Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sgnmulsgn Structured version   Unicode version

Theorem sgnmulsgn 29208
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 10716 . . . . 5  |-  -u 1  <  0
2 breq1 4429 . . . . 5  |-  ( (sgn
`  ( A  x.  B ) )  = 
-u 1  ->  (
(sgn `  ( A  x.  B ) )  <  0  <->  -u 1  <  0
) )
31, 2mpbiri 236 . . . 4  |-  ( (sgn
`  ( A  x.  B ) )  = 
-u 1  ->  (sgn `  ( A  x.  B
) )  <  0
)
43adantl 467 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B )
)  =  -u 1
)  ->  (sgn `  ( A  x.  B )
)  <  0 )
5 simpr 462 . . . 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 462 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  =  0 )
7 simplr 760 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  <  0 )
87lt0ne0d 10178 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  =/=  0 )
96, 8pm2.21ddne 2745 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  =  -u 1
)
10 simpr 462 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  (sgn `  ( A  x.  B )
)  =  1 )
11 simplr 760 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  (sgn `  ( A  x.  B )
)  <  0 )
1210, 11eqbrtrrd 4448 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  1  <  0
)
13 1nn0 10885 . . . . . 6  |-  1  e.  NN0
14 nn0nlt0 10896 . . . . . 6  |-  ( 1  e.  NN0  ->  -.  1  <  0 )
1513, 14mp1i 13 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  -.  1  <  0 )
1612, 15pm2.21dd 177 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  (sgn `  ( A  x.  B )
)  =  -u 1
)
17 remulcl 9623 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
1817rexrd 9689 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR* )
1918adantr 466 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B )
)  <  0 )  ->  ( A  x.  B )  e.  RR* )
20 sgncl 29197 . . . . 5  |-  ( ( A  x.  B )  e.  RR*  ->  (sgn `  ( A  x.  B
) )  e.  { -u 1 ,  0 ,  1 } )
21 eltpi 4047 . . . . 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 ) )
2219, 20, 213syl 18 . . . 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 ) )
235, 9, 16, 22mpjao3dan 1331 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B )
)  <  0 )  ->  (sgn `  ( A  x.  B )
)  =  -u 1
)
244, 23impbida 840 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( (sgn `  ( A  x.  B )
)  =  -u 1  <->  (sgn
`  ( A  x.  B ) )  <  0 ) )
25 sgnnbi 29204 . . 3  |-  ( ( A  x.  B )  e.  RR*  ->  ( (sgn
`  ( A  x.  B ) )  = 
-u 1  <->  ( A  x.  B )  <  0
) )
2618, 25syl 17 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( (sgn `  ( A  x.  B )
)  =  -u 1  <->  ( A  x.  B )  <  0 ) )
27 sgnmul 29201 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (sgn `  ( A  x.  B ) )  =  ( (sgn `  A
)  x.  (sgn `  B ) ) )
2827breq1d 4436 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( (sgn `  ( A  x.  B )
)  <  0  <->  ( (sgn `  A )  x.  (sgn `  B ) )  <  0 ) )
2924, 26, 283bitr3d 286 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 187    /\ wa 370    \/ w3o 981    = wceq 1437    e. wcel 1870   {ctp 4006   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543   RR*cxr 9673    < clt 9674   -ucneg 9860   NN0cn0 10869  sgncsgn 13128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-n0 10870  df-rp 11303  df-sgn 13129
This theorem is referenced by:  signsvfn  29259  signsvfnn  29263
  Copyright terms: Public domain W3C validator