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Theorem signswch 29443
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 3970 . . . . . 6  |-  { -u
1 ,  1 }  =  ( { -u
1 }  u.  {
1 } )
2 snsstp1 4122 . . . . . . 7  |-  { -u
1 }  C_  { -u
1 ,  0 ,  1 }
3 snsstp3 4124 . . . . . . 7  |-  { 1 }  C_  { -u 1 ,  0 ,  1 }
42, 3unssi 3608 . . . . . 6  |-  ( {
-u 1 }  u.  { 1 } )  C_  {
-u 1 ,  0 ,  1 }
51, 4eqsstri 3461 . . . . 5  |-  { -u
1 ,  1 } 
C_  { -u 1 ,  0 ,  1 }
65sseli 3427 . . . 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 29434 . . . 4  |-  ( ( X  e.  { -u
1 ,  0 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  ->  ( X  .+^ 
Y )  =  if ( Y  =  0 ,  X ,  Y
) )
96, 8sylan 474 . . 3  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( X  .+^  Y )  =  if ( Y  =  0 ,  X ,  Y ) )
109neeq1d 2682 . 2  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( ( X  .+^  Y )  =/=  X  <->  if ( Y  =  0 ,  X ,  Y )  =/=  X ) )
11 neeq1 2685 . . . 4  |-  ( X  =  if ( Y  =  0 ,  X ,  Y )  ->  ( X  =/=  X  <->  if ( Y  =  0 ,  X ,  Y )  =/=  X ) )
1211bibi1d 321 . . 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 2685 . . . 4  |-  ( Y  =  if ( Y  =  0 ,  X ,  Y )  ->  ( Y  =/=  X  <->  if ( Y  =  0 ,  X ,  Y )  =/=  X ) )
1413bibi1d 321 . . 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 2631 . . . . 5  |-  -.  X  =/=  X
1615a1i 11 . . . 4  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  -.  X  =/=  X )
17 0re 9640 . . . . . 6  |-  0  e.  RR
1817ltnri 9740 . . . . 5  |-  -.  0  <  0
19 simpr 463 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  Y  =  0 )
2019oveq2d 6304 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  ( X  x.  0 ) )
21 neg1cn 10710 . . . . . . . . . 10  |-  -u 1  e.  CC
22 ax-1cn 9594 . . . . . . . . . 10  |-  1  e.  CC
23 prssi 4127 . . . . . . . . . 10  |-  ( (
-u 1  e.  CC  /\  1  e.  CC )  ->  { -u 1 ,  1 }  C_  CC )
2421, 22, 23mp2an 677 . . . . . . . . 9  |-  { -u
1 ,  1 } 
C_  CC
25 simpll 759 . . . . . . . . 9  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  X  e.  { -u 1 ,  1 } )
2624, 25sseldi 3429 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  X  e.  CC )
2726mul01d 9829 . . . . . . 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 4411 . . . . 5  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( ( X  x.  Y )  <  0  <->  0  <  0
) )
3018, 29mtbiri 305 . . . 4  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  -.  ( X  x.  Y )  <  0
)
3116, 302falsed 353 . . 3  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  =/= 
X  <->  ( X  x.  Y )  <  0
) )
32 simplr 761 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  e.  {
-u 1 ,  0 ,  1 } )
33 tpcomb 4068 . . . . . . . 8  |-  { -u
1 ,  0 ,  1 }  =  { -u 1 ,  1 ,  0 }
3432, 33syl6eleq 2538 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  e.  {
-u 1 ,  1 ,  0 } )
35 simpr 463 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  -.  Y  =  0 )
3635neqned 2630 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  =/=  0 )
3734, 36jca 535 . . . . . 6  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  ( Y  e.  { -u 1 ,  1 ,  0 }  /\  Y  =/=  0
) )
38 eldifsn 4096 . . . . . . 7  |-  ( Y  e.  ( { -u
1 ,  1 ,  0 }  \  {
0 } )  <->  ( Y  e.  { -u 1 ,  1 ,  0 }  /\  Y  =/=  0
) )
39 neg1ne0 10712 . . . . . . . . 9  |-  -u 1  =/=  0
40 ax-1ne0 9605 . . . . . . . . 9  |-  1  =/=  0
41 diftpsn3 4109 . . . . . . . . 9  |-  ( (
-u 1  =/=  0  /\  1  =/=  0
)  ->  ( { -u 1 ,  1 ,  0 }  \  {
0 } )  =  { -u 1 ,  1 } )
4239, 40, 41mp2an 677 . . . . . . . 8  |-  ( {
-u 1 ,  1 ,  0 }  \  { 0 } )  =  { -u 1 ,  1 }
4342eleq2i 2520 . . . . . . 7  |-  ( Y  e.  ( { -u
1 ,  1 ,  0 }  \  {
0 } )  <->  Y  e.  {
-u 1 ,  1 } )
4438, 43bitr3i 255 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 ,  0 }  /\  Y  =/=  0 )  <->  Y  e.  {
-u 1 ,  1 } )
4537, 44sylib 200 . . . . 5  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  e.  {
-u 1 ,  1 } )
46 neirr 2631 . . . . . . . . . . 11  |-  -.  -u 1  =/=  -u 1
47 0le1 10134 . . . . . . . . . . . . 13  |-  0  <_  1
48 1re 9639 . . . . . . . . . . . . . 14  |-  1  e.  RR
4917, 48lenlti 9751 . . . . . . . . . . . . 13  |-  ( 0  <_  1  <->  -.  1  <  0 )
5047, 49mpbi 212 . . . . . . . . . . . 12  |-  -.  1  <  0
51 neg1mulneg1e1 10824 . . . . . . . . . . . . 13  |-  ( -u
1  x.  -u 1
)  =  1
5251breq1i 4408 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  -u 1
)  <  0  <->  1  <  0 )
5350, 52mtbir 301 . . . . . . . . . . 11  |-  -.  ( -u 1  x.  -u 1
)  <  0
5446, 532false 352 . . . . . . . . . 10  |-  ( -u
1  =/=  -u 1  <->  (
-u 1  x.  -u 1
)  <  0 )
55 neeq1 2685 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  ( Y  =/=  -u 1  <->  -u 1  =/=  -u 1 ) )
56 oveq2 6296 . . . . . . . . . . . 12  |-  ( Y  =  -u 1  ->  ( -u 1  x.  Y )  =  ( -u 1  x.  -u 1 ) )
5756breq1d 4411 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  (
( -u 1  x.  Y
)  <  0  <->  ( -u 1  x.  -u 1 )  <  0 ) )
5855, 57bibi12d 323 . . . . . . . . . 10  |-  ( Y  =  -u 1  ->  (
( Y  =/=  -u 1  <->  (
-u 1  x.  Y
)  <  0 )  <-> 
( -u 1  =/=  -u 1  <->  (
-u 1  x.  -u 1
)  <  0 ) ) )
5954, 58mpbiri 237 . . . . . . . . 9  |-  ( Y  =  -u 1  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
6059adantl 468 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  -u
1 )  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
61 neg1rr 10711 . . . . . . . . . . . 12  |-  -u 1  e.  RR
62 neg1lt0 10713 . . . . . . . . . . . . 13  |-  -u 1  <  0
63 0lt1 10133 . . . . . . . . . . . . 13  |-  0  <  1
6461, 17, 48lttri 9757 . . . . . . . . . . . . 13  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
6562, 63, 64mp2an 677 . . . . . . . . . . . 12  |-  -u 1  <  1
6661, 65gtneii 9743 . . . . . . . . . . 11  |-  1  =/=  -u 1
6721mulid1i 9642 . . . . . . . . . . . 12  |-  ( -u
1  x.  1 )  =  -u 1
6867, 62eqbrtri 4421 . . . . . . . . . . 11  |-  ( -u
1  x.  1 )  <  0
6966, 682th 243 . . . . . . . . . 10  |-  ( 1  =/=  -u 1  <->  ( -u 1  x.  1 )  <  0
)
70 neeq1 2685 . . . . . . . . . . 11  |-  ( Y  =  1  ->  ( Y  =/=  -u 1  <->  1  =/=  -u 1 ) )
71 oveq2 6296 . . . . . . . . . . . 12  |-  ( Y  =  1  ->  ( -u 1  x.  Y )  =  ( -u 1  x.  1 ) )
7271breq1d 4411 . . . . . . . . . . 11  |-  ( Y  =  1  ->  (
( -u 1  x.  Y
)  <  0  <->  ( -u 1  x.  1 )  <  0
) )
7370, 72bibi12d 323 . . . . . . . . . 10  |-  ( Y  =  1  ->  (
( Y  =/=  -u 1  <->  (
-u 1  x.  Y
)  <  0 )  <-> 
( 1  =/=  -u 1  <->  (
-u 1  x.  1 )  <  0 ) ) )
7469, 73mpbiri 237 . . . . . . . . 9  |-  ( Y  =  1  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
7574adantl 468 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  1 )  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
76 elpri 3984 . . . . . . . 8  |-  ( Y  e.  { -u 1 ,  1 }  ->  ( Y  =  -u 1  \/  Y  =  1
) )
7760, 75, 76mpjaodan 794 . . . . . . 7  |-  ( Y  e.  { -u 1 ,  1 }  ->  ( Y  =/=  -u 1  <->  (
-u 1  x.  Y
)  <  0 ) )
7877adantr 467 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  -u
1 )  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
79 neeq2 2686 . . . . . . . 8  |-  ( X  =  -u 1  ->  ( Y  =/=  X  <->  Y  =/=  -u 1 ) )
80 oveq1 6295 . . . . . . . . 9  |-  ( X  =  -u 1  ->  ( X  x.  Y )  =  ( -u 1  x.  Y ) )
8180breq1d 4411 . . . . . . . 8  |-  ( X  =  -u 1  ->  (
( X  x.  Y
)  <  0  <->  ( -u 1  x.  Y )  <  0
) )
8279, 81bibi12d 323 . . . . . . 7  |-  ( X  =  -u 1  ->  (
( Y  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) ) )
8382adantl 468 . . . . . 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 236 . . . . 5  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  -u
1 )  ->  ( Y  =/=  X  <->  ( X  x.  Y )  <  0
) )
8545, 84sylan 474 . . . 4  |-  ( ( ( ( X  e. 
{ -u 1 ,  1 }  /\  Y  e. 
{ -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0 )  /\  X  =  -u 1 )  -> 
( Y  =/=  X  <->  ( X  x.  Y )  <  0 ) )
8666necomi 2677 . . . . . . . . . . 11  |-  -u 1  =/=  1
8721, 22mulcomi 9646 . . . . . . . . . . . . 13  |-  ( -u
1  x.  1 )  =  ( 1  x.  -u 1 )
8887breq1i 4408 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  1 )  <  0  <->  (
1  x.  -u 1
)  <  0 )
8968, 88mpbi 212 . . . . . . . . . . 11  |-  ( 1  x.  -u 1 )  <  0
9086, 892th 243 . . . . . . . . . 10  |-  ( -u
1  =/=  1  <->  (
1  x.  -u 1
)  <  0 )
91 neeq1 2685 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  ( Y  =/=  1  <->  -u 1  =/=  1 ) )
92 oveq2 6296 . . . . . . . . . . . 12  |-  ( Y  =  -u 1  ->  (
1  x.  Y )  =  ( 1  x.  -u 1 ) )
9392breq1d 4411 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  (
( 1  x.  Y
)  <  0  <->  ( 1  x.  -u 1 )  <  0 ) )
9491, 93bibi12d 323 . . . . . . . . . 10  |-  ( Y  =  -u 1  ->  (
( Y  =/=  1  <->  ( 1  x.  Y )  <  0 )  <->  ( -u 1  =/=  1  <->  ( 1  x.  -u 1 )  <  0 ) ) )
9590, 94mpbiri 237 . . . . . . . . 9  |-  ( Y  =  -u 1  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0 ) )
9695adantl 468 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  -u
1 )  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0 ) )
97 neirr 2631 . . . . . . . . . . 11  |-  -.  1  =/=  1
9822mulid1i 9642 . . . . . . . . . . . . 13  |-  ( 1  x.  1 )  =  1
9998breq1i 4408 . . . . . . . . . . . 12  |-  ( ( 1  x.  1 )  <  0  <->  1  <  0 )
10050, 99mtbir 301 . . . . . . . . . . 11  |-  -.  (
1  x.  1 )  <  0
10197, 1002false 352 . . . . . . . . . 10  |-  ( 1  =/=  1  <->  ( 1  x.  1 )  <  0 )
102 neeq1 2685 . . . . . . . . . . 11  |-  ( Y  =  1  ->  ( Y  =/=  1  <->  1  =/=  1 ) )
103 oveq2 6296 . . . . . . . . . . . 12  |-  ( Y  =  1  ->  (
1  x.  Y )  =  ( 1  x.  1 ) )
104103breq1d 4411 . . . . . . . . . . 11  |-  ( Y  =  1  ->  (
( 1  x.  Y
)  <  0  <->  ( 1  x.  1 )  <  0 ) )
105102, 104bibi12d 323 . . . . . . . . . 10  |-  ( Y  =  1  ->  (
( Y  =/=  1  <->  ( 1  x.  Y )  <  0 )  <->  ( 1  =/=  1  <->  ( 1  x.  1 )  <  0 ) ) )
106101, 105mpbiri 237 . . . . . . . . 9  |-  ( Y  =  1  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0 ) )
107106adantl 468 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  1 )  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) )
10896, 107, 76mpjaodan 794 . . . . . . 7  |-  ( Y  e.  { -u 1 ,  1 }  ->  ( Y  =/=  1  <->  (
1  x.  Y )  <  0 ) )
109108adantr 467 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  1 )  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) )
110 neeq2 2686 . . . . . . . 8  |-  ( X  =  1  ->  ( Y  =/=  X  <->  Y  =/=  1 ) )
111 oveq1 6295 . . . . . . . . 9  |-  ( X  =  1  ->  ( X  x.  Y )  =  ( 1  x.  Y ) )
112111breq1d 4411 . . . . . . . 8  |-  ( X  =  1  ->  (
( X  x.  Y
)  <  0  <->  ( 1  x.  Y )  <  0 ) )
113110, 112bibi12d 323 . . . . . . 7  |-  ( X  =  1  ->  (
( Y  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) ) )
114113adantl 468 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  1 )  ->  ( ( Y  =/=  X  <->  ( X  x.  Y )  <  0
)  <->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) ) )
115109, 114mpbird 236 . . . . 5  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  1 )  ->  ( Y  =/=  X  <->  ( X  x.  Y )  <  0
) )
11645, 115sylan 474 . . . 4  |-  ( ( ( ( X  e. 
{ -u 1 ,  1 }  /\  Y  e. 
{ -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0 )  /\  X  =  1 )  -> 
( Y  =/=  X  <->  ( X  x.  Y )  <  0 ) )
117 elpri 3984 . . . . 5  |-  ( X  e.  { -u 1 ,  1 }  ->  ( X  =  -u 1  \/  X  =  1
) )
118117ad2antrr 731 . . . 4  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  ( X  =  -u 1  \/  X  =  1 ) )
11985, 116, 118mpjaodan 794 . . 3  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  ( Y  =/=  X  <->  ( X  x.  Y )  <  0
) )
12012, 14, 31, 119ifbothda 3915 . 2  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( if ( Y  =  0 ,  X ,  Y )  =/=  X  <->  ( X  x.  Y )  <  0 ) )
12110, 120bitrd 257 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 188    \/ wo 370    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621    \ cdif 3400    u. cun 3401    C_ wss 3403   ifcif 3880   {csn 3967   {cpr 3969   {ctp 3971   <.cop 3973   class class class wbr 4401   ` cfv 5581  (class class class)co 6288    |-> cmpt2 6290   CCcc 9534   0cc0 9536   1c1 9537    x. cmul 9541    < clt 9672    <_ cle 9673   -ucneg 9858   ndxcnx 15111   Basecbs 15114   +g cplusg 15183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860
This theorem is referenced by:  signsvfn  29464
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