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Theorem sgnmulsgp 26938
Description: If two real numbers are of different signs, so are their signs. (Contributed by Thierry Arnoux, 12-Oct-2018.)
Assertion
Ref Expression
sgnmulsgp  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  ( A  x.  B )  <->  0  <  ( (sgn `  A )  x.  (sgn `  B ) ) ) )

Proof of Theorem sgnmulsgp
StepHypRef Expression
1 0lt1 9867 . . . . 5  |-  0  <  1
2 breq2 4301 . . . . 5  |-  ( (sgn
`  ( A  x.  B ) )  =  1  ->  ( 0  <  (sgn `  ( A  x.  B )
)  <->  0  <  1
) )
31, 2mpbiri 233 . . . 4  |-  ( (sgn
`  ( A  x.  B ) )  =  1  ->  0  <  (sgn
`  ( A  x.  B ) ) )
43adantl 466 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  0  <  (sgn `  ( A  x.  B
) ) )
5 simplr 754 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  -u 1
)  ->  0  <  (sgn
`  ( A  x.  B ) ) )
6 simpr 461 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  -u 1
)  ->  (sgn `  ( A  x.  B )
)  =  -u 1
)
75, 6breqtrd 4321 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  -u 1
)  ->  0  <  -u 1 )
8 1nn0 10600 . . . . . . . 8  |-  1  e.  NN0
9 nn0nlt0 10611 . . . . . . . 8  |-  ( 1  e.  NN0  ->  -.  1  <  0 )
108, 9ax-mp 5 . . . . . . 7  |-  -.  1  <  0
11 1re 9390 . . . . . . . 8  |-  1  e.  RR
12 lt0neg1 9850 . . . . . . . 8  |-  ( 1  e.  RR  ->  (
1  <  0  <->  0  <  -u 1 ) )
1311, 12ax-mp 5 . . . . . . 7  |-  ( 1  <  0  <->  0  <  -u 1 )
1410, 13mtbi 298 . . . . . 6  |-  -.  0  <  -u 1
1514a1i 11 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  -u 1
)  ->  -.  0  <  -u 1 )
167, 15pm2.21dd 174 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  -u 1
)  ->  (sgn `  ( A  x.  B )
)  =  1 )
17 simpr 461 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  =  0 )
18 remulcl 9372 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
1918ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  ( A  x.  B )  e.  RR )
20 sgnclre 26927 . . . . . . . . . 10  |-  ( ( A  x.  B )  e.  RR  ->  (sgn `  ( A  x.  B
) )  e.  RR )
2119, 20syl 16 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  e.  RR )
2217, 21eqeltrrd 2518 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  0  e.  RR )
23 simplr 754 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  0  <  (sgn `  ( A  x.  B
) ) )
2422, 23ltned 9515 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  0  =/=  (sgn `  ( A  x.  B
) ) )
2524necomd 2700 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  =/=  0 )
2625neneqd 2629 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  -.  (sgn `  ( A  x.  B )
)  =  0 )
2717, 26pm2.21dd 174 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  =  1 )
28 simpr 461 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  (sgn `  ( A  x.  B )
)  =  1 )
2918rexrd 9438 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR* )
3029adantr 465 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B
) ) )  -> 
( A  x.  B
)  e.  RR* )
31 sgncl 26926 . . . . 5  |-  ( ( A  x.  B )  e.  RR*  ->  (sgn `  ( A  x.  B
) )  e.  { -u 1 ,  0 ,  1 } )
32 eltpi 3925 . . . . 5  |-  ( (sgn
`  ( A  x.  B ) )  e. 
{ -u 1 ,  0 ,  1 }  ->  ( (sgn `  ( A  x.  B ) )  = 
-u 1  \/  (sgn `  ( A  x.  B
) )  =  0  \/  (sgn `  ( A  x.  B )
)  =  1 ) )
3330, 31, 323syl 20 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B
) ) )  -> 
( (sgn `  ( A  x.  B )
)  =  -u 1  \/  (sgn `  ( A  x.  B ) )  =  0  \/  (sgn `  ( A  x.  B
) )  =  1 ) )
3416, 27, 28, 33mpjao3dan 1285 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B
) ) )  -> 
(sgn `  ( A  x.  B ) )  =  1 )
354, 34impbida 828 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( (sgn `  ( A  x.  B )
)  =  1  <->  0  <  (sgn `  ( A  x.  B )
) ) )
36 sgnpbi 26934 . . 3  |-  ( ( A  x.  B )  e.  RR*  ->  ( (sgn
`  ( A  x.  B ) )  =  1  <->  0  <  ( A  x.  B )
) )
3729, 36syl 16 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( (sgn `  ( A  x.  B )
)  =  1  <->  0  <  ( A  x.  B ) ) )
38 sgnmul 26930 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (sgn `  ( A  x.  B ) )  =  ( (sgn `  A
)  x.  (sgn `  B ) ) )
3938breq2d 4309 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  (sgn `  ( A  x.  B
) )  <->  0  <  ( (sgn `  A )  x.  (sgn `  B )
) ) )
4035, 37, 393bitr3d 283 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  ( A  x.  B )  <->  0  <  ( (sgn `  A )  x.  (sgn `  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 964    = wceq 1369    e. wcel 1756   {ctp 3886   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   RRcr 9286   0cc0 9287   1c1 9288    x. cmul 9292   RR*cxr 9422    < clt 9423   -ucneg 9601   NN0cn0 10584  sgncsgn 12580
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-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
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-reu 2727  df-rmo 2728  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-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  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-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-n0 10585  df-rp 10997  df-sgn 12581
This theorem is referenced by:  signsvfpn  26991
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