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

Theorem sgnmulsgn 26930
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 10426 . . . . 5  |-  -u 1  <  0
2 breq1 4293 . . . . 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 9365 . . . . . . . . 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 26920 . . . . . . . 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 9508 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  =/=  0 )
1312neneqd 2622 . . . . 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 4312 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B
) )  <  0
)  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  1  <  0
)
18 1nn0 10593 . . . . . . 7  |-  1  e.  NN0
19 nn0nlt0 10604 . . . . . . 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 9431 . . . . . 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 26919 . . . . 5  |-  ( ( A  x.  B )  e.  RR*  ->  (sgn `  ( A  x.  B
) )  e.  { -u 1 ,  0 ,  1 } )
26 eltpi 3918 . . . . 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 1285 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B )
)  <  0 )  ->  (sgn `  ( A  x.  B )
)  =  -u 1
)
294, 28impbida 828 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( (sgn `  ( A  x.  B )
)  =  -u 1  <->  (sgn
`  ( A  x.  B ) )  <  0 ) )
30 sgnnbi 26926 . . 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 26923 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (sgn `  ( A  x.  B ) )  =  ( (sgn `  A
)  x.  (sgn `  B ) ) )
3332breq1d 4300 . 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 964    = wceq 1369    e. wcel 1756   {ctp 3879   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   RRcr 9279   0cc0 9280   1c1 9281    x. cmul 9285   RR*cxr 9415    < clt 9416   -ucneg 9594   NN0cn0 10577  sgncsgn 12573
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-n0 10578  df-rp 10990  df-sgn 12574
This theorem is referenced by:  signsvfn  26981  signsvfnn  26985
  Copyright terms: Public domain W3C validator