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Theorem sgnmul 28748
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 9566 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
21rexrd 9632 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR* )
3 eqeq1 2458 . 2  |-  ( (sgn
`  ( A  x.  B ) )  =  0  ->  ( (sgn `  ( A  x.  B
) )  =  ( (sgn `  A )  x.  (sgn `  B )
)  <->  0  =  ( (sgn `  A )  x.  (sgn `  B )
) ) )
4 eqeq1 2458 . 2  |-  ( (sgn
`  ( A  x.  B ) )  =  1  ->  ( (sgn `  ( A  x.  B
) )  =  ( (sgn `  A )  x.  (sgn `  B )
)  <->  1  =  ( (sgn `  A )  x.  (sgn `  B )
) ) )
5 eqeq1 2458 . 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 5848 . . . . . . 7  |-  ( A  =  0  ->  (sgn `  A )  =  (sgn
`  0 ) )
7 sgn0 13007 . . . . . . 7  |-  (sgn ` 
0 )  =  0
86, 7syl6eq 2511 . . . . . 6  |-  ( A  =  0  ->  (sgn `  A )  =  0 )
98oveq1d 6285 . . . . 5  |-  ( A  =  0  ->  (
(sgn `  A )  x.  (sgn `  B )
)  =  ( 0  x.  (sgn `  B
) ) )
109adantl 464 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  /\  A  =  0 )  ->  ( (sgn `  A )  x.  (sgn `  B ) )  =  ( 0  x.  (sgn `  B ) ) )
11 sgnclre 28745 . . . . . . 7  |-  ( B  e.  RR  ->  (sgn `  B )  e.  RR )
1211recnd 9611 . . . . . 6  |-  ( B  e.  RR  ->  (sgn `  B )  e.  CC )
1312mul02d 9767 . . . . 5  |-  ( B  e.  RR  ->  (
0  x.  (sgn `  B ) )  =  0 )
1413ad3antlr 728 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  /\  A  =  0 )  ->  ( 0  x.  (sgn `  B )
)  =  0 )
1510, 14eqtr2d 2496 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  /\  A  =  0 )  ->  0  =  ( (sgn `  A )  x.  (sgn `  B )
) )
16 fveq2 5848 . . . . . . 7  |-  ( B  =  0  ->  (sgn `  B )  =  (sgn
`  0 ) )
1716, 7syl6eq 2511 . . . . . 6  |-  ( B  =  0  ->  (sgn `  B )  =  0 )
1817oveq2d 6286 . . . . 5  |-  ( B  =  0  ->  (
(sgn `  A )  x.  (sgn `  B )
)  =  ( (sgn
`  A )  x.  0 ) )
1918adantl 464 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  /\  B  =  0 )  ->  ( (sgn `  A )  x.  (sgn `  B ) )  =  ( (sgn `  A
)  x.  0 ) )
20 sgnclre 28745 . . . . . . 7  |-  ( A  e.  RR  ->  (sgn `  A )  e.  RR )
2120recnd 9611 . . . . . 6  |-  ( A  e.  RR  ->  (sgn `  A )  e.  CC )
2221mul01d 9768 . . . . 5  |-  ( A  e.  RR  ->  (
(sgn `  A )  x.  0 )  =  0 )
2322ad3antrrr 727 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  /\  B  =  0 )  ->  ( (sgn `  A )  x.  0 )  =  0 )
2419, 23eqtr2d 2496 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  =  0 )  /\  B  =  0 )  ->  0  =  ( (sgn `  A )  x.  (sgn `  B )
) )
25 simpl 455 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
2625recnd 9611 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
27 simpr 459 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
2827recnd 9611 . . . . 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 482 . . 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 )
) )
32 simpll 751 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  A  e.  RR )
3332rexrd 9632 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  A  e.  RR* )
34 oveq1 6277 . . . 4  |-  ( (sgn
`  A )  =  0  ->  ( (sgn `  A )  x.  (sgn `  B ) )  =  ( 0  x.  (sgn `  B ) ) )
3534eqeq2d 2468 . . 3  |-  ( (sgn
`  A )  =  0  ->  ( 1  =  ( (sgn `  A )  x.  (sgn `  B ) )  <->  1  =  ( 0  x.  (sgn `  B ) ) ) )
36 oveq1 6277 . . . 4  |-  ( (sgn
`  A )  =  1  ->  ( (sgn `  A )  x.  (sgn `  B ) )  =  ( 1  x.  (sgn `  B ) ) )
3736eqeq2d 2468 . . 3  |-  ( (sgn
`  A )  =  1  ->  ( 1  =  ( (sgn `  A )  x.  (sgn `  B ) )  <->  1  =  ( 1  x.  (sgn `  B ) ) ) )
38 oveq1 6277 . . . 4  |-  ( (sgn
`  A )  = 
-u 1  ->  (
(sgn `  A )  x.  (sgn `  B )
)  =  ( -u
1  x.  (sgn `  B ) ) )
3938eqeq2d 2468 . . 3  |-  ( (sgn
`  A )  = 
-u 1  ->  (
1  =  ( (sgn
`  A )  x.  (sgn `  B )
)  <->  1  =  (
-u 1  x.  (sgn `  B ) ) ) )
40 simpr 459 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  =  0 )  ->  A  =  0 )
4126adantr 463 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  A  e.  CC )
4228adantr 463 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  B  e.  CC )
43 simpr 459 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  0  <  ( A  x.  B ) )
4443gt0ne0d 10113 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  ( A  x.  B )  =/=  0
)
4541, 42, 44mulne0bad 10200 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  A  =/=  0 )
4645neneqd 2656 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  -.  A  =  0 )
4746adantr 463 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  =  0 )  ->  -.  A  =  0
)
4840, 47pm2.21dd 174 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  =  0 )  -> 
1  =  ( 0  x.  (sgn `  B
) ) )
4927ad2antrr 723 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  B  e.  RR )
5049rexrd 9632 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  B  e.  RR* )
51 simpll 751 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  ( A  e.  RR  /\  B  e.  RR ) )
52 0red 9586 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  0  e.  RR )
53 simplll 757 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  A  e.  RR )
54 simpr 459 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  0  <  A )
5552, 53, 54ltled 9722 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  0  <_  A )
56 simplr 753 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  0  <  ( A  x.  B
) )
57 prodgt0 10383 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  B )
5851, 55, 56, 57syl12anc 1224 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  0  <  B )
59 sgnp 13008 . . . . . 6  |-  ( ( B  e.  RR*  /\  0  <  B )  ->  (sgn `  B )  =  1 )
6050, 58, 59syl2anc 659 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  (sgn `  B )  =  1 )
6160oveq2d 6286 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  (
1  x.  (sgn `  B ) )  =  ( 1  x.  1 ) )
62 1t1e1 10679 . . . 4  |-  ( 1  x.  1 )  =  1
6361, 62syl6req 2512 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  0  <  A )  ->  1  =  ( 1  x.  (sgn `  B )
) )
6427ad2antrr 723 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  B  e.  RR )
6564rexrd 9632 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  B  e.  RR* )
66 simplll 757 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  A  e.  RR )
6766renegcld 9982 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  -u A  e.  RR )
6864renegcld 9982 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  -u B  e.  RR )
69 0red 9586 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  e.  RR )
70 simpr 459 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  A  <  0 )
7125lt0neg1d 10118 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  0  <->  0  <  -u A ) )
7271ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  ( A  <  0  <->  0  <  -u A ) )
7370, 72mpbid 210 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  <  -u A )
7469, 67, 73ltled 9722 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  <_ 
-u A )
75 simplr 753 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  <  ( A  x.  B
) )
7626ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  A  e.  CC )
7728ad2antrr 723 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  B  e.  CC )
7876, 77mul2negd 10007 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  ( -u A  x.  -u B
)  =  ( A  x.  B ) )
7975, 78breqtrrd 4465 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  <  ( -u A  x.  -u B ) )
80 prodgt0 10383 . . . . . . . 8  |-  ( ( ( -u A  e.  RR  /\  -u B  e.  RR )  /\  (
0  <_  -u A  /\  0  <  ( -u A  x.  -u B ) ) )  ->  0  <  -u B )
8167, 68, 74, 79, 80syl22anc 1227 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  0  <  -u B )
8227lt0neg1d 10118 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  <  0  <->  0  <  -u B ) )
8382ad2antrr 723 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  ( B  <  0  <->  0  <  -u B ) )
8481, 83mpbird 232 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  B  <  0 )
85 sgnn 13012 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  <  0 )  ->  (sgn `  B )  =  -u
1 )
8665, 84, 85syl2anc 659 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  (sgn `  B )  =  -u
1 )
8786oveq2d 6286 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  ( -u 1  x.  (sgn `  B ) )  =  ( -u 1  x.  -u 1 ) )
88 neg1mulneg1e1 10749 . . . 4  |-  ( -u
1  x.  -u 1
)  =  1
8987, 88syl6req 2512 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B
) )  /\  A  <  0 )  ->  1  =  ( -u 1  x.  (sgn `  B )
) )
9033, 35, 37, 39, 48, 63, 89sgn3da 28747 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  ( A  x.  B )
)  ->  1  =  ( (sgn `  A )  x.  (sgn `  B )
) )
91 simpll 751 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0
)  ->  A  e.  RR )
9291rexrd 9632 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0
)  ->  A  e.  RR* )
9334eqeq2d 2468 . . 3  |-  ( (sgn
`  A )  =  0  ->  ( -u 1  =  ( (sgn `  A )  x.  (sgn `  B ) )  <->  -u 1  =  ( 0  x.  (sgn `  B ) ) ) )
9436eqeq2d 2468 . . 3  |-  ( (sgn
`  A )  =  1  ->  ( -u 1  =  ( (sgn `  A )  x.  (sgn `  B ) )  <->  -u 1  =  ( 1  x.  (sgn `  B ) ) ) )
9538eqeq2d 2468 . . 3  |-  ( (sgn
`  A )  = 
-u 1  ->  ( -u 1  =  ( (sgn
`  A )  x.  (sgn `  B )
)  <->  -u 1  =  (
-u 1  x.  (sgn `  B ) ) ) )
96 simpr 459 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  A  =  0 )
9726ad2antrr 723 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  A  e.  CC )
9828ad2antrr 723 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  B  e.  CC )
99 simplr 753 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  -> 
( A  x.  B
)  <  0 )
10099lt0ne0d 10114 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  -> 
( A  x.  B
)  =/=  0 )
10197, 98, 100mulne0bad 10200 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  A  =/=  0 )
102101neneqd 2656 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  -.  A  =  0
)
10396, 102pm2.21dd 174 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  =  0 )  ->  -u 1  =  ( 0  x.  (sgn `  B
) ) )
10427ad2antrr 723 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  0  <  A )  ->  B  e.  RR )
105104rexrd 9632 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  0  <  A )  ->  B  e.  RR* )
106 simplr 753 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0
)  ->  B  e.  RR )
10726, 28mulcomd 9606 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
108107breq1d 4449 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  <  0  <->  ( B  x.  A )  <  0 ) )
109108biimpa 482 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0
)  ->  ( B  x.  A )  <  0
)
110106, 91, 109mul2lt0rgt0 27800 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  0  <  A )  ->  B  <  0 )
111105, 110, 85syl2anc 659 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  0  <  A )  ->  (sgn `  B )  =  -u
1 )
112111oveq2d 6286 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  0  <  A )  ->  (
1  x.  (sgn `  B ) )  =  ( 1  x.  -u 1
) )
113 neg1cn 10635 . . . . 5  |-  -u 1  e.  CC
114113mulid2i 9588 . . . 4  |-  ( 1  x.  -u 1 )  = 
-u 1
115112, 114syl6req 2512 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  0  <  A )  ->  -u 1  =  ( 1  x.  (sgn `  B )
) )
116106adantr 463 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  B  e.  RR )
117116rexrd 9632 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  B  e.  RR* )
118106, 91, 109mul2lt0rlt0 27799 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  0  <  B )
119117, 118, 59syl2anc 659 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  (sgn `  B )  =  1 )
120119oveq2d 6286 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  ( -u 1  x.  (sgn `  B ) )  =  ( -u 1  x.  1 ) )
121113mulid1i 9587 . . . 4  |-  ( -u
1  x.  1 )  =  -u 1
122120, 121syl6req 2512 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  -u 1  =  ( -u 1  x.  (sgn `  B )
) )
12392, 93, 94, 95, 103, 115, 122sgn3da 28747 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  x.  B )  <  0
)  ->  -u 1  =  ( (sgn `  A
)  x.  (sgn `  B ) ) )
1242, 3, 4, 5, 31, 90, 123sgn3da 28747 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 366    /\ wa 367    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    x. cmul 9486   RR*cxr 9616    < clt 9617    <_ cle 9618   -ucneg 9797  sgncsgn 13004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-rp 11222  df-sgn 13005
This theorem is referenced by:  sgnmulrp2  28749  sgnmulsgn  28755  sgnmulsgp  28756
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