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Theorem sgnmul 28118
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 9573 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
21rexrd 9639 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR* )
3 eqeq1 2471 . 2  |-  ( (sgn
`  ( A  x.  B ) )  =  0  ->  ( (sgn `  ( A  x.  B
) )  =  ( (sgn `  A )  x.  (sgn `  B )
)  <->  0  =  ( (sgn `  A )  x.  (sgn `  B )
) ) )
4 eqeq1 2471 . 2  |-  ( (sgn
`  ( A  x.  B ) )  =  1  ->  ( (sgn `  ( A  x.  B
) )  =  ( (sgn `  A )  x.  (sgn `  B )
)  <->  1  =  ( (sgn `  A )  x.  (sgn `  B )
) ) )
5 eqeq1 2471 . 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 5864 . . . . . . 7  |-  ( A  =  0  ->  (sgn `  A )  =  (sgn
`  0 ) )
7 sgn0 12879 . . . . . . 7  |-  (sgn ` 
0 )  =  0
86, 7syl6eq 2524 . . . . . 6  |-  ( A  =  0  ->  (sgn `  A )  =  0 )
98oveq1d 6297 . . . . 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 28115 . . . . . . 7  |-  ( B  e.  RR  ->  (sgn `  B )  e.  RR )
1211recnd 9618 . . . . . 6  |-  ( B  e.  RR  ->  (sgn `  B )  e.  CC )
1312mul02d 9773 . . . . 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 2509 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  /\  A  =  0 )  ->  0  =  ( (sgn `  A )  x.  (sgn `  B )
) )
16 fveq2 5864 . . . . . . 7  |-  ( B  =  0  ->  (sgn `  B )  =  (sgn
`  0 ) )
1716, 7syl6eq 2524 . . . . . 6  |-  ( B  =  0  ->  (sgn `  B )  =  0 )
1817oveq2d 6298 . . . . 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 28115 . . . . . . 7  |-  ( A  e.  RR  ->  (sgn `  A )  e.  RR )
2120recnd 9618 . . . . . 6  |-  ( A  e.  RR  ->  (sgn `  A )  e.  CC )
2221mul01d 9774 . . . . 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 2509 . . 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 9618 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
27 simpr 461 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
2827recnd 9618 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
2926, 28mul0ord 10195 . . . 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 9639 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  A  e.  RR* )
34 oveq1 6289 . . . 4  |-  ( (sgn
`  A )  =  0  ->  ( (sgn `  A )  x.  (sgn `  B ) )  =  ( 0  x.  (sgn `  B ) ) )
3534eqeq2d 2481 . . 3  |-  ( (sgn
`  A )  =  0  ->  ( 1  =  ( (sgn `  A )  x.  (sgn `  B ) )  <->  1  =  ( 0  x.  (sgn `  B ) ) ) )
36 oveq1 6289 . . . 4  |-  ( (sgn
`  A )  =  1  ->  ( (sgn `  A )  x.  (sgn `  B ) )  =  ( 1  x.  (sgn `  B ) ) )
3736eqeq2d 2481 . . 3  |-  ( (sgn
`  A )  =  1  ->  ( 1  =  ( (sgn `  A )  x.  (sgn `  B ) )  <->  1  =  ( 1  x.  (sgn `  B ) ) ) )
38 oveq1 6289 . . . 4  |-  ( (sgn
`  A )  = 
-u 1  ->  (
(sgn `  A )  x.  (sgn `  B )
)  =  ( -u
1  x.  (sgn `  B ) ) )
3938eqeq2d 2481 . . 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 9592 . . . . . . . . . 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 9716 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  0  =/=  ( A  x.  B
) )
4746necomd 2738 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  ( A  x.  B )  =/=  0
)
4841, 42, 47mulne0bad 10200 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  A  =/=  0 )
4948neneqd 2669 . . . . 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 9639 . . . . . 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 9728 . . . . . . 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 10383 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  B )
6152, 58, 59, 60syl12anc 1226 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  0  <  B )
62 sgnp 12880 . . . . . 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 6298 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  (
1  x.  (sgn `  B ) )  =  ( 1  x.  1 ) )
65 1t1e1 10679 . . . 4  |-  ( 1  x.  1 )  =  1
6664, 65syl6req 2525 . . 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 9639 . . . . . 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 9982 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  -u A  e.  RR )
7167renegcld 9982 . . . . . . . 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 10118 . . . . . . . . . . 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 9728 . . . . . . . 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 10007 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  ( -u A  x.  -u B
)  =  ( A  x.  B ) )
8278, 81breqtrrd 4473 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  <  ( -u A  x.  -u B ) )
83 prodgt0 10383 . . . . . . . 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 1229 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  <  -u B )
8527lt0neg1d 10118 . . . . . . . 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 12884 . . . . . 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 6298 . . . 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 10749 . . . 4  |-  ( -u
1  x.  -u 1
)  =  1
9290, 91syl6req 2525 . . 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 28117 . 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 9639 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0
)  ->  A  e.  RR* )
9634eqeq2d 2481 . . 3  |-  ( (sgn
`  A )  =  0  ->  ( -u 1  =  ( (sgn `  A )  x.  (sgn `  B ) )  <->  -u 1  =  ( 0  x.  (sgn `  B ) ) ) )
9736eqeq2d 2481 . . 3  |-  ( (sgn
`  A )  =  1  ->  ( -u 1  =  ( (sgn `  A )  x.  (sgn `  B ) )  <->  -u 1  =  ( 1  x.  (sgn `  B ) ) ) )
9838eqeq2d 2481 . . 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 9716 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  -> 
( A  x.  B
)  =/=  0 )
107100, 101, 106mulne0bad 10200 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  A  =/=  0 )
108107neneqd 2669 . . . 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 9639 . . . . . 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 9613 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
114113breq1d 4457 . . . . . . . 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 27231 . . . . . 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 6298 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  0  <  A )  ->  (
1  x.  (sgn `  B ) )  =  ( 1  x.  -u 1
) )
119 neg1cn 10635 . . . . 5  |-  -u 1  e.  CC
120119mulid2i 9595 . . . 4  |-  ( 1  x.  -u 1 )  = 
-u 1
121118, 120syl6req 2525 . . 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 9639 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  B  e.  RR* )
124112, 94, 115mul2lt0rlt0 27230 . . . . . 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 6298 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  ( -u 1  x.  (sgn `  B ) )  =  ( -u 1  x.  1 ) )
127119mulid1i 9594 . . . 4  |-  ( -u
1  x.  1 )  =  -u 1
128126, 127syl6req 2525 . . 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 28117 . 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 28117 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 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    x. cmul 9493   RR*cxr 9623    < clt 9624    <_ cle 9625   -ucneg 9802  sgncsgn 12876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-rp 11217  df-sgn 12877
This theorem is referenced by:  sgnmulrp2  28119  sgnmulsgn  28125  sgnmulsgp  28126
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