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Theorem sgnmulsgn 28156
Description: If two real numbers are of different signs, so are their signs. (Contributed by Thierry Arnoux, 12-Oct-2018.)
Assertion
Ref Expression
sgnmulsgn  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  <  0  <->  ( (sgn `  A )  x.  (sgn `  B )
)  <  0 ) )

Proof of Theorem sgnmulsgn
StepHypRef Expression
1 neg1lt0 10642 . . . . 5  |-  -u 1  <  0
2 breq1 4450 . . . . 5  |-  ( (sgn
`  ( A  x.  B ) )  = 
-u 1  ->  (
(sgn `  ( A  x.  B ) )  <  0  <->  -u 1  <  0
) )
31, 2mpbiri 233 . . . 4  |-  ( (sgn
`  ( A  x.  B ) )  = 
-u 1  ->  (sgn `  ( A  x.  B
) )  <  0
)
43adantl 466 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B )
)  =  -u 1
)  ->  (sgn `  ( A  x.  B )
)  <  0 )
5 simpr 461 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  -u 1
)  ->  (sgn `  ( A  x.  B )
)  =  -u 1
)
6 simpr 461 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  =  0 )
7 remulcl 9577 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
87ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  ( A  x.  B )  e.  RR )
9 sgnclre 28146 . . . . . . . 8  |-  ( ( A  x.  B )  e.  RR  ->  (sgn `  ( A  x.  B
) )  e.  RR )
108, 9syl 16 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  e.  RR )
11 simplr 754 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  <  0 )
1210, 11ltned 9720 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  =/=  0 )
1312neneqd 2669 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  -.  (sgn `  ( A  x.  B )
)  =  0 )
146, 13pm2.21dd 174 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  =  -u 1
)
15 simpr 461 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  (sgn `  ( A  x.  B )
)  =  1 )
16 simplr 754 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  (sgn `  ( A  x.  B )
)  <  0 )
1715, 16eqbrtrrd 4469 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  1  <  0
)
18 1nn0 10811 . . . . . . 7  |-  1  e.  NN0
19 nn0nlt0 10822 . . . . . . 7  |-  ( 1  e.  NN0  ->  -.  1  <  0 )
2018, 19ax-mp 5 . . . . . 6  |-  -.  1  <  0
2120a1i 11 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  -.  1  <  0 )
2217, 21pm2.21dd 174 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  (sgn `  ( A  x.  B )
)  =  -u 1
)
237rexrd 9643 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR* )
2423adantr 465 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B )
)  <  0 )  ->  ( A  x.  B )  e.  RR* )
25 sgncl 28145 . . . . 5  |-  ( ( A  x.  B )  e.  RR*  ->  (sgn `  ( A  x.  B
) )  e.  { -u 1 ,  0 ,  1 } )
26 eltpi 4071 . . . . 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 ) )
2724, 25, 263syl 20 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B )
)  <  0 )  ->  ( (sgn `  ( A  x.  B
) )  =  -u
1  \/  (sgn `  ( A  x.  B
) )  =  0  \/  (sgn `  ( A  x.  B )
)  =  1 ) )
285, 14, 22, 27mpjao3dan 1295 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B )
)  <  0 )  ->  (sgn `  ( A  x.  B )
)  =  -u 1
)
294, 28impbida 830 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( (sgn `  ( A  x.  B )
)  =  -u 1  <->  (sgn
`  ( A  x.  B ) )  <  0 ) )
30 sgnnbi 28152 . . 3  |-  ( ( A  x.  B )  e.  RR*  ->  ( (sgn
`  ( A  x.  B ) )  = 
-u 1  <->  ( A  x.  B )  <  0
) )
3123, 30syl 16 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( (sgn `  ( A  x.  B )
)  =  -u 1  <->  ( A  x.  B )  <  0 ) )
32 sgnmul 28149 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (sgn `  ( A  x.  B ) )  =  ( (sgn `  A
)  x.  (sgn `  B ) ) )
3332breq1d 4457 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( (sgn `  ( A  x.  B )
)  <  0  <->  ( (sgn `  A )  x.  (sgn `  B ) )  <  0 ) )
3429, 31, 333bitr3d 283 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  <  0  <->  ( (sgn `  A )  x.  (sgn `  B )
)  <  0 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767   {ctp 4031   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   RRcr 9491   0cc0 9492   1c1 9493    x. cmul 9497   RR*cxr 9627    < clt 9628   -ucneg 9806   NN0cn0 10795  sgncsgn 12882
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 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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-pss 3492  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-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-n0 10796  df-rp 11221  df-sgn 12883
This theorem is referenced by:  signsvfn  28207  signsvfnn  28211
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