (sgn `  B )  =  (sgn
`  0 ) )
1716, 7syl6eq 2489 . . . . . 6  |-  ( B  =  0  ->  (sgn `  B )  =  0 )
1817oveq2d 6105 . . . . 5  |-  ( B  =  0  ->  (
(sgn `  A )  x.  (sgn `  B )
)  =  ( (sgn
`  A )  x.  0 ) )
1918adantl 466 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  /\  B  =  0 )  ->  ( (sgn `  A )  x.  (sgn `  B ) )  =  ( (sgn `  A
)  x.  0 ) )
20 sgnclre 26920 . . . . . . 7  |-  ( A  e.  RR  ->  (sgn `  A )  e.  RR )
2120recnd 9410 . . . . . 6  |-  ( A  e.  RR  ->  (sgn `  A )  e.  CC )
2221mul01d 9566 . . . . 5  |-  ( A  e.  RR  ->  (
(sgn `  A )  x.  0 )  =  0 )
2322ad3antrrr 729 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  /\  B  =  0 )  ->  ( (sgn `  A )  x.  0 )  =  0 )
2419, 23eqtr2d 2474 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  /\  B  =  0 )  ->  0  =  ( (sgn `  A )  x.  (sgn `  B )
) )
25 simpl 457 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
2625recnd 9410 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
27 simpr 461 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
2827recnd 9410 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
2926, 28mul0ord 9984 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  =  0  <-> 
( A  =  0  \/  B  =  0 ) ) )
3029biimpa 484 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  ->  ( A  =  0  \/  B  =  0 ) )
3115, 24, 30mpjaodan 784 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  ->  0  =  ( (sgn `  A )  x.  (sgn `  B )
) )
3225adantr 465 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  A  e.  RR )
3332rexrd 9431 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  A  e.  RR* )
34 oveq1 6096 . . . 4  |-  ( (sgn
`  A )  =  0  ->  ( (sgn `  A )  x.  (sgn `  B ) )  =  ( 0  x.  (sgn `  B ) ) )
3534eqeq2d 2452 . . 3  |-  ( (sgn
`  A )  =  0  ->  ( 1  =  ( (sgn `  A )  x.  (sgn `  B ) )  <->  1  =  ( 0  x.  (sgn `  B ) ) ) )
36 oveq1 6096 . . . 4  |-  ( (sgn
`  A )  =  1  ->  ( (sgn `  A )  x.  (sgn `  B ) )  =  ( 1  x.  (sgn `  B ) ) )
3736eqeq2d 2452 . . 3  |-  ( (sgn
`  A )  =  1  ->  ( 1  =  ( (sgn `  A )  x.  (sgn `  B ) )  <->  1  =  ( 1  x.  (sgn `  B ) ) ) )
38 oveq1 6096 . . . 4  |-  ( (sgn
`  A )  = 
-u 1  ->  (
(sgn `  A )  x.  (sgn `  B )
)  =  ( -u
1  x.  (sgn `  B ) ) )
3938eqeq2d 2452 . . 3  |-  ( (sgn
`  A )  = 
-u 1  ->  (
1  =  ( (sgn
`  A )  x.  (sgn `  B )
)  <->  1  =  (
-u 1  x.  (sgn `  B ) ) ) )
40 simpr 461 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  =  0 )  ->  A  =  0 )
4126adantr 465 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  A  e.  CC )
4228adantr 465 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  B  e.  CC )
43 0re 9384 . . . . . . . . . 10  |-  0  e.  RR
4443a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  0  e.  RR )
45 simpr 461 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  0  <  ( A  x.  B ) )
4644, 45ltned 9508 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  0  =/=  ( A  x.  B
) )
4746necomd 2693 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  ( A  x.  B )  =/=  0
)
4841, 42, 47mulne0bad 9989 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  A  =/=  0 )
4948neneqd 2622 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  -.  A  =  0 )
5049adantr 465 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  =  0 )  ->  -.  A  =  0
)
5140, 50pm2.21dd 174 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  =  0 )  -> 
1  =  ( 0  x.  (sgn `  B
) ) )
52 simpll 753 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  ( A  e.  RR  /\  B  e.  RR ) )
5352, 27syl 16 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  B  e.  RR )
5453rexrd 9431 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  B  e.  RR* )
5543a1i 11 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  0  e.  RR )
5632adantr 465 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  A  e.  RR )
57 simpr 461 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  0  <  A )
5855, 56, 57ltled 9520 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  0  <_  A )
5945adantr 465 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  0  <  ( A  x.  B
) )
60 prodgt0 10172 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  B )
6152, 58, 59, 60syl12anc 1216 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  0  <  B )
62 sgnp 12577 . . . . . 6  |-  ( ( B  e.  RR*  /\  0  <  B )  ->  (sgn `  B )  =  1 )
6354, 61, 62syl2anc 661 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  (sgn `  B )  =  1 )
6463oveq2d 6105 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  (
1  x.  (sgn `  B ) )  =  ( 1  x.  1 ) )
65 1t1e1 10467 . . . 4  |-  ( 1  x.  1 )  =  1
6664, 65syl6req 2490 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  1  =  ( 1  x.  (sgn `  B )
) )
6727ad2antrr 725 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  B  e.  RR )
6867rexrd 9431 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  B  e.  RR* )
69 simplll 757 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  A  e.  RR )
7069renegcld 9773 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  -u A  e.  RR )
7167renegcld 9773 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  -u B  e.  RR )
7243a1i 11 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  e.  RR )
73 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  A  <  0 )
7425lt0neg1d 9907 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  0  <->  0  <  -u A ) )
7574ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  ( A  <  0  <->  0  <  -u A ) )
7673, 75mpbid 210 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  <  -u A )
7772, 70, 76ltled 9520 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  <_ 
-u A )
7845adantr 465 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  <  ( A  x.  B
) )
7941adantr 465 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  A  e.  CC )
8042adantr 465 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  B  e.  CC )
8179, 80mul2negd 9797 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  ( -u A  x.  -u B
)  =  ( A  x.  B ) )
8278, 81breqtrrd 4316 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  <  ( -u A  x.  -u B ) )
83 prodgt0 10172 . . . . . . . 8  |-  ( ( ( -u A  e.  RR  /\  -u B  e.  RR )  /\  (
0  <_  -u A  /\  0  <  ( -u A  x.  -u B ) ) )  ->  0  <  -u B )
8470, 71, 77, 82, 83syl22anc 1219 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  <  -u B )
8527lt0neg1d 9907 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  <  0  <->  0  <  -u B ) )
8685ad2antrr 725 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  ( B  <  0  <->  0  <  -u B ) )
8784, 86mpbird 232 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  B  <  0 )
88 sgnn 12581 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  <  0 )  ->  (sgn `  B )  =  -u
1 )
8968, 87, 88syl2anc 661 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  (sgn `  B )  =  -u
1 )
9089oveq2d 6105 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  ( -u 1  x.  (sgn `  B ) )  =  ( -u 1  x.  -u 1 ) )
91 neg1mulneg1e1 10537 . . . 4  |-  ( -u
1  x.  -u 1
)  =  1
9290, 91syl6req 2490 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  1  =  ( -u 1  x.  (sgn `  B )
) )
9333, 35, 37, 39, 51, 66, 92sgn3da 26922 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  1  =  ( (sgn `  A )  x.  (sgn `  B )
) )
9425adantr 465 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0
)  ->  A  e.  RR )
9594rexrd 9431 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0
)  ->  A  e.  RR* )
9634eqeq2d 2452 . . 3  |-  ( (sgn
`  A )  =  0  ->  ( -u 1  =  ( (sgn `  A )  x.  (sgn `  B ) )  <->  -u 1  =  ( 0  x.  (sgn `  B ) ) ) )
9736eqeq2d 2452 . . 3  |-  ( (sgn
`  A )  =  1  ->  ( -u 1  =  ( (sgn `  A )  x.  (sgn `  B ) )  <->  -u 1  =  ( 1  x.  (sgn `  B ) ) ) )
9838eqeq2d 2452 . . 3  |-  ( (sgn
`  A )  = 
-u 1  ->  ( -u 1  =  ( (sgn
`  A )  x.  (sgn `  B )
)  <->  -u 1  =  (
-u 1  x.  (sgn `  B ) ) ) )
99 simpr 461 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  A  =  0 )
10026ad2antrr 725 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  A  e.  CC )
10128ad2antrr 725 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  B  e.  CC )
102 simplll 757 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  A  e.  RR )
103 simpllr 758 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  B  e.  RR )
104102, 103, 1syl2anc 661 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  -> 
( A  x.  B
)  e.  RR )
105 simplr 754 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  -> 
( A  x.  B
)  <  0 )
106104, 105ltned 9508 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  -> 
( A  x.  B
)  =/=  0 )
107100, 101, 106mulne0bad 9989 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  A  =/=  0 )
108107neneqd 2622 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  -.  A  =  0
)
10999, 108pm2.21dd 174 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  -u 1  =  ( 0  x.  (sgn `  B
) ) )
11027ad2antrr 725 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  0  <  A )  ->  B  e.  RR )
111110rexrd 9431 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  0  <  A )  ->  B  e.  RR* )
112 simplr 754 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0
)  ->  B  e.  RR )
11326, 28mulcomd 9405 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
114113breq1d 4300 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  <  0  <->  ( B  x.  A )  <  0 ) )
115114biimpa 484 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0
)  ->  ( B  x.  A )  <  0
)
116112, 94, 115mul2lt0rgt0 26037 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  0  <  A )  ->  B  <  0 )
117111, 116, 88syl2anc 661 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  0  <  A )  ->  (sgn `  B )  =  -u
1 )
118117oveq2d 6105 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  0  <  A )  ->  (
1  x.  (sgn `  B ) )  =  ( 1  x.  -u 1
) )
119 neg1cn 10423 . . . . 5  |-  -u 1  e.  CC
120119mulid2i 9387 . . . 4  |-  ( 1  x.  -u 1 )  = 
-u 1
121118, 120syl6req 2490 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  0  <  A )  ->  -u 1  =  ( 1  x.  (sgn `  B )
) )
122112adantr 465 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  B  e.  RR )
123122rexrd 9431 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  B  e.  RR* )
124112, 94, 115mul2lt0rlt0 26036 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  0  <  B )
125123, 124, 62syl2anc 661 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  (sgn `  B )  =  1 )
126125oveq2d 6105 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  ( -u 1  x.  (sgn `  B ) )  =  ( -u 1  x.  1 ) )
127119mulid1i 9386 . . . 4  |-  ( -u
1  x.  1 )  =  -u 1
128126, 127syl6req 2490 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  -u 1  =  ( -u 1  x.  (sgn `  B )
) )
12995, 96, 97, 98, 109, 121, 128sgn3da 26922 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0
)  ->  -u 1  =  ( (sgn `  A
)  x.  (sgn `  B ) ) )
1302, 3, 4, 5, 31, 93, 129sgn3da 26922 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (sgn `  ( A  x.  B ) )  =  ( (sgn `  A
)  x.  (sgn `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   CCcc 9278   RRcr 9279   0cc0 9280   1c1 9281    x. cmul 9285   RR*cxr 9415    < clt 9416    <_ cle 9417   -ucneg 9594  sgncsgn 12573
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-rp 10990  df-sgn 12574
This theorem is referenced by:  sgnmulrp2  26924  sgnmulsgn  26930  sgnmulsgp  26931
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