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Theorem sgn3da 29414
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 13145 . . . . . . . . 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 2437 . . . . . . 7  |-  ( ph  ->  ( 0  =  (sgn
`  A )  <->  0  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) ) )
54pm5.32i 642 . . . . . 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 2435 . . . . . . . 8  |-  ( 0  =  (sgn `  A
)  ->  ( ps  <->  ch ) )
87bicomd 205 . . . . . . 7  |-  ( 0  =  (sgn `  A
)  ->  ( ch  <->  ps ) )
98adantl 468 . . . . . 6  |-  ( (
ph  /\  0  =  (sgn `  A ) )  ->  ( ch  <->  ps )
)
105, 9sylbir 217 . . . . 5  |-  ( (
ph  /\  0  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )  -> 
( ch  <->  ps )
)
1110expcom 437 . . . 4  |-  ( 0  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u
1 ,  1 ) )  ->  ( ph  ->  ( ch  <->  ps )
) )
1211pm5.74d 251 . . 3  |-  ( 0  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u
1 ,  1 ) )  ->  ( ( ph  ->  ch )  <->  ( ph  ->  ps ) ) )
133eqeq2d 2437 . . . . . . 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 642 . . . . . 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 2427 . . . . . . . . 9  |-  ( -u
1  =  if ( A  <  0 , 
-u 1 ,  1 )  ->  ( -u 1  =  (sgn `  A )  <->  if ( A  <  0 ,  -u 1 ,  1 )  =  (sgn `  A ) ) )
1615imbi1d 319 . . . . . . . 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 2427 . . . . . . . . 9  |-  ( 1  =  if ( A  <  0 ,  -u
1 ,  1 )  ->  ( 1  =  (sgn `  A )  <->  if ( A  <  0 ,  -u 1 ,  1 )  =  (sgn `  A ) ) )
1817imbi1d 319 . . . . . . . 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 467 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  <  0 )  /\  ( A  <  0  ->  ta ) )  ->  ta )
21 simp2 1007 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  <  0 )  /\  ta  /\  A  <  0 )  ->  ta )
22213expia 1208 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  <  0 )  /\  ta )  ->  ( A  <  0  ->  ta )
)
2320, 22impbida 841 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  <  0 )  ->  (
( A  <  0  ->  ta )  <->  ta )
)
24 pm3.24 891 . . . . . . . . . . . . . . . . 17  |-  -.  ( A  <  0  /\  -.  A  <  0 )
2524pm2.21i 135 . . . . . . . . . . . . . . . 16  |-  ( ( A  <  0  /\ 
-.  A  <  0
)  ->  th )
2625adantl 468 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( A  <  0  /\  -.  A  <  0 ) )  ->  th )
2726expr 619 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  <  0 )  ->  ( -.  A  <  0  ->  th ) )
28 tbtru 1448 . . . . . . . . . . . . . 14  |-  ( ( -.  A  <  0  ->  th )  <->  ( ( -.  A  <  0  ->  th )  <-> T.  )
)
2927, 28sylib 200 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  <  0 )  ->  (
( -.  A  <  0  ->  th )  <-> T.  ) )
3023, 29anbi12d 716 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  0 )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ( ta  /\ T.  ) ) )
31 ancom 452 . . . . . . . . . . . . 13  |-  ( ( ta  /\ T.  )  <-> 
( T.  /\  ta ) )
32 truan 1455 . . . . . . . . . . . . 13  |-  ( ( T.  /\  ta )  <->  ta )
3331, 32bitri 253 . . . . . . . . . . . 12  |-  ( ( ta  /\ T.  )  <->  ta )
3430, 33syl6bb 265 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  0 )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ta ) )
35343adant3 1026 . . . . . . . . . 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 2435 . . . . . . . . . . 11  |-  ( -u
1  =  (sgn `  A )  ->  ( ps 
<->  ta ) )
38373ad2ant3 1029 . . . . . . . . . 10  |-  ( (
ph  /\  A  <  0  /\  -u 1  =  (sgn
`  A ) )  ->  ( ps  <->  ta )
)
3935, 38bitr4d 260 . . . . . . . . 9  |-  ( (
ph  /\  A  <  0  /\  -u 1  =  (sgn
`  A ) )  ->  ( ( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) )  <->  ps )
)
40393expia 1208 . . . . . . . 8  |-  ( (
ph  /\  A  <  0 )  ->  ( -u 1  =  (sgn `  A )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ps ) ) )
41193adant2 1025 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  A  <  0  /\  A  <  0 )  ->  ta )
42413expia 1208 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  A  <  0 )  ->  ( A  <  0  ->  ta ) )
43 tbtru 1448 . . . . . . . . . . . . . 14  |-  ( ( A  <  0  ->  ta )  <->  ( ( A  <  0  ->  ta ) 
<-> T.  ) )
4442, 43sylib 200 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
( A  <  0  ->  ta )  <-> T.  )
)
45 pm3.35 590 . . . . . . . . . . . . . . 15  |-  ( ( -.  A  <  0  /\  ( -.  A  <  0  ->  th )
)  ->  th )
4645adantll 719 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  A  <  0 )  /\  ( -.  A  <  0  ->  th ) )  ->  th )
47 simp2 1007 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  -.  A  <  0 )  /\  th 
/\  -.  A  <  0 )  ->  th )
48473expia 1208 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  A  <  0 )  /\  th )  ->  ( -.  A  <  0  ->  th )
)
4946, 48impbida 841 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
( -.  A  <  0  ->  th )  <->  th ) )
5044, 49anbi12d 716 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ( T.  /\  th ) ) )
51 truan 1455 . . . . . . . . . . . 12  |-  ( ( T.  /\  th )  <->  th )
5250, 51syl6bb 265 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  th ) )
53523adant3 1026 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  <  0  /\  1  =  (sgn `  A )
)  ->  ( (
( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) )  <->  th )
)
54 sgn3da.2 . . . . . . . . . . . 12  |-  ( (sgn
`  A )  =  1  ->  ( ps  <->  th ) )
5554eqcoms 2435 . . . . . . . . . . 11  |-  ( 1  =  (sgn `  A
)  ->  ( ps  <->  th ) )
56553ad2ant3 1029 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  <  0  /\  1  =  (sgn `  A )
)  ->  ( ps  <->  th ) )
5753, 56bitr4d 260 . . . . . . . . 9  |-  ( (
ph  /\  -.  A  <  0  /\  1  =  (sgn `  A )
)  ->  ( (
( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) )  <->  ps )
)
58573expia 1208 . . . . . . . 8  |-  ( (
ph  /\  -.  A  <  0 )  ->  (
1  =  (sgn `  A )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ps ) ) )
5916, 18, 40, 58ifbothda 3945 . . . . . . 7  |-  ( ph  ->  ( if ( A  <  0 ,  -u
1 ,  1 )  =  (sgn `  A
)  ->  ( (
( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) )  <->  ps )
) )
6059imp 431 . . . . . 6  |-  ( (
ph  /\  if ( A  <  0 ,  -u
1 ,  1 )  =  (sgn `  A
) )  ->  (
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
)  <->  ps ) )
6114, 60sylbir 217 . . . . 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 437 . . . 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 251 . . 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 437 . . . 4  |-  ( A  =  0  ->  ( ph  ->  ch ) )
6665adantl 468 . . 3  |-  ( ( T.  /\  A  =  0 )  ->  ( ph  ->  ch ) )
6719ex 436 . . . . . . 7  |-  ( ph  ->  ( A  <  0  ->  ta ) )
6867adantr 467 . . . . . 6  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( A  <  0  ->  ta ) )
69 simp1 1006 . . . . . . . 8  |-  ( (
ph  /\  -.  A  =  0  /\  -.  A  <  0 )  ->  ph )
70 df-ne 2621 . . . . . . . . . . . 12  |-  ( A  =/=  0  <->  -.  A  =  0 )
71 0xr 9689 . . . . . . . . . . . . 13  |-  0  e.  RR*
72 xrlttri2 11443 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  0  e.  RR* )  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  < 
A ) ) )
731, 71, 72sylancl 667 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  =/=  0  <->  ( A  <  0  \/  0  <  A ) ) )
7470, 73syl5bbr 263 . . . . . . . . . . 11  |-  ( ph  ->  ( -.  A  =  0  <->  ( A  <  0  \/  0  < 
A ) ) )
7574biimpa 487 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( A  <  0  \/  0  <  A ) )
7675ord 379 . . . . . . . . 9  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( -.  A  <  0  ->  0  <  A ) )
77763impia 1203 . . . . . . . 8  |-  ( (
ph  /\  -.  A  =  0  /\  -.  A  <  0 )  -> 
0  <  A )
78 sgn3da.5 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  th )
7969, 77, 78syl2anc 666 . . . . . . 7  |-  ( (
ph  /\  -.  A  =  0  /\  -.  A  <  0 )  ->  th )
80793expia 1208 . . . . . 6  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( -.  A  <  0  ->  th )
)
8168, 80jca 535 . . . . 5  |-  ( (
ph  /\  -.  A  =  0 )  -> 
( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th )
) )
8281expcom 437 . . . 4  |-  ( -.  A  =  0  -> 
( ph  ->  ( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) ) ) )
8382adantl 468 . . 3  |-  ( ( T.  /\  -.  A  =  0 )  -> 
( ph  ->  ( ( A  <  0  ->  ta )  /\  ( -.  A  <  0  ->  th ) ) ) )
8412, 63, 66, 83ifbothda 3945 . 2  |-  ( T. 
->  ( ph  ->  ps ) )
8584trud 1447 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 983    = wceq 1438   T. wtru 1439    e. wcel 1869    =/= wne 2619   ifcif 3910   class class class wbr 4421   ` cfv 5599   0cc0 9541   1c1 9542   RR*cxr 9676    < clt 9677   -ucneg 9863  sgncsgn 13143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-i2m1 9609  ax-1ne0 9610  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-po 4772  df-so 4773  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-neg 9865  df-sgn 13144
This theorem is referenced by:  sgnmul  29415  sgnsub  29417  sgnnbi  29418  sgnpbi  29419  sgn0bi  29420  sgnsgn  29421
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