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Theorem signswch 28155
Description: The zero-skipping operation changes value when the operands change signs (Contributed by Thierry Arnoux, 9-Oct-2018.)
Hypotheses
Ref Expression
signsw.p  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
signsw.w  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
Assertion
Ref Expression
signswch  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( ( X  .+^  Y )  =/=  X  <->  ( X  x.  Y )  <  0
) )
Distinct variable groups:    a, b, X    Y, a, b
Allowed substitution hints:    .+^ ( a, b)    W( a, b)

Proof of Theorem signswch
StepHypRef Expression
1 df-pr 4030 . . . . . . 7  |-  { -u
1 ,  1 }  =  ( { -u
1 }  u.  {
1 } )
2 snsstp1 4178 . . . . . . . 8  |-  { -u
1 }  C_  { -u
1 ,  0 ,  1 }
3 snsstp3 4180 . . . . . . . 8  |-  { 1 }  C_  { -u 1 ,  0 ,  1 }
42, 3unssi 3679 . . . . . . 7  |-  ( {
-u 1 }  u.  { 1 } )  C_  {
-u 1 ,  0 ,  1 }
51, 4eqsstri 3534 . . . . . 6  |-  { -u
1 ,  1 } 
C_  { -u 1 ,  0 ,  1 }
65sseli 3500 . . . . 5  |-  ( X  e.  { -u 1 ,  1 }  ->  X  e.  { -u 1 ,  0 ,  1 } )
76adantr 465 . . . 4  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  ->  X  e.  { -u 1 ,  0 ,  1 } )
8 signsw.p . . . . 5  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
98signspval 28146 . . . 4  |-  ( ( X  e.  { -u
1 ,  0 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  ->  ( X  .+^ 
Y )  =  if ( Y  =  0 ,  X ,  Y
) )
107, 9sylancom 667 . . 3  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( X  .+^  Y )  =  if ( Y  =  0 ,  X ,  Y ) )
1110neeq1d 2744 . 2  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( ( X  .+^  Y )  =/=  X  <->  if ( Y  =  0 ,  X ,  Y )  =/=  X ) )
12 neeq1 2748 . . . 4  |-  ( X  =  if ( Y  =  0 ,  X ,  Y )  ->  ( X  =/=  X  <->  if ( Y  =  0 ,  X ,  Y )  =/=  X ) )
1312bibi1d 319 . . 3  |-  ( X  =  if ( Y  =  0 ,  X ,  Y )  ->  (
( X  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( if ( Y  =  0 ,  X ,  Y )  =/=  X  <->  ( X  x.  Y )  <  0
) ) )
14 neeq1 2748 . . . 4  |-  ( Y  =  if ( Y  =  0 ,  X ,  Y )  ->  ( Y  =/=  X  <->  if ( Y  =  0 ,  X ,  Y )  =/=  X ) )
1514bibi1d 319 . . 3  |-  ( Y  =  if ( Y  =  0 ,  X ,  Y )  ->  (
( Y  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( if ( Y  =  0 ,  X ,  Y )  =/=  X  <->  ( X  x.  Y )  <  0
) ) )
16 neirr 2671 . . . . 5  |-  -.  X  =/=  X
1716a1i 11 . . . 4  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  -.  X  =/=  X )
18 0re 9592 . . . . . 6  |-  0  e.  RR
1918ltnri 9689 . . . . 5  |-  -.  0  <  0
20 simpr 461 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  Y  =  0 )
2120oveq2d 6298 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  ( X  x.  0 ) )
22 neg1cn 10635 . . . . . . . . . 10  |-  -u 1  e.  CC
23 ax-1cn 9546 . . . . . . . . . 10  |-  1  e.  CC
24 prssi 4183 . . . . . . . . . 10  |-  ( (
-u 1  e.  CC  /\  1  e.  CC )  ->  { -u 1 ,  1 }  C_  CC )
2522, 23, 24mp2an 672 . . . . . . . . 9  |-  { -u
1 ,  1 } 
C_  CC
26 simpll 753 . . . . . . . . 9  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  X  e.  { -u 1 ,  1 } )
2725, 26sseldi 3502 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  X  e.  CC )
2827mul01d 9774 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  x.  0 )  =  0 )
2921, 28eqtrd 2508 . . . . . 6  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  0 )
3029breq1d 4457 . . . . 5  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( ( X  x.  Y )  <  0  <->  0  <  0
) )
3119, 30mtbiri 303 . . . 4  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  -.  ( X  x.  Y )  <  0
)
3217, 312falsed 351 . . 3  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  =/= 
X  <->  ( X  x.  Y )  <  0
) )
33 simplr 754 . . . . . . . . 9  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  e.  {
-u 1 ,  0 ,  1 } )
34 tpcomb 4124 . . . . . . . . 9  |-  { -u
1 ,  0 ,  1 }  =  { -u 1 ,  1 ,  0 }
3533, 34syl6eleq 2565 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  e.  {
-u 1 ,  1 ,  0 } )
36 simpr 461 . . . . . . . . 9  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  -.  Y  =  0 )
3736neqned 2670 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  =/=  0 )
3835, 37jca 532 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  ( Y  e.  { -u 1 ,  1 ,  0 }  /\  Y  =/=  0
) )
39 eldifsn 4152 . . . . . . . 8  |-  ( Y  e.  ( { -u
1 ,  1 ,  0 }  \  {
0 } )  <->  ( Y  e.  { -u 1 ,  1 ,  0 }  /\  Y  =/=  0
) )
40 neg1ne0 10637 . . . . . . . . . 10  |-  -u 1  =/=  0
41 ax-1ne0 9557 . . . . . . . . . 10  |-  1  =/=  0
42 diftpsn3 4165 . . . . . . . . . 10  |-  ( (
-u 1  =/=  0  /\  1  =/=  0
)  ->  ( { -u 1 ,  1 ,  0 }  \  {
0 } )  =  { -u 1 ,  1 } )
4340, 41, 42mp2an 672 . . . . . . . . 9  |-  ( {
-u 1 ,  1 ,  0 }  \  { 0 } )  =  { -u 1 ,  1 }
4443eleq2i 2545 . . . . . . . 8  |-  ( Y  e.  ( { -u
1 ,  1 ,  0 }  \  {
0 } )  <->  Y  e.  {
-u 1 ,  1 } )
4539, 44bitr3i 251 . . . . . . 7  |-  ( ( Y  e.  { -u
1 ,  1 ,  0 }  /\  Y  =/=  0 )  <->  Y  e.  {
-u 1 ,  1 } )
4638, 45sylib 196 . . . . . 6  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  e.  {
-u 1 ,  1 } )
4746adantr 465 . . . . 5  |-  ( ( ( ( X  e. 
{ -u 1 ,  1 }  /\  Y  e. 
{ -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0 )  /\  X  =  -u 1 )  ->  Y  e.  { -u 1 ,  1 } )
48 neirr 2671 . . . . . . . . . . 11  |-  -.  -u 1  =/=  -u 1
49 0le1 10072 . . . . . . . . . . . . 13  |-  0  <_  1
50 1re 9591 . . . . . . . . . . . . . 14  |-  1  e.  RR
5118, 50lenlti 9700 . . . . . . . . . . . . 13  |-  ( 0  <_  1  <->  -.  1  <  0 )
5249, 51mpbi 208 . . . . . . . . . . . 12  |-  -.  1  <  0
53 neg1mulneg1e1 10749 . . . . . . . . . . . . 13  |-  ( -u
1  x.  -u 1
)  =  1
5453breq1i 4454 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  -u 1
)  <  0  <->  1  <  0 )
5552, 54mtbir 299 . . . . . . . . . . 11  |-  -.  ( -u 1  x.  -u 1
)  <  0
5648, 552false 350 . . . . . . . . . 10  |-  ( -u
1  =/=  -u 1  <->  (
-u 1  x.  -u 1
)  <  0 )
57 neeq1 2748 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  ( Y  =/=  -u 1  <->  -u 1  =/=  -u 1 ) )
58 oveq2 6290 . . . . . . . . . . . 12  |-  ( Y  =  -u 1  ->  ( -u 1  x.  Y )  =  ( -u 1  x.  -u 1 ) )
5958breq1d 4457 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  (
( -u 1  x.  Y
)  <  0  <->  ( -u 1  x.  -u 1 )  <  0 ) )
6057, 59bibi12d 321 . . . . . . . . . 10  |-  ( Y  =  -u 1  ->  (
( Y  =/=  -u 1  <->  (
-u 1  x.  Y
)  <  0 )  <-> 
( -u 1  =/=  -u 1  <->  (
-u 1  x.  -u 1
)  <  0 ) ) )
6156, 60mpbiri 233 . . . . . . . . 9  |-  ( Y  =  -u 1  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
6261adantl 466 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  -u
1 )  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
63 neg1lt0 10638 . . . . . . . . . . . . 13  |-  -u 1  <  0
64 0lt1 10071 . . . . . . . . . . . . 13  |-  0  <  1
65 neg1rr 10636 . . . . . . . . . . . . . 14  |-  -u 1  e.  RR
6665, 18, 50lttri 9706 . . . . . . . . . . . . 13  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
6763, 64, 66mp2an 672 . . . . . . . . . . . 12  |-  -u 1  <  1
6865, 50ltnei 9704 . . . . . . . . . . . 12  |-  ( -u
1  <  1  ->  1  =/=  -u 1 )
6967, 68ax-mp 5 . . . . . . . . . . 11  |-  1  =/=  -u 1
7022mulid1i 9594 . . . . . . . . . . . 12  |-  ( -u
1  x.  1 )  =  -u 1
7170, 63eqbrtri 4466 . . . . . . . . . . 11  |-  ( -u
1  x.  1 )  <  0
7269, 712th 239 . . . . . . . . . 10  |-  ( 1  =/=  -u 1  <->  ( -u 1  x.  1 )  <  0
)
73 neeq1 2748 . . . . . . . . . . 11  |-  ( Y  =  1  ->  ( Y  =/=  -u 1  <->  1  =/=  -u 1 ) )
74 oveq2 6290 . . . . . . . . . . . 12  |-  ( Y  =  1  ->  ( -u 1  x.  Y )  =  ( -u 1  x.  1 ) )
7574breq1d 4457 . . . . . . . . . . 11  |-  ( Y  =  1  ->  (
( -u 1  x.  Y
)  <  0  <->  ( -u 1  x.  1 )  <  0
) )
7673, 75bibi12d 321 . . . . . . . . . 10  |-  ( Y  =  1  ->  (
( Y  =/=  -u 1  <->  (
-u 1  x.  Y
)  <  0 )  <-> 
( 1  =/=  -u 1  <->  (
-u 1  x.  1 )  <  0 ) ) )
7772, 76mpbiri 233 . . . . . . . . 9  |-  ( Y  =  1  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
7877adantl 466 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  1 )  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
79 elpri 4047 . . . . . . . 8  |-  ( Y  e.  { -u 1 ,  1 }  ->  ( Y  =  -u 1  \/  Y  =  1
) )
8062, 78, 79mpjaodan 784 . . . . . . 7  |-  ( Y  e.  { -u 1 ,  1 }  ->  ( Y  =/=  -u 1  <->  (
-u 1  x.  Y
)  <  0 ) )
8180adantr 465 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  -u
1 )  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
82 neeq2 2750 . . . . . . . 8  |-  ( X  =  -u 1  ->  ( Y  =/=  X  <->  Y  =/=  -u 1 ) )
83 oveq1 6289 . . . . . . . . 9  |-  ( X  =  -u 1  ->  ( X  x.  Y )  =  ( -u 1  x.  Y ) )
8483breq1d 4457 . . . . . . . 8  |-  ( X  =  -u 1  ->  (
( X  x.  Y
)  <  0  <->  ( -u 1  x.  Y )  <  0
) )
8582, 84bibi12d 321 . . . . . . 7  |-  ( X  =  -u 1  ->  (
( Y  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) ) )
8685adantl 466 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  -u
1 )  ->  (
( Y  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) ) )
8781, 86mpbird 232 . . . . 5  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  -u
1 )  ->  ( Y  =/=  X  <->  ( X  x.  Y )  <  0
) )
8847, 87sylancom 667 . . . 4  |-  ( ( ( ( X  e. 
{ -u 1 ,  1 }  /\  Y  e. 
{ -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0 )  /\  X  =  -u 1 )  -> 
( Y  =/=  X  <->  ( X  x.  Y )  <  0 ) )
8946adantr 465 . . . . 5  |-  ( ( ( ( X  e. 
{ -u 1 ,  1 }  /\  Y  e. 
{ -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0 )  /\  X  =  1 )  ->  Y  e.  { -u 1 ,  1 } )
9069necomi 2737 . . . . . . . . . . 11  |-  -u 1  =/=  1
9122, 23mulcomi 9598 . . . . . . . . . . . . 13  |-  ( -u
1  x.  1 )  =  ( 1  x.  -u 1 )
9291breq1i 4454 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  1 )  <  0  <->  (
1  x.  -u 1
)  <  0 )
9371, 92mpbi 208 . . . . . . . . . . 11  |-  ( 1  x.  -u 1 )  <  0
9490, 932th 239 . . . . . . . . . 10  |-  ( -u
1  =/=  1  <->  (
1  x.  -u 1
)  <  0 )
95 neeq1 2748 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  ( Y  =/=  1  <->  -u 1  =/=  1 ) )
96 oveq2 6290 . . . . . . . . . . . 12  |-  ( Y  =  -u 1  ->  (
1  x.  Y )  =  ( 1  x.  -u 1 ) )
9796breq1d 4457 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  (
( 1  x.  Y
)  <  0  <->  ( 1  x.  -u 1 )  <  0 ) )
9895, 97bibi12d 321 . . . . . . . . . 10  |-  ( Y  =  -u 1  ->  (
( Y  =/=  1  <->  ( 1  x.  Y )  <  0 )  <->  ( -u 1  =/=  1  <->  ( 1  x.  -u 1 )  <  0 ) ) )
9994, 98mpbiri 233 . . . . . . . . 9  |-  ( Y  =  -u 1  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0 ) )
10099adantl 466 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  -u
1 )  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0 ) )
101 neirr 2671 . . . . . . . . . . 11  |-  -.  1  =/=  1
10223mulid1i 9594 . . . . . . . . . . . . 13  |-  ( 1  x.  1 )  =  1
103102breq1i 4454 . . . . . . . . . . . 12  |-  ( ( 1  x.  1 )  <  0  <->  1  <  0 )
10452, 103mtbir 299 . . . . . . . . . . 11  |-  -.  (
1  x.  1 )  <  0
105101, 1042false 350 . . . . . . . . . 10  |-  ( 1  =/=  1  <->  ( 1  x.  1 )  <  0 )
106 neeq1 2748 . . . . . . . . . . 11  |-  ( Y  =  1  ->  ( Y  =/=  1  <->  1  =/=  1 ) )
107 oveq2 6290 . . . . . . . . . . . 12  |-  ( Y  =  1  ->  (
1  x.  Y )  =  ( 1  x.  1 ) )
108107breq1d 4457 . . . . . . . . . . 11  |-  ( Y  =  1  ->  (
( 1  x.  Y
)  <  0  <->  ( 1  x.  1 )  <  0 ) )
109106, 108bibi12d 321 . . . . . . . . . 10  |-  ( Y  =  1  ->  (
( Y  =/=  1  <->  ( 1  x.  Y )  <  0 )  <->  ( 1  =/=  1  <->  ( 1  x.  1 )  <  0 ) ) )
110105, 109mpbiri 233 . . . . . . . . 9  |-  ( Y  =  1  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0 ) )
111110adantl 466 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  1 )  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) )
112100, 111, 79mpjaodan 784 . . . . . . 7  |-  ( Y  e.  { -u 1 ,  1 }  ->  ( Y  =/=  1  <->  (
1  x.  Y )  <  0 ) )
113112adantr 465 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  1 )  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) )
114 neeq2 2750 . . . . . . . 8  |-  ( X  =  1  ->  ( Y  =/=  X  <->  Y  =/=  1 ) )
115 oveq1 6289 . . . . . . . . 9  |-  ( X  =  1  ->  ( X  x.  Y )  =  ( 1  x.  Y ) )
116115breq1d 4457 . . . . . . . 8  |-  ( X  =  1  ->  (
( X  x.  Y
)  <  0  <->  ( 1  x.  Y )  <  0 ) )
117114, 116bibi12d 321 . . . . . . 7  |-  ( X  =  1  ->  (
( Y  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) ) )
118117adantl 466 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  1 )  ->  ( ( Y  =/=  X  <->  ( X  x.  Y )  <  0
)  <->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) ) )
119113, 118mpbird 232 . . . . 5  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  1 )  ->  ( Y  =/=  X  <->  ( X  x.  Y )  <  0
) )
12089, 119sylancom 667 . . . 4  |-  ( ( ( ( X  e. 
{ -u 1 ,  1 }  /\  Y  e. 
{ -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0 )  /\  X  =  1 )  -> 
( Y  =/=  X  <->  ( X  x.  Y )  <  0 ) )
121 elpri 4047 . . . . 5  |-  ( X  e.  { -u 1 ,  1 }  ->  ( X  =  -u 1  \/  X  =  1
) )
122121ad2antrr 725 . . . 4  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  ( X  =  -u 1  \/  X  =  1 ) )
12388, 120, 122mpjaodan 784 . . 3  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  ( Y  =/=  X  <->  ( X  x.  Y )  <  0
) )
12413, 15, 32, 123ifbothda 3974 . 2  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( if ( Y  =  0 ,  X ,  Y )  =/=  X  <->  ( X  x.  Y )  <  0 ) )
12511, 124bitrd 253 1  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( ( X  .+^  Y )  =/=  X  <->  ( X  x.  Y )  <  0
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473    u. cun 3474    C_ wss 3476   ifcif 3939   {csn 4027   {cpr 4029   {ctp 4031   <.cop 4033   class class class wbr 4447   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   CCcc 9486   0cc0 9488   1c1 9489    x. cmul 9493    < clt 9624    <_ cle 9625   -ucneg 9802   ndxcnx 14480   Basecbs 14483   +g cplusg 14548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804
This theorem is referenced by:  signsvfn  28176
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