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

Theorem sgnmulsgp 28753
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 10071 . . . . 5  |-  0  <  1
2 breq2 4443 . . . . 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 464 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  0  <  (sgn `  ( A  x.  B
) ) )
5 simplr 753 . . . . . 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 459 . . . . . 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 4463 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  -u 1
)  ->  0  <  -u 1 )
8 1nn0 10807 . . . . . . . 8  |-  1  e.  NN0
9 nn0nlt0 10818 . . . . . . . 8  |-  ( 1  e.  NN0  ->  -.  1  <  0 )
108, 9ax-mp 5 . . . . . . 7  |-  -.  1  <  0
11 1re 9584 . . . . . . . 8  |-  1  e.  RR
12 lt0neg1 10054 . . . . . . . 8  |-  ( 1  e.  RR  ->  (
1  <  0  <->  0  <  -u 1 ) )
1311, 12ax-mp 5 . . . . . . 7  |-  ( 1  <  0  <->  0  <  -u 1 )
1410, 13mtbi 296 . . . . . 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 459 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  =  0 )
18 simplr 753 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  0  <  (sgn `  ( A  x.  B
) ) )
1918gt0ne0d 10113 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  =/=  0 )
2017, 19pm2.21ddne 2768 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  0 )  ->  (sgn `  ( A  x.  B )
)  =  1 )
21 simpr 459 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B ) ) )  /\  (sgn `  ( A  x.  B )
)  =  1 )  ->  (sgn `  ( A  x.  B )
)  =  1 )
22 remulcl 9566 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
2322rexrd 9632 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR* )
2423adantr 463 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B
) ) )  -> 
( A  x.  B
)  e.  RR* )
25 sgncl 28741 . . . . 5  |-  ( ( A  x.  B )  e.  RR*  ->  (sgn `  ( A  x.  B
) )  e.  { -u 1 ,  0 ,  1 } )
26 eltpi 4060 . . . . 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 )  /\  0  <  (sgn `  ( A  x.  B
) ) )  -> 
( (sgn `  ( A  x.  B )
)  =  -u 1  \/  (sgn `  ( A  x.  B ) )  =  0  \/  (sgn `  ( A  x.  B
) )  =  1 ) )
2816, 20, 21, 27mpjao3dan 1293 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  (sgn `  ( A  x.  B
) ) )  -> 
(sgn `  ( A  x.  B ) )  =  1 )
294, 28impbida 830 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( (sgn `  ( A  x.  B )
)  =  1  <->  0  <  (sgn `  ( A  x.  B )
) ) )
30 sgnpbi 28749 . . 3  |-  ( ( A  x.  B )  e.  RR*  ->  ( (sgn
`  ( A  x.  B ) )  =  1  <->  0  <  ( A  x.  B )
) )
3123, 30syl 16 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( (sgn `  ( A  x.  B )
)  =  1  <->  0  <  ( A  x.  B ) ) )
32 sgnmul 28745 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (sgn `  ( A  x.  B ) )  =  ( (sgn `  A
)  x.  (sgn `  B ) ) )
3332breq2d 4451 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  (sgn `  ( A  x.  B
) )  <->  0  <  ( (sgn `  A )  x.  (sgn `  B )
) ) )
3429, 31, 333bitr3d 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 367    \/ w3o 970    = wceq 1398    e. wcel 1823   {ctp 4020   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482    x. cmul 9486   RR*cxr 9616    < clt 9617   -ucneg 9797   NN0cn0 10791  sgncsgn 13001
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-pss 3477  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-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  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-om 6674  df-recs 7034  df-rdg 7068  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-nn 10532  df-n0 10792  df-rp 11222  df-sgn 13002
This theorem is referenced by:  signsvfpn  28806
  Copyright terms: Public domain W3C validator