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Theorem signswch 27098
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 3980 . . . . . . 7  |-  { -u
1 ,  1 }  =  ( { -u
1 }  u.  {
1 } )
2 snsstp1 4124 . . . . . . . 8  |-  { -u
1 }  C_  { -u
1 ,  0 ,  1 }
3 snsstp3 4126 . . . . . . . 8  |-  { 1 }  C_  { -u 1 ,  0 ,  1 }
42, 3unssi 3631 . . . . . . 7  |-  ( {
-u 1 }  u.  { 1 } )  C_  {
-u 1 ,  0 ,  1 }
51, 4eqsstri 3486 . . . . . 6  |-  { -u
1 ,  1 } 
C_  { -u 1 ,  0 ,  1 }
65sseli 3452 . . . . 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 27089 . . . 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 2725 . 2  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( ( X  .+^  Y )  =/=  X  <->  if ( Y  =  0 ,  X ,  Y )  =/=  X ) )
12 neeq1 2729 . . . 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 2729 . . . 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 2653 . . . . 5  |-  -.  X  =/=  X
1716a1i 11 . . . 4  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  -.  X  =/=  X )
18 0re 9489 . . . . . 6  |-  0  e.  RR
1918ltnri 9586 . . . . 5  |-  -.  0  <  0
20 simpr 461 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  Y  =  0 )
2120oveq2d 6208 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  ( X  x.  0 ) )
22 neg1cn 10528 . . . . . . . . . 10  |-  -u 1  e.  CC
23 ax-1cn 9443 . . . . . . . . . 10  |-  1  e.  CC
24 prssi 4129 . . . . . . . . . 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 3454 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  X  e.  CC )
2827mul01d 9671 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  x.  0 )  =  0 )
2921, 28eqtrd 2492 . . . . . 6  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  0 )
3029breq1d 4402 . . . . 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 4072 . . . . . . . . 9  |-  { -u
1 ,  0 ,  1 }  =  { -u 1 ,  1 ,  0 }
3533, 34syl6eleq 2549 . . . . . . . 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 )
3736neneqad 2652 . . . . . . . 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 4100 . . . . . . . 8  |-  ( Y  e.  ( { -u
1 ,  1 ,  0 }  \  {
0 } )  <->  ( Y  e.  { -u 1 ,  1 ,  0 }  /\  Y  =/=  0
) )
40 neg1ne0 10530 . . . . . . . . . 10  |-  -u 1  =/=  0
41 ax-1ne0 9454 . . . . . . . . . 10  |-  1  =/=  0
42 diftpsn3 4112 . . . . . . . . . 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 2529 . . . . . . . 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 2653 . . . . . . . . . . 11  |-  -.  -u 1  =/=  -u 1
49 0le1 9966 . . . . . . . . . . . . 13  |-  0  <_  1
50 1re 9488 . . . . . . . . . . . . . 14  |-  1  e.  RR
5118, 50lenlti 9597 . . . . . . . . . . . . 13  |-  ( 0  <_  1  <->  -.  1  <  0 )
5249, 51mpbi 208 . . . . . . . . . . . 12  |-  -.  1  <  0
53 neg1mulneg1e1 10642 . . . . . . . . . . . . 13  |-  ( -u
1  x.  -u 1
)  =  1
5453breq1i 4399 . . . . . . . . . . . 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 2729 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  ( Y  =/=  -u 1  <->  -u 1  =/=  -u 1 ) )
58 oveq2 6200 . . . . . . . . . . . 12  |-  ( Y  =  -u 1  ->  ( -u 1  x.  Y )  =  ( -u 1  x.  -u 1 ) )
5958breq1d 4402 . . . . . . . . . . 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 10531 . . . . . . . . . . . . 13  |-  -u 1  <  0
64 0lt1 9965 . . . . . . . . . . . . 13  |-  0  <  1
65 neg1rr 10529 . . . . . . . . . . . . . 14  |-  -u 1  e.  RR
6665, 18, 50lttri 9603 . . . . . . . . . . . . 13  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
6763, 64, 66mp2an 672 . . . . . . . . . . . 12  |-  -u 1  <  1
6865, 50ltnei 9601 . . . . . . . . . . . 12  |-  ( -u
1  <  1  ->  1  =/=  -u 1 )
6967, 68ax-mp 5 . . . . . . . . . . 11  |-  1  =/=  -u 1
7022mulid1i 9491 . . . . . . . . . . . 12  |-  ( -u
1  x.  1 )  =  -u 1
7170, 63eqbrtri 4411 . . . . . . . . . . 11  |-  ( -u
1  x.  1 )  <  0
7269, 712th 239 . . . . . . . . . 10  |-  ( 1  =/=  -u 1  <->  ( -u 1  x.  1 )  <  0
)
73 neeq1 2729 . . . . . . . . . . 11  |-  ( Y  =  1  ->  ( Y  =/=  -u 1  <->  1  =/=  -u 1 ) )
74 oveq2 6200 . . . . . . . . . . . 12  |-  ( Y  =  1  ->  ( -u 1  x.  Y )  =  ( -u 1  x.  1 ) )
7574breq1d 4402 . . . . . . . . . . 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 3997 . . . . . . . 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 2731 . . . . . . . 8  |-  ( X  =  -u 1  ->  ( Y  =/=  X  <->  Y  =/=  -u 1 ) )
83 oveq1 6199 . . . . . . . . 9  |-  ( X  =  -u 1  ->  ( X  x.  Y )  =  ( -u 1  x.  Y ) )
8483breq1d 4402 . . . . . . . 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 2718 . . . . . . . . . . 11  |-  -u 1  =/=  1
9122, 23mulcomi 9495 . . . . . . . . . . . . 13  |-  ( -u
1  x.  1 )  =  ( 1  x.  -u 1 )
9291breq1i 4399 . . . . . . . . . . . 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 2729 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  ( Y  =/=  1  <->  -u 1  =/=  1 ) )
96 oveq2 6200 . . . . . . . . . . . 12  |-  ( Y  =  -u 1  ->  (
1  x.  Y )  =  ( 1  x.  -u 1 ) )
9796breq1d 4402 . . . . . . . . . . 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 2653 . . . . . . . . . . 11  |-  -.  1  =/=  1
10223mulid1i 9491 . . . . . . . . . . . . 13  |-  ( 1  x.  1 )  =  1
103102breq1i 4399 . . . . . . . . . . . 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 2729 . . . . . . . . . . 11  |-  ( Y  =  1  ->  ( Y  =/=  1  <->  1  =/=  1 ) )
107 oveq2 6200 . . . . . . . . . . . 12  |-  ( Y  =  1  ->  (
1  x.  Y )  =  ( 1  x.  1 ) )
108107breq1d 4402 . . . . . . . . . . 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 2731 . . . . . . . 8  |-  ( X  =  1  ->  ( Y  =/=  X  <->  Y  =/=  1 ) )
115 oveq1 6199 . . . . . . . . 9  |-  ( X  =  1  ->  ( X  x.  Y )  =  ( 1  x.  Y ) )
116115breq1d 4402 . . . . . . . 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 3997 . . . . 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 3924 . 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 1370    e. wcel 1758    =/= wne 2644    \ cdif 3425    u. cun 3426    C_ wss 3428   ifcif 3891   {csn 3977   {cpr 3979   {ctp 3981   <.cop 3983   class class class wbr 4392   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   CCcc 9383   0cc0 9385   1c1 9386    x. cmul 9390    < clt 9521    <_ cle 9522   -ucneg 9699   ndxcnx 14275   Basecbs 14278   +g cplusg 14342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701
This theorem is referenced by:  signsvfn  27119
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