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Theorem signswch 29235
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 3996 . . . . . 6  |-  { -u
1 ,  1 }  =  ( { -u
1 }  u.  {
1 } )
2 snsstp1 4145 . . . . . . 7  |-  { -u
1 }  C_  { -u
1 ,  0 ,  1 }
3 snsstp3 4147 . . . . . . 7  |-  { 1 }  C_  { -u 1 ,  0 ,  1 }
42, 3unssi 3638 . . . . . 6  |-  ( {
-u 1 }  u.  { 1 } )  C_  {
-u 1 ,  0 ,  1 }
51, 4eqsstri 3491 . . . . 5  |-  { -u
1 ,  1 } 
C_  { -u 1 ,  0 ,  1 }
65sseli 3457 . . . 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 29226 . . . 4  |-  ( ( X  e.  { -u
1 ,  0 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  ->  ( X  .+^ 
Y )  =  if ( Y  =  0 ,  X ,  Y
) )
96, 8sylan 473 . . 3  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( X  .+^  Y )  =  if ( Y  =  0 ,  X ,  Y ) )
109neeq1d 2699 . 2  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( ( X  .+^  Y )  =/=  X  <->  if ( Y  =  0 ,  X ,  Y )  =/=  X ) )
11 neeq1 2703 . . . 4  |-  ( X  =  if ( Y  =  0 ,  X ,  Y )  ->  ( X  =/=  X  <->  if ( Y  =  0 ,  X ,  Y )  =/=  X ) )
1211bibi1d 320 . . 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 2703 . . . 4  |-  ( Y  =  if ( Y  =  0 ,  X ,  Y )  ->  ( Y  =/=  X  <->  if ( Y  =  0 ,  X ,  Y )  =/=  X ) )
1413bibi1d 320 . . 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 2626 . . . . 5  |-  -.  X  =/=  X
1615a1i 11 . . . 4  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  -.  X  =/=  X )
17 0re 9632 . . . . . 6  |-  0  e.  RR
1817ltnri 9732 . . . . 5  |-  -.  0  <  0
19 simpr 462 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  Y  =  0 )
2019oveq2d 6312 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  ( X  x.  0 ) )
21 neg1cn 10702 . . . . . . . . . 10  |-  -u 1  e.  CC
22 ax-1cn 9586 . . . . . . . . . 10  |-  1  e.  CC
23 prssi 4150 . . . . . . . . . 10  |-  ( (
-u 1  e.  CC  /\  1  e.  CC )  ->  { -u 1 ,  1 }  C_  CC )
2421, 22, 23mp2an 676 . . . . . . . . 9  |-  { -u
1 ,  1 } 
C_  CC
25 simpll 758 . . . . . . . . 9  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  X  e.  { -u 1 ,  1 } )
2624, 25sseldi 3459 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  X  e.  CC )
2726mul01d 9821 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  x.  0 )  =  0 )
2820, 27eqtrd 2461 . . . . . 6  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  0 )
2928breq1d 4427 . . . . 5  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( ( X  x.  Y )  <  0  <->  0  <  0
) )
3018, 29mtbiri 304 . . . 4  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  -.  ( X  x.  Y )  <  0
)
3116, 302falsed 352 . . 3  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  Y  =  0 )  ->  ( X  =/= 
X  <->  ( X  x.  Y )  <  0
) )
32 simplr 760 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  e.  {
-u 1 ,  0 ,  1 } )
33 tpcomb 4091 . . . . . . . 8  |-  { -u
1 ,  0 ,  1 }  =  { -u 1 ,  1 ,  0 }
3432, 33syl6eleq 2518 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  e.  {
-u 1 ,  1 ,  0 } )
35 simpr 462 . . . . . . . 8  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  -.  Y  =  0 )
3635neqned 2625 . . . . . . 7  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  =/=  0 )
3734, 36jca 534 . . . . . 6  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  ( Y  e.  { -u 1 ,  1 ,  0 }  /\  Y  =/=  0
) )
38 eldifsn 4119 . . . . . . 7  |-  ( Y  e.  ( { -u
1 ,  1 ,  0 }  \  {
0 } )  <->  ( Y  e.  { -u 1 ,  1 ,  0 }  /\  Y  =/=  0
) )
39 neg1ne0 10704 . . . . . . . . 9  |-  -u 1  =/=  0
40 ax-1ne0 9597 . . . . . . . . 9  |-  1  =/=  0
41 diftpsn3 4132 . . . . . . . . 9  |-  ( (
-u 1  =/=  0  /\  1  =/=  0
)  ->  ( { -u 1 ,  1 ,  0 }  \  {
0 } )  =  { -u 1 ,  1 } )
4239, 40, 41mp2an 676 . . . . . . . 8  |-  ( {
-u 1 ,  1 ,  0 }  \  { 0 } )  =  { -u 1 ,  1 }
4342eleq2i 2498 . . . . . . 7  |-  ( Y  e.  ( { -u
1 ,  1 ,  0 }  \  {
0 } )  <->  Y  e.  {
-u 1 ,  1 } )
4438, 43bitr3i 254 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 ,  0 }  /\  Y  =/=  0 )  <->  Y  e.  {
-u 1 ,  1 } )
4537, 44sylib 199 . . . . 5  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  Y  e.  {
-u 1 ,  1 } )
46 neirr 2626 . . . . . . . . . . 11  |-  -.  -u 1  =/=  -u 1
47 0le1 10126 . . . . . . . . . . . . 13  |-  0  <_  1
48 1re 9631 . . . . . . . . . . . . . 14  |-  1  e.  RR
4917, 48lenlti 9743 . . . . . . . . . . . . 13  |-  ( 0  <_  1  <->  -.  1  <  0 )
5047, 49mpbi 211 . . . . . . . . . . . 12  |-  -.  1  <  0
51 neg1mulneg1e1 10816 . . . . . . . . . . . . 13  |-  ( -u
1  x.  -u 1
)  =  1
5251breq1i 4424 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  -u 1
)  <  0  <->  1  <  0 )
5350, 52mtbir 300 . . . . . . . . . . 11  |-  -.  ( -u 1  x.  -u 1
)  <  0
5446, 532false 351 . . . . . . . . . 10  |-  ( -u
1  =/=  -u 1  <->  (
-u 1  x.  -u 1
)  <  0 )
55 neeq1 2703 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  ( Y  =/=  -u 1  <->  -u 1  =/=  -u 1 ) )
56 oveq2 6304 . . . . . . . . . . . 12  |-  ( Y  =  -u 1  ->  ( -u 1  x.  Y )  =  ( -u 1  x.  -u 1 ) )
5756breq1d 4427 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  (
( -u 1  x.  Y
)  <  0  <->  ( -u 1  x.  -u 1 )  <  0 ) )
5855, 57bibi12d 322 . . . . . . . . . 10  |-  ( Y  =  -u 1  ->  (
( Y  =/=  -u 1  <->  (
-u 1  x.  Y
)  <  0 )  <-> 
( -u 1  =/=  -u 1  <->  (
-u 1  x.  -u 1
)  <  0 ) ) )
5954, 58mpbiri 236 . . . . . . . . 9  |-  ( Y  =  -u 1  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
6059adantl 467 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  -u
1 )  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
61 neg1rr 10703 . . . . . . . . . . . 12  |-  -u 1  e.  RR
62 neg1lt0 10705 . . . . . . . . . . . . 13  |-  -u 1  <  0
63 0lt1 10125 . . . . . . . . . . . . 13  |-  0  <  1
6461, 17, 48lttri 9749 . . . . . . . . . . . . 13  |-  ( (
-u 1  <  0  /\  0  <  1
)  ->  -u 1  <  1 )
6562, 63, 64mp2an 676 . . . . . . . . . . . 12  |-  -u 1  <  1
6661, 65gtneii 9735 . . . . . . . . . . 11  |-  1  =/=  -u 1
6721mulid1i 9634 . . . . . . . . . . . 12  |-  ( -u
1  x.  1 )  =  -u 1
6867, 62eqbrtri 4436 . . . . . . . . . . 11  |-  ( -u
1  x.  1 )  <  0
6966, 682th 242 . . . . . . . . . 10  |-  ( 1  =/=  -u 1  <->  ( -u 1  x.  1 )  <  0
)
70 neeq1 2703 . . . . . . . . . . 11  |-  ( Y  =  1  ->  ( Y  =/=  -u 1  <->  1  =/=  -u 1 ) )
71 oveq2 6304 . . . . . . . . . . . 12  |-  ( Y  =  1  ->  ( -u 1  x.  Y )  =  ( -u 1  x.  1 ) )
7271breq1d 4427 . . . . . . . . . . 11  |-  ( Y  =  1  ->  (
( -u 1  x.  Y
)  <  0  <->  ( -u 1  x.  1 )  <  0
) )
7370, 72bibi12d 322 . . . . . . . . . 10  |-  ( Y  =  1  ->  (
( Y  =/=  -u 1  <->  (
-u 1  x.  Y
)  <  0 )  <-> 
( 1  =/=  -u 1  <->  (
-u 1  x.  1 )  <  0 ) ) )
7469, 73mpbiri 236 . . . . . . . . 9  |-  ( Y  =  1  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
7574adantl 467 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  1 )  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
76 elpri 4013 . . . . . . . 8  |-  ( Y  e.  { -u 1 ,  1 }  ->  ( Y  =  -u 1  \/  Y  =  1
) )
7760, 75, 76mpjaodan 793 . . . . . . 7  |-  ( Y  e.  { -u 1 ,  1 }  ->  ( Y  =/=  -u 1  <->  (
-u 1  x.  Y
)  <  0 ) )
7877adantr 466 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  -u
1 )  ->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) )
79 neeq2 2705 . . . . . . . 8  |-  ( X  =  -u 1  ->  ( Y  =/=  X  <->  Y  =/=  -u 1 ) )
80 oveq1 6303 . . . . . . . . 9  |-  ( X  =  -u 1  ->  ( X  x.  Y )  =  ( -u 1  x.  Y ) )
8180breq1d 4427 . . . . . . . 8  |-  ( X  =  -u 1  ->  (
( X  x.  Y
)  <  0  <->  ( -u 1  x.  Y )  <  0
) )
8279, 81bibi12d 322 . . . . . . 7  |-  ( X  =  -u 1  ->  (
( Y  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( Y  =/=  -u 1  <->  ( -u 1  x.  Y )  <  0
) ) )
8382adantl 467 . . . . . 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 235 . . . . 5  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  -u
1 )  ->  ( Y  =/=  X  <->  ( X  x.  Y )  <  0
) )
8545, 84sylan 473 . . . 4  |-  ( ( ( ( X  e. 
{ -u 1 ,  1 }  /\  Y  e. 
{ -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0 )  /\  X  =  -u 1 )  -> 
( Y  =/=  X  <->  ( X  x.  Y )  <  0 ) )
8666necomi 2692 . . . . . . . . . . 11  |-  -u 1  =/=  1
8721, 22mulcomi 9638 . . . . . . . . . . . . 13  |-  ( -u
1  x.  1 )  =  ( 1  x.  -u 1 )
8887breq1i 4424 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  1 )  <  0  <->  (
1  x.  -u 1
)  <  0 )
8968, 88mpbi 211 . . . . . . . . . . 11  |-  ( 1  x.  -u 1 )  <  0
9086, 892th 242 . . . . . . . . . 10  |-  ( -u
1  =/=  1  <->  (
1  x.  -u 1
)  <  0 )
91 neeq1 2703 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  ( Y  =/=  1  <->  -u 1  =/=  1 ) )
92 oveq2 6304 . . . . . . . . . . . 12  |-  ( Y  =  -u 1  ->  (
1  x.  Y )  =  ( 1  x.  -u 1 ) )
9392breq1d 4427 . . . . . . . . . . 11  |-  ( Y  =  -u 1  ->  (
( 1  x.  Y
)  <  0  <->  ( 1  x.  -u 1 )  <  0 ) )
9491, 93bibi12d 322 . . . . . . . . . 10  |-  ( Y  =  -u 1  ->  (
( Y  =/=  1  <->  ( 1  x.  Y )  <  0 )  <->  ( -u 1  =/=  1  <->  ( 1  x.  -u 1 )  <  0 ) ) )
9590, 94mpbiri 236 . . . . . . . . 9  |-  ( Y  =  -u 1  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0 ) )
9695adantl 467 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  -u
1 )  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0 ) )
97 neirr 2626 . . . . . . . . . . 11  |-  -.  1  =/=  1
9822mulid1i 9634 . . . . . . . . . . . . 13  |-  ( 1  x.  1 )  =  1
9998breq1i 4424 . . . . . . . . . . . 12  |-  ( ( 1  x.  1 )  <  0  <->  1  <  0 )
10050, 99mtbir 300 . . . . . . . . . . 11  |-  -.  (
1  x.  1 )  <  0
10197, 1002false 351 . . . . . . . . . 10  |-  ( 1  =/=  1  <->  ( 1  x.  1 )  <  0 )
102 neeq1 2703 . . . . . . . . . . 11  |-  ( Y  =  1  ->  ( Y  =/=  1  <->  1  =/=  1 ) )
103 oveq2 6304 . . . . . . . . . . . 12  |-  ( Y  =  1  ->  (
1  x.  Y )  =  ( 1  x.  1 ) )
104103breq1d 4427 . . . . . . . . . . 11  |-  ( Y  =  1  ->  (
( 1  x.  Y
)  <  0  <->  ( 1  x.  1 )  <  0 ) )
105102, 104bibi12d 322 . . . . . . . . . 10  |-  ( Y  =  1  ->  (
( Y  =/=  1  <->  ( 1  x.  Y )  <  0 )  <->  ( 1  =/=  1  <->  ( 1  x.  1 )  <  0 ) ) )
106101, 105mpbiri 236 . . . . . . . . 9  |-  ( Y  =  1  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0 ) )
107106adantl 467 . . . . . . . 8  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  Y  =  1 )  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) )
10896, 107, 76mpjaodan 793 . . . . . . 7  |-  ( Y  e.  { -u 1 ,  1 }  ->  ( Y  =/=  1  <->  (
1  x.  Y )  <  0 ) )
109108adantr 466 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  1 )  ->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) )
110 neeq2 2705 . . . . . . . 8  |-  ( X  =  1  ->  ( Y  =/=  X  <->  Y  =/=  1 ) )
111 oveq1 6303 . . . . . . . . 9  |-  ( X  =  1  ->  ( X  x.  Y )  =  ( 1  x.  Y ) )
112111breq1d 4427 . . . . . . . 8  |-  ( X  =  1  ->  (
( X  x.  Y
)  <  0  <->  ( 1  x.  Y )  <  0 ) )
113110, 112bibi12d 322 . . . . . . 7  |-  ( X  =  1  ->  (
( Y  =/=  X  <->  ( X  x.  Y )  <  0 )  <->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) ) )
114113adantl 467 . . . . . 6  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  1 )  ->  ( ( Y  =/=  X  <->  ( X  x.  Y )  <  0
)  <->  ( Y  =/=  1  <->  ( 1  x.  Y )  <  0
) ) )
115109, 114mpbird 235 . . . . 5  |-  ( ( Y  e.  { -u
1 ,  1 }  /\  X  =  1 )  ->  ( Y  =/=  X  <->  ( X  x.  Y )  <  0
) )
11645, 115sylan 473 . . . 4  |-  ( ( ( ( X  e. 
{ -u 1 ,  1 }  /\  Y  e. 
{ -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0 )  /\  X  =  1 )  -> 
( Y  =/=  X  <->  ( X  x.  Y )  <  0 ) )
117 elpri 4013 . . . . 5  |-  ( X  e.  { -u 1 ,  1 }  ->  ( X  =  -u 1  \/  X  =  1
) )
118117ad2antrr 730 . . . 4  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  ( X  =  -u 1  \/  X  =  1 ) )
11985, 116, 118mpjaodan 793 . . 3  |-  ( ( ( X  e.  { -u 1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  /\  -.  Y  =  0
)  ->  ( Y  =/=  X  <->  ( X  x.  Y )  <  0
) )
12012, 14, 31, 119ifbothda 3941 . 2  |-  ( ( X  e.  { -u
1 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  -> 
( if ( Y  =  0 ,  X ,  Y )  =/=  X  <->  ( X  x.  Y )  <  0 ) )
12110, 120bitrd 256 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 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616    \ cdif 3430    u. cun 3431    C_ wss 3433   ifcif 3906   {csn 3993   {cpr 3995   {ctp 3997   <.cop 3999   class class class wbr 4417   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   CCcc 9526   0cc0 9528   1c1 9529    x. cmul 9533    < clt 9664    <_ cle 9665   -ucneg 9850   ndxcnx 15070   Basecbs 15073   +g cplusg 15142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852
This theorem is referenced by:  signsvfn  29256
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