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Theorem signswch 28496
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 4017 . . . . . 6  |-  { -u
1 ,  1 }  =  ( { -u
1 }  u.  {
1 } )
2 snsstp1 4166 . . . . . . 7  |-  { -u
1 }  C_  { -u
1 ,  0 ,  1 }
3 snsstp3 4168 . . . . . . 7  |-  { 1 }  C_  { -u 1 ,  0 ,  1 }
42, 3unssi 3664 . . . . . 6  |-  ( {
-u 1 }  u.  { 1 } )  C_  {
-u 1 ,  0 ,  1 }
51, 4eqsstri 3519 . . . . 5  |-  { -u
1 ,  1 } 
C_  { -u 1 ,  0 ,  1 }
65sseli 3485 . . . 4  |-  ( X  e.  { -u 1 ,  1 }  ->  X  e.  { -u 1 ,  0 ,  1 } )
7 signsw.p . . . . 5  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
87signspval 28487 . . . 4  |-  ( ( X  e.  { -u
1 ,  0 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  ->  ( X  .+^ 
Y )  =  if ( Y  =  0 ,  X ,  Y
) )
96, 8sylan 471 . . 3  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( X  .+^  Y )  =  if ( Y  =  0 ,  X ,  Y ) )
109neeq1d 2720 . 2  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( ( X  .+^  Y )  =/=  X  <->  if ( Y  =  0 ,  X ,  Y )  =/=  X ) )
11 neeq1 2724 . . . 4  |-  ( X  =  if ( Y  =  0 ,  X ,  Y )  ->  ( X  =/=  X  <->  if ( Y  =  0 ,  X ,  Y )  =/=  X ) )
1211bibi1d 319 . . 3  |-  ( X  =  if ( Y  =  0 ,  X ,  Y )  ->  (
( X  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( if ( Y  =  0 ,  X ,  Y )  =/=  X  <->  ( X  x.  Y )  <  0
) ) )
13 neeq1 2724 . . . 4  |-  ( Y  =  if ( Y  =  0 ,  X ,  Y )  ->  ( Y  =/=  X  <->  if ( Y  =  0 ,  X ,  Y )  =/=  X ) )
1413bibi1d 319 . . 3  |-  ( Y  =  if ( Y  =  0 ,  X ,  Y )  ->  (
( Y  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( if ( Y  =  0 ,  X ,  Y )  =/=  X  <->  ( X  x.  Y )  <  0
) ) )
15 neirr 2647 . . . . 5  |-  -.  X  =/=  X
1615a1i 11 . . . 4  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  -.  X  =/=  X )
17 0re 9599 . . . . . 6  |-  0  e.  RR
1817ltnri 9696 . . . . 5  |-  -.  0  <  0
19 simpr 461 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  Y  =  0 )
2019oveq2d 6297 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  ( X  x.  0 ) )
21 neg1cn 10646 . . . . . . . . . 10  |-  -u 1  e.  CC
22 ax-1cn 9553 . . . . . . . . . 10  |-  1  e.  CC
23 prssi 4171 . . . . . . . . . 10  |-  ( (
-u 1  e.  CC  /\  1  e.  CC )  ->  { -u 1 ,  1 }  C_  CC )
2421, 22, 23mp2an 672 . . . . . . . . 9  |-  { -u
1 ,  1 } 
C_  CC
25 simpll 753 . . . . . . . . 9  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  X  e.  { -u 1 ,  1 } )
2624, 25sseldi 3487 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  X  e.  CC )
2726mul01d 9782 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  x.  0 )  =  0 )
2820, 27eqtrd 2484 . . . . . 6  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  0 )
2928breq1d 4447 . . . . 5  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( ( X  x.  Y )  <  0  <->  0  <  0
) )
3018, 29mtbiri 303 . . . 4  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  -.  ( X  x.  Y )  <  0
)
3116, 302falsed 351 . . 3  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  =/= 
X  <->  ( X  x.  Y )  <  0
) )
32 simplr 755 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  e.  {
-u 1 ,  0 ,  1 } )
33 tpcomb 4112 . . . . . . . 8  |-  { -u
1 ,  0 ,  1 }  =  { -u 1 ,  1 ,  0 }
3432, 33syl6eleq 2541 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  e.  {
-u 1 ,  1 ,  0 } )
35 simpr 461 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  -.  Y  =  0 )
3635neqned 2646 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  =/=  0 )
3734, 36jca 532 . . . . . 6  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  ( Y  e.  { -u 1 ,  1 ,  0 }  /\  Y  =/=  0
) )
38 eldifsn 4140 . . . . . . 7  |-  ( Y  e.  ( { -u
1 ,  1 ,  0 }  \  {
0 } )  <->  ( Y  e.  { -u 1 ,  1 ,  0 }  /\  Y  =/=  0
) )
39 neg1ne0 10648 . . . . . . . . 9  |-  -u 1  =/=  0
40 ax-1ne0 9564 . . . . . . . . 9  |-  1  =/=  0
41 diftpsn3 4153 . . . . . . . . 9  |-  ( (
-u 1  =/=  0  /\  1  =/=  0
)  ->  ( { -u 1 ,  1 ,  0 }  \  {
0 } )  =  { -u 1 ,  1 } )
4239, 40, 41mp2an 672 . . . . . . . 8  |-  ( {
-u 1 ,  1 ,  0 }  \  { 0 } )  =  { -u 1 ,  1 }
4342eleq2i 2521 . . . . . . 7  |-  ( Y  e.  ( { -u
1 ,  1 ,  0 }  \  {
0 } )  <->  Y  e.  {
-u 1 ,  1 } )
4438, 43bitr3i 251 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 ,  0 }  /\  Y  =/=  0 )  <->  Y  e.  {
-u 1 ,  1 } )
4537, 44sylib 196 . . . . 5  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  e.  {
-u 1 ,  1 } )
46 neirr 2647 . . . . . . . . . . 11  |-  -.  -u 1  =/=  -u 1
47 0le1 10083 . . . . . . . . . . . . 13  |-  0  <_  1
48 1re 9598 . . . . . . . . . . . . . 14  |-  1  e.  RR
4917, 48lenlti 9707 . . . . . . . . . . . . 13  |-  ( 0  <_  1  <->  -.  1  <  0 )
5047, 49mpbi 208 . . . . . . . . . . . 12  |-  -.  1  <  0
51 neg1mulneg1e1 10760 . . . . . . . . . . . . 13  |-  ( -u
1  x.  -u 1
)  =  1
5251breq1i 4444 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  -u 1
)  <  0  <->  1  <  0 )
5350, 52mtbir 299 . . . . . . . . . . 11  |-  -.  ( -u 1  x.  -u 1
)  <  0
5446, 532false 350 . . . . . . . . . 10  |-  ( -u
1  =/=  -u 1  <->  (
-u 1  x.  -u 1
)  <  0 )
55 neeq1 2724 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  ( Y  =/=  -u 1  <->  -u 1  =/=  -u 1 ) )
56 oveq2 6289 . . . . . . . . . . . 12  |-  ( Y  =  -u 1  ->  ( -u 1  x.  Y )  =  ( -u 1  x.  -u 1 ) )
5756breq1d 4447 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  (
( -u 1  x.  Y
)  <  0  <->  ( -u 1  x.  -u 1 )  <  0 ) )
5855, 57bibi12d 321 . . . . . . . . . 10  |-  ( Y  =  -u 1  ->  (
( Y  =/=  -u 1  <->  (
-u 1  x.  Y
)  <  0 )  <-> 
( -u 1  =/=  -u 1  <->  (
-u 1  x.  -u 1
)  <  0 ) ) )
5954, 58mpbiri 233 . . . . . . . . 9  |-  ( Y  =  -u 1  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
6059adantl 466 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  -u
1 )  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
61 neg1rr 10647 . . . . . . . . . . . 12  |-  -u 1  e.  RR
62 neg1lt0 10649 . . . . . . . . . . . . 13  |-  -u 1  <  0
63 0lt1 10082 . . . . . . . . . . . . 13  |-  0  <  1
6461, 17, 48lttri 9713 . . . . . . . . . . . . 13  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
6562, 63, 64mp2an 672 . . . . . . . . . . . 12  |-  -u 1  <  1
6661, 65gtneii 9699 . . . . . . . . . . 11  |-  1  =/=  -u 1
6721mulid1i 9601 . . . . . . . . . . . 12  |-  ( -u
1  x.  1 )  =  -u 1
6867, 62eqbrtri 4456 . . . . . . . . . . 11  |-  ( -u
1  x.  1 )  <  0
6966, 682th 239 . . . . . . . . . 10  |-  ( 1  =/=  -u 1  <->  ( -u 1  x.  1 )  <  0
)
70 neeq1 2724 . . . . . . . . . . 11  |-  ( Y  =  1  ->  ( Y  =/=  -u 1  <->  1  =/=  -u 1 ) )
71 oveq2 6289 . . . . . . . . . . . 12  |-  ( Y  =  1  ->  ( -u 1  x.  Y )  =  ( -u 1  x.  1 ) )
7271breq1d 4447 . . . . . . . . . . 11  |-  ( Y  =  1  ->  (
( -u 1  x.  Y
)  <  0  <->  ( -u 1  x.  1 )  <  0
) )
7370, 72bibi12d 321 . . . . . . . . . 10  |-  ( Y  =  1  ->  (
( Y  =/=  -u 1  <->  (
-u 1  x.  Y
)  <  0 )  <-> 
( 1  =/=  -u 1  <->  (
-u 1  x.  1 )  <  0 ) ) )
7469, 73mpbiri 233 . . . . . . . . 9  |-  ( Y  =  1  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
7574adantl 466 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  1 )  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
76 elpri 4034 . . . . . . . 8  |-  ( Y  e.  { -u 1 ,  1 }  ->  ( Y  =  -u 1  \/  Y  =  1
) )
7760, 75, 76mpjaodan 786 . . . . . . 7  |-  ( Y  e.  { -u 1 ,  1 }  ->  ( Y  =/=  -u 1  <->  (
-u 1  x.  Y
)  <  0 ) )
7877adantr 465 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  -u
1 )  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
79 neeq2 2726 . . . . . . . 8  |-  ( X  =  -u 1  ->  ( Y  =/=  X  <->  Y  =/=  -u 1 ) )
80 oveq1 6288 . . . . . . . . 9  |-  ( X  =  -u 1  ->  ( X  x.  Y )  =  ( -u 1  x.  Y ) )
8180breq1d 4447 . . . . . . . 8  |-  ( X  =  -u 1  ->  (
( X  x.  Y
)  <  0  <->  ( -u 1  x.  Y )  <  0
) )
8279, 81bibi12d 321 . . . . . . 7  |-  ( X  =  -u 1  ->  (
( Y  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) ) )
8382adantl 466 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  -u
1 )  ->  (
( Y  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) ) )
8478, 83mpbird 232 . . . . 5  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  -u
1 )  ->  ( Y  =/=  X  <->  ( X  x.  Y )  <  0
) )
8545, 84sylan 471 . . . 4  |-  ( ( ( ( X  e. 
{ -u 1 ,  1 }  /\  Y  e. 
{ -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0 )  /\  X  =  -u 1 )  -> 
( Y  =/=  X  <->  ( X  x.  Y )  <  0 ) )
8666necomi 2713 . . . . . . . . . . 11  |-  -u 1  =/=  1
8721, 22mulcomi 9605 . . . . . . . . . . . . 13  |-  ( -u
1  x.  1 )  =  ( 1  x.  -u 1 )
8887breq1i 4444 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  1 )  <  0  <->  (
1  x.  -u 1
)  <  0 )
8968, 88mpbi 208 . . . . . . . . . . 11  |-  ( 1  x.  -u 1 )  <  0
9086, 892th 239 . . . . . . . . . 10  |-  ( -u
1  =/=  1  <->  (
1  x.  -u 1
)  <  0 )
91 neeq1 2724 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  ( Y  =/=  1  <->  -u 1  =/=  1 ) )
92 oveq2 6289 . . . . . . . . . . . 12  |-  ( Y  =  -u 1  ->  (
1  x.  Y )  =  ( 1  x.  -u 1 ) )
9392breq1d 4447 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  (
( 1  x.  Y
)  <  0  <->  ( 1  x.  -u 1 )  <  0 ) )
9491, 93bibi12d 321 . . . . . . . . . 10  |-  ( Y  =  -u 1  ->  (
( Y  =/=  1  <->  ( 1  x.  Y )  <  0 )  <->  ( -u 1  =/=  1  <->  ( 1  x.  -u 1 )  <  0 ) ) )
9590, 94mpbiri 233 . . . . . . . . 9  |-  ( Y  =  -u 1  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0 ) )
9695adantl 466 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  -u
1 )  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0 ) )
97 neirr 2647 . . . . . . . . . . 11  |-  -.  1  =/=  1
9822mulid1i 9601 . . . . . . . . . . . . 13  |-  ( 1  x.  1 )  =  1
9998breq1i 4444 . . . . . . . . . . . 12  |-  ( ( 1  x.  1 )  <  0  <->  1  <  0 )
10050, 99mtbir 299 . . . . . . . . . . 11  |-  -.  (
1  x.  1 )  <  0
10197, 1002false 350 . . . . . . . . . 10  |-  ( 1  =/=  1  <->  ( 1  x.  1 )  <  0 )
102 neeq1 2724 . . . . . . . . . . 11  |-  ( Y  =  1  ->  ( Y  =/=  1  <->  1  =/=  1 ) )
103 oveq2 6289 . . . . . . . . . . . 12  |-  ( Y  =  1  ->  (
1  x.  Y )  =  ( 1  x.  1 ) )
104103breq1d 4447 . . . . . . . . . . 11  |-  ( Y  =  1  ->  (
( 1  x.  Y
)  <  0  <->  ( 1  x.  1 )  <  0 ) )
105102, 104bibi12d 321 . . . . . . . . . 10  |-  ( Y  =  1  ->  (
( Y  =/=  1  <->  ( 1  x.  Y )  <  0 )  <->  ( 1  =/=  1  <->  ( 1  x.  1 )  <  0 ) ) )
106101, 105mpbiri 233 . . . . . . . . 9  |-  ( Y  =  1  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0 ) )
107106adantl 466 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  1 )  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) )
10896, 107, 76mpjaodan 786 . . . . . . 7  |-  ( Y  e.  { -u 1 ,  1 }  ->  ( Y  =/=  1  <->  (
1  x.  Y )  <  0 ) )
109108adantr 465 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  1 )  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) )
110 neeq2 2726 . . . . . . . 8  |-  ( X  =  1  ->  ( Y  =/=  X  <->  Y  =/=  1 ) )
111 oveq1 6288 . . . . . . . . 9  |-  ( X  =  1  ->  ( X  x.  Y )  =  ( 1  x.  Y ) )
112111breq1d 4447 . . . . . . . 8  |-  ( X  =  1  ->  (
( X  x.  Y
)  <  0  <->  ( 1  x.  Y )  <  0 ) )
113110, 112bibi12d 321 . . . . . . 7  |-  ( X  =  1  ->  (
( Y  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) ) )
114113adantl 466 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  1 )  ->  ( ( Y  =/=  X  <->  ( X  x.  Y )  <  0
)  <->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) ) )
115109, 114mpbird 232 . . . . 5  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  1 )  ->  ( Y  =/=  X  <->  ( X  x.  Y )  <  0
) )
11645, 115sylan 471 . . . 4  |-  ( ( ( ( X  e. 
{ -u 1 ,  1 }  /\  Y  e. 
{ -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0 )  /\  X  =  1 )  -> 
( Y  =/=  X  <->  ( X  x.  Y )  <  0 ) )
117 elpri 4034 . . . . 5  |-  ( X  e.  { -u 1 ,  1 }  ->  ( X  =  -u 1  \/  X  =  1
) )
118117ad2antrr 725 . . . 4  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  ( X  =  -u 1  \/  X  =  1 ) )
11985, 116, 118mpjaodan 786 . . 3  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  ( Y  =/=  X  <->  ( X  x.  Y )  <  0
) )
12012, 14, 31, 119ifbothda 3961 . 2  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( if ( Y  =  0 ,  X ,  Y )  =/=  X  <->  ( X  x.  Y )  <  0 ) )
12110, 120bitrd 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 1383    e. wcel 1804    =/= wne 2638    \ cdif 3458    u. cun 3459    C_ wss 3461   ifcif 3926   {csn 4014   {cpr 4016   {ctp 4018   <.cop 4020   class class class wbr 4437   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   CCcc 9493   0cc0 9495   1c1 9496    x. cmul 9500    < clt 9631    <_ cle 9632   -ucneg 9811   ndxcnx 14611   Basecbs 14614   +g cplusg 14679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813
This theorem is referenced by:  signsvfn  28517
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