Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sgn3da Structured version   Unicode version

Theorem sgn3da 28986
Description: A conditional containing a signum is true if it is true in all three possible cases. (Contributed by Thierry Arnoux, 1-Oct-2018.)
Hypotheses
Ref Expression
sgn3da.0  |-  ( ph  ->  A  e.  RR* )
sgn3da.1  |-  ( (sgn
`  A )  =  0  ->  ( ps  <->  ch ) )
sgn3da.2  |-  ( (sgn
`  A )  =  1  ->  ( ps  <->  th ) )
sgn3da.3  |-  ( (sgn
`  A )  = 
-u 1  ->  ( ps 
<->  ta ) )
sgn3da.4  |-  ( (
ph  /\  A  = 
0 )  ->  ch )
sgn3da.5  |-  ( (
ph  /\  0  <  A )  ->  th )
sgn3da.6  |-  ( (
ph  /\  A  <  0 )  ->  ta )
Assertion
Ref Expression
sgn3da  |-  ( ph  ->  ps )

Proof of Theorem sgn3da
StepHypRef Expression
1 sgn3da.0 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR* )
2 sgnval 13070 . . . . . . . . 9  |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
31, 2syl 17 . . . . . . . 8  |-  ( ph  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
43eqeq2d 2416 . . . . . . 7  |-  ( ph  ->  ( 0  =  (sgn
`  A )  <->  0  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) ) )
54pm5.32i 635 . . . . . 6  |-  ( (
ph  /\  0  =  (sgn `  A ) )  <-> 
( ph  /\  0  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) ) )
6 sgn3da.1 . . . . . . . . 9  |-  ( (sgn
`  A )  =  0  ->  ( ps  <->  ch ) )
76eqcoms 2414 . . . . . . . 8  |-  ( 0  =  (sgn `  A
)  ->  ( ps  <->  ch ) )
87bicomd 201 . . . . . . 7  |-  ( 0  =  (sgn `  A
)  ->  ( ch  <->  ps ) )
98adantl 464 . . . . . 6  |-  ( (
ph  /\  0  =  (sgn `  A ) )  ->  ( ch  <->  ps )
)
105, 9sylbir 213 . . . . 5  |-  ( (
ph  /\  0  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )  -> 
( ch  <->  ps )
)
1110expcom 433 . . . 4  |-  ( 0  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u
1 ,  1 ) )  ->  ( ph  ->  ( ch  <->  ps )
) )
1211pm5.74d 247 . . 3  |-  ( 0  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u
1 ,  1 ) )  ->  ( ( ph  ->  ch )  <->  ( ph  ->  ps ) ) )
133eqeq2d 2416 . . . . . . 7  |-  ( ph  ->  ( if ( A  <  0 ,  -u
1 ,  1 )  =  (sgn `  A
)  <->  if ( A  <  0 ,  -u 1 ,  1 )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) ) )
1413pm5.32i 635 . . . . . 6  |-  ( (
ph  /\  if ( A  <  0 ,  -u
1 ,  1 )  =  (sgn `  A
) )  <->  ( ph  /\  if ( A  <  0 ,  -u 1 ,  1 )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) ) )
15 eqeq1 2406 . . . . . . . . 9  |-  ( -u
1  =  if ( A  <  0 , 
-u 1 ,  1 )  ->  ( -u 1  =  (sgn `  A )  <->  if ( A  <  0 ,  -u 1 ,  1 )  =  (sgn `  A ) ) )
1615imbi1d 315 . . . . . . . 8  |-  ( -u
1  =  if ( A  <  0 , 
-u 1 ,  1 )  ->  ( ( -u 1  =  (sgn `  A )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ps ) )  <->  ( if ( A  <  0 ,  -u 1 ,  1 )  =  (sgn `  A )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ps ) ) ) )
17 eqeq1 2406 . . . . . . . . 9  |-  ( 1  =  if ( A  <  0 ,  -u
1 ,  1 )  ->  ( 1  =  (sgn `  A )  <->  if ( A  <  0 ,  -u 1 ,  1 )  =  (sgn `  A ) ) )
1817imbi1d 315 . . . . . . . 8  |-  ( 1  =  if ( A  <  0 ,  -u
1 ,  1 )  ->  ( ( 1  =  (sgn `  A
)  ->  ( (
( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) )  <->  ps )
)  <->  ( if ( A  <  0 , 
-u 1 ,  1 )  =  (sgn `  A )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ps ) ) ) )
19 sgn3da.6 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  A  <  0 )  ->  ta )
2019adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  <  0 )  /\  ( A  <  0  ->  ta ) )  ->  ta )
21 simp2 998 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  <  0 )  /\  ta  /\  A  <  0 )  ->  ta )
22213expia 1199 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  <  0 )  /\  ta )  ->  ( A  <  0  ->  ta )
)
2320, 22impbida 833 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  <  0 )  ->  (
( A  <  0  ->  ta )  <->  ta )
)
24 pm3.24 883 . . . . . . . . . . . . . . . . 17  |-  -.  ( A  <  0  /\  -.  A  <  0 )
2524pm2.21i 131 . . . . . . . . . . . . . . . 16  |-  ( ( A  <  0  /\ 
-.  A  <  0
)  ->  th )
2625adantl 464 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( A  <  0  /\  -.  A  <  0 ) )  ->  th )
2726expr 613 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  <  0 )  ->  ( -.  A  <  0  ->  th ) )
28 tbtru 1415 . . . . . . . . . . . . . 14  |-  ( ( -.  A  <  0  ->  th )  <->  ( ( -.  A  <  0  ->  th )  <-> T.  )
)
2927, 28sylib 196 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  <  0 )  ->  (
( -.  A  <  0  ->  th )  <-> T.  ) )
3023, 29anbi12d 709 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  0 )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ( ta  /\ T.  ) ) )
31 ancom 448 . . . . . . . . . . . . 13  |-  ( ( ta  /\ T.  )  <-> 
( T.  /\  ta ) )
32 truan 1422 . . . . . . . . . . . . 13  |-  ( ( T.  /\  ta )  <->  ta )
3331, 32bitri 249 . . . . . . . . . . . 12  |-  ( ( ta  /\ T.  )  <->  ta )
3430, 33syl6bb 261 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  0 )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ta ) )
35343adant3 1017 . . . . . . . . . 10  |-  ( (
ph  /\  A  <  0  /\  -u 1  =  (sgn
`  A ) )  ->  ( ( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) )  <->  ta )
)
36 sgn3da.3 . . . . . . . . . . . 12  |-  ( (sgn
`  A )  = 
-u 1  ->  ( ps 
<->  ta ) )
3736eqcoms 2414 . . . . . . . . . . 11  |-  ( -u
1  =  (sgn `  A )  ->  ( ps 
<->  ta ) )
38373ad2ant3 1020 . . . . . . . . . 10  |-  ( (
ph  /\  A  <  0  /\  -u 1  =  (sgn
`  A ) )  ->  ( ps  <->  ta )
)
3935, 38bitr4d 256 . . . . . . . . 9  |-  ( (
ph  /\  A  <  0  /\  -u 1  =  (sgn
`  A ) )  ->  ( ( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) )  <->  ps )
)
40393expia 1199 . . . . . . . 8  |-  ( (
ph  /\  A  <  0 )  ->  ( -u 1  =  (sgn `  A )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ps ) ) )
41193adant2 1016 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  A  <  0  /\  A  <  0 )  ->  ta )
42413expia 1199 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  A  <  0 )  ->  ( A  <  0  ->  ta ) )
43 tbtru 1415 . . . . . . . . . . . . . 14  |-  ( ( A  <  0  ->  ta )  <->  ( ( A  <  0  ->  ta ) 
<-> T.  ) )
4442, 43sylib 196 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
( A  <  0  ->  ta )  <-> T.  )
)
45 pm3.35 585 . . . . . . . . . . . . . . 15  |-  ( ( -.  A  <  0  /\  ( -.  A  <  0  ->  th )
)  ->  th )
4645adantll 712 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  A  <  0 )  /\  ( -.  A  <  0  ->  th ) )  ->  th )
47 simp2 998 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  -.  A  <  0 )  /\  th 
/\  -.  A  <  0 )  ->  th )
48473expia 1199 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  A  <  0 )  /\  th )  ->  ( -.  A  <  0  ->  th )
)
4946, 48impbida 833 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
( -.  A  <  0  ->  th )  <->  th ) )
5044, 49anbi12d 709 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ( T.  /\  th ) ) )
51 truan 1422 . . . . . . . . . . . 12  |-  ( ( T.  /\  th )  <->  th )
5250, 51syl6bb 261 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  th ) )
53523adant3 1017 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  <  0  /\  1  =  (sgn `  A )
)  ->  ( (
( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) )  <->  th )
)
54 sgn3da.2 . . . . . . . . . . . 12  |-  ( (sgn
`  A )  =  1  ->  ( ps  <->  th ) )
5554eqcoms 2414 . . . . . . . . . . 11  |-  ( 1  =  (sgn `  A
)  ->  ( ps  <->  th ) )
56553ad2ant3 1020 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  <  0  /\  1  =  (sgn `  A )
)  ->  ( ps  <->  th ) )
5753, 56bitr4d 256 . . . . . . . . 9  |-  ( (
ph  /\  -.  A  <  0  /\  1  =  (sgn `  A )
)  ->  ( (
( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) )  <->  ps )
)
58573expia 1199 . . . . . . . 8  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
1  =  (sgn `  A )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ps ) ) )
5916, 18, 40, 58ifbothda 3920 . . . . . . 7  |-  ( ph  ->  ( if ( A  <  0 ,  -u
1 ,  1 )  =  (sgn `  A
)  ->  ( (
( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) )  <->  ps )
) )
6059imp 427 . . . . . 6  |-  ( (
ph  /\  if ( A  <  0 ,  -u
1 ,  1 )  =  (sgn `  A
) )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ps ) )
6114, 60sylbir 213 . . . . 5  |-  ( (
ph  /\  if ( A  <  0 ,  -u
1 ,  1 )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u
1 ,  1 ) ) )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ps ) )
6261expcom 433 . . . 4  |-  ( if ( A  <  0 ,  -u 1 ,  1 )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 , 
-u 1 ,  1 ) )  ->  ( ph  ->  ( ( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) )  <->  ps )
) )
6362pm5.74d 247 . . 3  |-  ( if ( A  <  0 ,  -u 1 ,  1 )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 , 
-u 1 ,  1 ) )  ->  (
( ph  ->  ( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) ) )  <-> 
( ph  ->  ps )
) )
64 sgn3da.4 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  ch )
6564expcom 433 . . . 4  |-  ( A  =  0  ->  ( ph  ->  ch ) )
6665adantl 464 . . 3  |-  ( ( T.  /\  A  =  0 )  ->  ( ph  ->  ch ) )
6719ex 432 . . . . . . 7  |-  ( ph  ->  ( A  <  0  ->  ta ) )
6867adantr 463 . . . . . 6  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( A  <  0  ->  ta ) )
69 simp1 997 . . . . . . . 8  |-  ( (
ph  /\  -.  A  =  0  /\  -.  A  <  0 )  ->  ph )
70 df-ne 2600 . . . . . . . . . . . 12  |-  ( A  =/=  0  <->  -.  A  =  0 )
71 0xr 9670 . . . . . . . . . . . . 13  |-  0  e.  RR*
72 xrlttri2 11401 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  0  e.  RR* )  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  < 
A ) ) )
731, 71, 72sylancl 660 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  <  A ) ) )
7470, 73syl5bbr 259 . . . . . . . . . . 11  |-  ( ph  ->  ( -.  A  =  0  <->  ( A  <  0  \/  0  < 
A ) ) )
7574biimpa 482 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( A  <  0  \/  0  <  A ) )
7675ord 375 . . . . . . . . 9  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( -.  A  <  0  ->  0  <  A ) )
77763impia 1194 . . . . . . . 8  |-  ( (
ph  /\  -.  A  =  0  /\  -.  A  <  0 )  -> 
0  <  A )
78 sgn3da.5 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  th )
7969, 77, 78syl2anc 659 . . . . . . 7  |-  ( (
ph  /\  -.  A  =  0  /\  -.  A  <  0 )  ->  th )
80793expia 1199 . . . . . 6  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( -.  A  <  0  ->  th )
)
8168, 80jca 530 . . . . 5  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
) )
8281expcom 433 . . . 4  |-  ( -.  A  =  0  -> 
( ph  ->  ( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) ) ) )
8382adantl 464 . . 3  |-  ( ( T.  /\  -.  A  =  0 )  -> 
( ph  ->  ( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) ) ) )
8412, 63, 66, 83ifbothda 3920 . 2  |-  ( T. 
->  ( ph  ->  ps ) )
8584trud 1414 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 974    = wceq 1405   T. wtru 1406    e. wcel 1842    =/= wne 2598   ifcif 3885   class class class wbr 4395   ` cfv 5569   0cc0 9522   1c1 9523   RR*cxr 9657    < clt 9658   -ucneg 9842  sgncsgn 13068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-i2m1 9590  ax-1ne0 9591  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-neg 9844  df-sgn 13069
This theorem is referenced by:  sgnmul  28987  sgnsub  28989  sgnnbi  28990  sgnpbi  28991  sgn0bi  28992  sgnsgn  28993
  Copyright terms: Public domain W3C validator