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Theorem sgn3da 28346
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 12895 . . . . . . . . 9  |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
31, 2syl 16 . . . . . . . 8  |-  ( ph  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
43eqeq2d 2455 . . . . . . 7  |-  ( ph  ->  ( 0  =  (sgn
`  A )  <->  0  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) ) )
54pm5.32i 637 . . . . . 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 2453 . . . . . . . 8  |-  ( 0  =  (sgn `  A
)  ->  ( ps  <->  ch ) )
87bicomd 201 . . . . . . 7  |-  ( 0  =  (sgn `  A
)  ->  ( ch  <->  ps ) )
98adantl 466 . . . . . 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 435 . . . 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 2455 . . . . . . 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 637 . . . . . 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 2445 . . . . . . . . 9  |-  ( -u
1  =  if ( A  <  0 , 
-u 1 ,  1 )  ->  ( -u 1  =  (sgn `  A )  <->  if ( A  <  0 ,  -u 1 ,  1 )  =  (sgn `  A ) ) )
1615imbi1d 317 . . . . . . . 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 2445 . . . . . . . . 9  |-  ( 1  =  if ( A  <  0 ,  -u
1 ,  1 )  ->  ( 1  =  (sgn `  A )  <->  if ( A  <  0 ,  -u 1 ,  1 )  =  (sgn `  A ) ) )
1817imbi1d 317 . . . . . . . 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 465 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  <  0 )  /\  ( A  <  0  ->  ta ) )  ->  ta )
21 simp2 996 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  <  0 )  /\  ta  /\  A  <  0 )  ->  ta )
22213expia 1197 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  <  0 )  /\  ta )  ->  ( A  <  0  ->  ta )
)
2320, 22impbida 830 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  <  0 )  ->  (
( A  <  0  ->  ta )  <->  ta )
)
24 pm3.24 880 . . . . . . . . . . . . . . . . 17  |-  -.  ( A  <  0  /\  -.  A  <  0 )
2524pm2.21i 131 . . . . . . . . . . . . . . . 16  |-  ( ( A  <  0  /\ 
-.  A  <  0
)  ->  th )
2625adantl 466 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( A  <  0  /\  -.  A  <  0 ) )  ->  th )
2726expr 615 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  <  0 )  ->  ( -.  A  <  0  ->  th ) )
28 tbtru 1391 . . . . . . . . . . . . . 14  |-  ( ( -.  A  <  0  ->  th )  <->  ( ( -.  A  <  0  ->  th )  <-> T.  )
)
2927, 28sylib 196 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  <  0 )  ->  (
( -.  A  <  0  ->  th )  <-> T.  ) )
3023, 29anbi12d 710 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  0 )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ( ta  /\ T.  ) ) )
31 ancom 450 . . . . . . . . . . . . 13  |-  ( ( ta  /\ T.  )  <-> 
( T.  /\  ta ) )
32 truan 1398 . . . . . . . . . . . . 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 1015 . . . . . . . . . 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 2453 . . . . . . . . . . 11  |-  ( -u
1  =  (sgn `  A )  ->  ( ps 
<->  ta ) )
38373ad2ant3 1018 . . . . . . . . . 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 1197 . . . . . . . 8  |-  ( (
ph  /\  A  <  0 )  ->  ( -u 1  =  (sgn `  A )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ps ) ) )
41193adant2 1014 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  A  <  0  /\  A  <  0 )  ->  ta )
42413expia 1197 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  A  <  0 )  ->  ( A  <  0  ->  ta ) )
43 tbtru 1391 . . . . . . . . . . . . . 14  |-  ( ( A  <  0  ->  ta )  <->  ( ( A  <  0  ->  ta ) 
<-> T.  ) )
4442, 43sylib 196 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
( A  <  0  ->  ta )  <-> T.  )
)
45 pm3.35 587 . . . . . . . . . . . . . . 15  |-  ( ( -.  A  <  0  /\  ( -.  A  <  0  ->  th )
)  ->  th )
4645adantll 713 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  A  <  0 )  /\  ( -.  A  <  0  ->  th ) )  ->  th )
47 simp2 996 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  -.  A  <  0 )  /\  th 
/\  -.  A  <  0 )  ->  th )
48473expia 1197 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  A  <  0 )  /\  th )  ->  ( -.  A  <  0  ->  th )
)
4946, 48impbida 830 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
( -.  A  <  0  ->  th )  <->  th ) )
5044, 49anbi12d 710 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ( T.  /\  th ) ) )
51 truan 1398 . . . . . . . . . . . 12  |-  ( ( T.  /\  th )  <->  th )
5250, 51syl6bb 261 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  th ) )
53523adant3 1015 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  <  0  /\  1  =  (sgn `  A )
)  ->  ( (
( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) )  <->  th )
)
54 sgn3da.2 . . . . . . . . . . . 12  |-  ( (sgn
`  A )  =  1  ->  ( ps  <->  th ) )
5554eqcoms 2453 . . . . . . . . . . 11  |-  ( 1  =  (sgn `  A
)  ->  ( ps  <->  th ) )
56553ad2ant3 1018 . . . . . . . . . 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 1197 . . . . . . . 8  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
1  =  (sgn `  A )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ps ) ) )
5916, 18, 40, 58ifbothda 3957 . . . . . . 7  |-  ( ph  ->  ( if ( A  <  0 ,  -u
1 ,  1 )  =  (sgn `  A
)  ->  ( (
( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) )  <->  ps )
) )
6059imp 429 . . . . . 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 435 . . . 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 435 . . . 4  |-  ( A  =  0  ->  ( ph  ->  ch ) )
6665adantl 466 . . 3  |-  ( ( T.  /\  A  =  0 )  ->  ( ph  ->  ch ) )
6719ex 434 . . . . . . 7  |-  ( ph  ->  ( A  <  0  ->  ta ) )
6867adantr 465 . . . . . 6  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( A  <  0  ->  ta ) )
69 simp1 995 . . . . . . . 8  |-  ( (
ph  /\  -.  A  =  0  /\  -.  A  <  0 )  ->  ph )
70 df-ne 2638 . . . . . . . . . . . 12  |-  ( A  =/=  0  <->  -.  A  =  0 )
71 0xr 9638 . . . . . . . . . . . . 13  |-  0  e.  RR*
72 xrlttri2 11352 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  0  e.  RR* )  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  < 
A ) ) )
731, 71, 72sylancl 662 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  <  A ) ) )
7470, 73syl5bbr 259 . . . . . . . . . . 11  |-  ( ph  ->  ( -.  A  =  0  <->  ( A  <  0  \/  0  < 
A ) ) )
7574biimpa 484 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( A  <  0  \/  0  <  A ) )
7675ord 377 . . . . . . . . 9  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( -.  A  <  0  ->  0  <  A ) )
77763impia 1192 . . . . . . . 8  |-  ( (
ph  /\  -.  A  =  0  /\  -.  A  <  0 )  -> 
0  <  A )
78 sgn3da.5 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  th )
7969, 77, 78syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  -.  A  =  0  /\  -.  A  <  0 )  ->  th )
80793expia 1197 . . . . . 6  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( -.  A  <  0  ->  th )
)
8168, 80jca 532 . . . . 5  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
) )
8281expcom 435 . . . 4  |-  ( -.  A  =  0  -> 
( ph  ->  ( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) ) ) )
8382adantl 466 . . 3  |-  ( ( T.  /\  -.  A  =  0 )  -> 
( ph  ->  ( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) ) ) )
8412, 63, 66, 83ifbothda 3957 . 2  |-  ( T. 
->  ( ph  ->  ps ) )
8584trud 1390 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 972    = wceq 1381   T. wtru 1382    e. wcel 1802    =/= wne 2636   ifcif 3922   class class class wbr 4433   ` cfv 5574   0cc0 9490   1c1 9491   RR*cxr 9625    < clt 9626   -ucneg 9806  sgncsgn 12893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-i2m1 9558  ax-1ne0 9559  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-po 4786  df-so 4787  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-neg 9808  df-sgn 12894
This theorem is referenced by:  sgnmul  28347  sgnsub  28349  sgnnbi  28350  sgnpbi  28351  sgn0bi  28352  sgnsgn  28353
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