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Theorem expdioph 27116
Description: The exponential function is Diophantine. This result completes and encapsulates our development using Pell equation solution sequences and is sometimes regarded as Matiyasevich's theorem properly. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Assertion
Ref Expression
expdioph  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) }  e.  (Dioph `  3 )

Proof of Theorem expdioph
StepHypRef Expression
1 pm4.42 926 . . . 4  |-  ( ( a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) )  <->  ( (
( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) )  /\  ( a `  2
)  e.  NN )  \/  ( ( a `
 3 )  =  ( ( a ` 
1 ) ^ (
a `  2 )
)  /\  -.  (
a `  2 )  e.  NN ) ) )
2 ancom 437 . . . . . 6  |-  ( ( ( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) )  /\  ( a `  2
)  e.  NN )  <-> 
( ( a ` 
2 )  e.  NN  /\  ( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) ) ) )
3 elmapi 6792 . . . . . . . . . . . . 13  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  a : ( 1 ... 3 ) --> NN0 )
4 df-2 9804 . . . . . . . . . . . . . . 15  |-  2  =  ( 1  +  1 )
5 df-3 9805 . . . . . . . . . . . . . . . 16  |-  3  =  ( 2  +  1 )
6 ssid 3197 . . . . . . . . . . . . . . . 16  |-  ( 1 ... 3 )  C_  ( 1 ... 3
)
75, 6jm2.27dlem5 27106 . . . . . . . . . . . . . . 15  |-  ( 1 ... 2 )  C_  ( 1 ... 3
)
84, 7jm2.27dlem5 27106 . . . . . . . . . . . . . 14  |-  ( 1 ... 1 )  C_  ( 1 ... 3
)
9 1nn 9757 . . . . . . . . . . . . . . 15  |-  1  e.  NN
109jm2.27dlem3 27104 . . . . . . . . . . . . . 14  |-  1  e.  ( 1 ... 1
)
118, 10sselii 3177 . . . . . . . . . . . . 13  |-  1  e.  ( 1 ... 3
)
12 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( a : ( 1 ... 3 ) --> NN0 
/\  1  e.  ( 1 ... 3 ) )  ->  ( a `  1 )  e. 
NN0 )
133, 11, 12sylancl 643 . . . . . . . . . . . 12  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
a `  1 )  e.  NN0 )
1413adantr 451 . . . . . . . . . . 11  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( a ` 
1 )  e.  NN0 )
15 elnn0 9967 . . . . . . . . . . 11  |-  ( ( a `  1 )  e.  NN0  <->  ( ( a `
 1 )  e.  NN  \/  ( a `
 1 )  =  0 ) )
1614, 15sylib 188 . . . . . . . . . 10  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( a `
 1 )  e.  NN  \/  ( a `
 1 )  =  0 ) )
17 elnn1uz2 10294 . . . . . . . . . . . 12  |-  ( ( a `  1 )  e.  NN  <->  ( (
a `  1 )  =  1  \/  (
a `  1 )  e.  ( ZZ>= `  2 )
) )
1817biimpi 186 . . . . . . . . . . 11  |-  ( ( a `  1 )  e.  NN  ->  (
( a `  1
)  =  1  \/  ( a `  1
)  e.  ( ZZ>= ` 
2 ) ) )
1918orim1i 503 . . . . . . . . . 10  |-  ( ( ( a `  1
)  e.  NN  \/  ( a `  1
)  =  0 )  ->  ( ( ( a `  1 )  =  1  \/  (
a `  1 )  e.  ( ZZ>= `  2 )
)  \/  ( a `
 1 )  =  0 ) )
2016, 19syl 15 . . . . . . . . 9  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( ( a `  1 )  =  1  \/  (
a `  1 )  e.  ( ZZ>= `  2 )
)  \/  ( a `
 1 )  =  0 ) )
2120biantrurd 494 . . . . . . . 8  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( a `
 3 )  =  ( ( a ` 
1 ) ^ (
a `  2 )
)  <->  ( ( ( ( a `  1
)  =  1  \/  ( a `  1
)  e.  ( ZZ>= ` 
2 ) )  \/  ( a `  1
)  =  0 )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) ) )
22 andir 838 . . . . . . . . . 10  |-  ( ( ( ( ( a `
 1 )  =  1  \/  ( a `
 1 )  e.  ( ZZ>= `  2 )
)  \/  ( a `
 1 )  =  0 )  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) )  <->  ( (
( ( a ` 
1 )  =  1  \/  ( a ` 
1 )  e.  (
ZZ>= `  2 ) )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) )  \/  ( ( a `  1 )  =  0  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) ) ) )
23 andir 838 . . . . . . . . . . 11  |-  ( ( ( ( a ` 
1 )  =  1  \/  ( a ` 
1 )  e.  (
ZZ>= `  2 ) )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) )  <->  ( ( ( a `  1 )  =  1  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) )  \/  ( ( a ` 
1 )  e.  (
ZZ>= `  2 )  /\  ( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) ) ) ) )
2423orbi1i 506 . . . . . . . . . 10  |-  ( ( ( ( ( a `
 1 )  =  1  \/  ( a `
 1 )  e.  ( ZZ>= `  2 )
)  /\  ( a `  3 )  =  ( ( a ` 
1 ) ^ (
a `  2 )
) )  \/  (
( a `  1
)  =  0  /\  ( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) ) ) )  <->  ( ( ( ( a `  1
)  =  1  /\  ( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) ) )  \/  ( ( a `
 1 )  e.  ( ZZ>= `  2 )  /\  ( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) ) ) )  \/  ( ( a `  1 )  =  0  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) ) ) )
2522, 24bitri 240 . . . . . . . . 9  |-  ( ( ( ( ( a `
 1 )  =  1  \/  ( a `
 1 )  e.  ( ZZ>= `  2 )
)  \/  ( a `
 1 )  =  0 )  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) )  <->  ( (
( ( a ` 
1 )  =  1  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) )  \/  ( ( a `  1 )  e.  ( ZZ>= `  2
)  /\  ( a `  3 )  =  ( ( a ` 
1 ) ^ (
a `  2 )
) ) )  \/  ( ( a ` 
1 )  =  0  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) ) )
26 nnz 10045 . . . . . . . . . . . . . . . 16  |-  ( ( a `  2 )  e.  NN  ->  (
a `  2 )  e.  ZZ )
27 1exp 11131 . . . . . . . . . . . . . . . 16  |-  ( ( a `  2 )  e.  ZZ  ->  (
1 ^ ( a `
 2 ) )  =  1 )
2826, 27syl 15 . . . . . . . . . . . . . . 15  |-  ( ( a `  2 )  e.  NN  ->  (
1 ^ ( a `
 2 ) )  =  1 )
2928adantl 452 . . . . . . . . . . . . . 14  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( 1 ^ ( a `  2
) )  =  1 )
3029eqeq2d 2294 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( a `
 3 )  =  ( 1 ^ (
a `  2 )
)  <->  ( a ` 
3 )  =  1 ) )
31 oveq1 5865 . . . . . . . . . . . . . . 15  |-  ( ( a `  1 )  =  1  ->  (
( a `  1
) ^ ( a `
 2 ) )  =  ( 1 ^ ( a `  2
) ) )
3231eqeq2d 2294 . . . . . . . . . . . . . 14  |-  ( ( a `  1 )  =  1  ->  (
( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) )  <->  ( a `  3 )  =  ( 1 ^ (
a `  2 )
) ) )
3332bibi1d 310 . . . . . . . . . . . . 13  |-  ( ( a `  1 )  =  1  ->  (
( ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) )  <-> 
( a `  3
)  =  1 )  <-> 
( ( a ` 
3 )  =  ( 1 ^ ( a `
 2 ) )  <-> 
( a `  3
)  =  1 ) ) )
3430, 33syl5ibrcom 213 . . . . . . . . . . . 12  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( a `
 1 )  =  1  ->  ( (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) )  <->  ( a `  3 )  =  1 ) ) )
3534pm5.32d 620 . . . . . . . . . . 11  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( ( a `  1 )  =  1  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) )  <->  ( (
a `  1 )  =  1  /\  (
a `  3 )  =  1 ) ) )
36 iba 489 . . . . . . . . . . . . 13  |-  ( ( a `  2 )  e.  NN  ->  (
( a `  1
)  e.  ( ZZ>= ` 
2 )  <->  ( (
a `  1 )  e.  ( ZZ>= `  2 )  /\  ( a `  2
)  e.  NN ) ) )
3736adantl 452 . . . . . . . . . . . 12  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( a `
 1 )  e.  ( ZZ>= `  2 )  <->  ( ( a `  1
)  e.  ( ZZ>= ` 
2 )  /\  (
a `  2 )  e.  NN ) ) )
3837anbi1d 685 . . . . . . . . . . 11  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( ( a `  1 )  e.  ( ZZ>= `  2
)  /\  ( a `  3 )  =  ( ( a ` 
1 ) ^ (
a `  2 )
) )  <->  ( (
( a `  1
)  e.  ( ZZ>= ` 
2 )  /\  (
a `  2 )  e.  NN )  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) ) ) )
3935, 38orbi12d 690 . . . . . . . . . 10  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( ( ( a `  1
)  =  1  /\  ( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) ) )  \/  ( ( a `
 1 )  e.  ( ZZ>= `  2 )  /\  ( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) ) ) )  <->  ( ( ( a `  1 )  =  1  /\  (
a `  3 )  =  1 )  \/  ( ( ( a `
 1 )  e.  ( ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) ) ) )
40 0exp 11137 . . . . . . . . . . . . . 14  |-  ( ( a `  2 )  e.  NN  ->  (
0 ^ ( a `
 2 ) )  =  0 )
4140adantl 452 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( 0 ^ ( a `  2
) )  =  0 )
4241eqeq2d 2294 . . . . . . . . . . . 12  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( a `
 3 )  =  ( 0 ^ (
a `  2 )
)  <->  ( a ` 
3 )  =  0 ) )
43 oveq1 5865 . . . . . . . . . . . . . 14  |-  ( ( a `  1 )  =  0  ->  (
( a `  1
) ^ ( a `
 2 ) )  =  ( 0 ^ ( a `  2
) ) )
4443eqeq2d 2294 . . . . . . . . . . . . 13  |-  ( ( a `  1 )  =  0  ->  (
( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) )  <->  ( a `  3 )  =  ( 0 ^ (
a `  2 )
) ) )
4544bibi1d 310 . . . . . . . . . . . 12  |-  ( ( a `  1 )  =  0  ->  (
( ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) )  <-> 
( a `  3
)  =  0 )  <-> 
( ( a ` 
3 )  =  ( 0 ^ ( a `
 2 ) )  <-> 
( a `  3
)  =  0 ) ) )
4642, 45syl5ibrcom 213 . . . . . . . . . . 11  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( a `
 1 )  =  0  ->  ( (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) )  <->  ( a `  3 )  =  0 ) ) )
4746pm5.32d 620 . . . . . . . . . 10  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( ( a `  1 )  =  0  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) )  <->  ( (
a `  1 )  =  0  /\  (
a `  3 )  =  0 ) ) )
4839, 47orbi12d 690 . . . . . . . . 9  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( ( ( ( a ` 
1 )  =  1  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) )  \/  ( ( a `  1 )  e.  ( ZZ>= `  2
)  /\  ( a `  3 )  =  ( ( a ` 
1 ) ^ (
a `  2 )
) ) )  \/  ( ( a ` 
1 )  =  0  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) )  <->  ( (
( ( a ` 
1 )  =  1  /\  ( a ` 
3 )  =  1 )  \/  ( ( ( a `  1
)  e.  ( ZZ>= ` 
2 )  /\  (
a `  2 )  e.  NN )  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) ) )  \/  ( ( a `
 1 )  =  0  /\  ( a `
 3 )  =  0 ) ) ) )
4925, 48syl5bb 248 . . . . . . . 8  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( ( ( ( a ` 
1 )  =  1  \/  ( a ` 
1 )  e.  (
ZZ>= `  2 ) )  \/  ( a ` 
1 )  =  0 )  /\  ( a `
 3 )  =  ( ( a ` 
1 ) ^ (
a `  2 )
) )  <->  ( (
( ( a ` 
1 )  =  1  /\  ( a ` 
3 )  =  1 )  \/  ( ( ( a `  1
)  e.  ( ZZ>= ` 
2 )  /\  (
a `  2 )  e.  NN )  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) ) )  \/  ( ( a `
 1 )  =  0  /\  ( a `
 3 )  =  0 ) ) ) )
5021, 49bitrd 244 . . . . . . 7  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( a `
 3 )  =  ( ( a ` 
1 ) ^ (
a `  2 )
)  <->  ( ( ( ( a `  1
)  =  1  /\  ( a `  3
)  =  1 )  \/  ( ( ( a `  1 )  e.  ( ZZ>= `  2
)  /\  ( a `  2 )  e.  NN )  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) ) )  \/  ( ( a `
 1 )  =  0  /\  ( a `
 3 )  =  0 ) ) ) )
5150pm5.32da 622 . . . . . 6  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
( ( a ` 
2 )  e.  NN  /\  ( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) ) )  <-> 
( ( a ` 
2 )  e.  NN  /\  ( ( ( ( a `  1 )  =  1  /\  (
a `  3 )  =  1 )  \/  ( ( ( a `
 1 )  e.  ( ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) )  \/  (
( a `  1
)  =  0  /\  ( a `  3
)  =  0 ) ) ) ) )
522, 51syl5bb 248 . . . . 5  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
( ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) )  /\  ( a ` 
2 )  e.  NN ) 
<->  ( ( a ` 
2 )  e.  NN  /\  ( ( ( ( a `  1 )  =  1  /\  (
a `  3 )  =  1 )  \/  ( ( ( a `
 1 )  e.  ( ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) )  \/  (
( a `  1
)  =  0  /\  ( a `  3
)  =  0 ) ) ) ) )
53 ancom 437 . . . . . 6  |-  ( ( ( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) )  /\  -.  ( a `  2
)  e.  NN )  <-> 
( -.  ( a `
 2 )  e.  NN  /\  ( a `
 3 )  =  ( ( a ` 
1 ) ^ (
a `  2 )
) ) )
54 2nn 9877 . . . . . . . . . . . 12  |-  2  e.  NN
5554jm2.27dlem3 27104 . . . . . . . . . . 11  |-  2  e.  ( 1 ... 2
)
567, 55sselii 3177 . . . . . . . . . 10  |-  2  e.  ( 1 ... 3
)
57 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( a : ( 1 ... 3 ) --> NN0 
/\  2  e.  ( 1 ... 3 ) )  ->  ( a `  2 )  e. 
NN0 )
583, 56, 57sylancl 643 . . . . . . . . 9  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
a `  2 )  e.  NN0 )
59 elnn0 9967 . . . . . . . . . . 11  |-  ( ( a `  2 )  e.  NN0  <->  ( ( a `
 2 )  e.  NN  \/  ( a `
 2 )  =  0 ) )
60 pm2.53 362 . . . . . . . . . . 11  |-  ( ( ( a `  2
)  e.  NN  \/  ( a `  2
)  =  0 )  ->  ( -.  (
a `  2 )  e.  NN  ->  ( a `  2 )  =  0 ) )
6159, 60sylbi 187 . . . . . . . . . 10  |-  ( ( a `  2 )  e.  NN0  ->  ( -.  ( a `  2
)  e.  NN  ->  ( a `  2 )  =  0 ) )
62 0nnn 9777 . . . . . . . . . . 11  |-  -.  0  e.  NN
63 eleq1 2343 . . . . . . . . . . 11  |-  ( ( a `  2 )  =  0  ->  (
( a `  2
)  e.  NN  <->  0  e.  NN ) )
6462, 63mtbiri 294 . . . . . . . . . 10  |-  ( ( a `  2 )  =  0  ->  -.  ( a `  2
)  e.  NN )
6561, 64impbid1 194 . . . . . . . . 9  |-  ( ( a `  2 )  e.  NN0  ->  ( -.  ( a `  2
)  e.  NN  <->  ( a `  2 )  =  0 ) )
6658, 65syl 15 . . . . . . . 8  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  ( -.  ( a `  2
)  e.  NN  <->  ( a `  2 )  =  0 ) )
6766anbi1d 685 . . . . . . 7  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
( -.  ( a `
 2 )  e.  NN  /\  ( a `
 3 )  =  ( ( a ` 
1 ) ^ (
a `  2 )
) )  <->  ( (
a `  2 )  =  0  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) ) ) )
6813nn0cnd 10020 . . . . . . . . . . 11  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
a `  1 )  e.  CC )
6968exp0d 11239 . . . . . . . . . 10  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
( a `  1
) ^ 0 )  =  1 )
7069eqeq2d 2294 . . . . . . . . 9  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
( a `  3
)  =  ( ( a `  1 ) ^ 0 )  <->  ( a `  3 )  =  1 ) )
71 oveq2 5866 . . . . . . . . . . 11  |-  ( ( a `  2 )  =  0  ->  (
( a `  1
) ^ ( a `
 2 ) )  =  ( ( a `
 1 ) ^
0 ) )
7271eqeq2d 2294 . . . . . . . . . 10  |-  ( ( a `  2 )  =  0  ->  (
( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) )  <->  ( a `  3 )  =  ( ( a ` 
1 ) ^ 0 ) ) )
7372bibi1d 310 . . . . . . . . 9  |-  ( ( a `  2 )  =  0  ->  (
( ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) )  <-> 
( a `  3
)  =  1 )  <-> 
( ( a ` 
3 )  =  ( ( a `  1
) ^ 0 )  <-> 
( a `  3
)  =  1 ) ) )
7470, 73syl5ibrcom 213 . . . . . . . 8  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
( a `  2
)  =  0  -> 
( ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) )  <-> 
( a `  3
)  =  1 ) ) )
7574pm5.32d 620 . . . . . . 7  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
( ( a ` 
2 )  =  0  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) )  <->  ( ( a `
 2 )  =  0  /\  ( a `
 3 )  =  1 ) ) )
7667, 75bitrd 244 . . . . . 6  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
( -.  ( a `
 2 )  e.  NN  /\  ( a `
 3 )  =  ( ( a ` 
1 ) ^ (
a `  2 )
) )  <->  ( (
a `  2 )  =  0  /\  (
a `  3 )  =  1 ) ) )
7753, 76syl5bb 248 . . . . 5  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
( ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) )  /\  -.  ( a `
 2 )  e.  NN )  <->  ( (
a `  2 )  =  0  /\  (
a `  3 )  =  1 ) ) )
7852, 77orbi12d 690 . . . 4  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
( ( ( a `
 3 )  =  ( ( a ` 
1 ) ^ (
a `  2 )
)  /\  ( a `  2 )  e.  NN )  \/  (
( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) )  /\  -.  ( a `  2
)  e.  NN ) )  <->  ( ( ( a `  2 )  e.  NN  /\  (
( ( ( a `
 1 )  =  1  /\  ( a `
 3 )  =  1 )  \/  (
( ( a ` 
1 )  e.  (
ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) )  \/  (
( a `  1
)  =  0  /\  ( a `  3
)  =  0 ) ) )  \/  (
( a `  2
)  =  0  /\  ( a `  3
)  =  1 ) ) ) )
791, 78syl5bb 248 . . 3  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
( a `  3
)  =  ( ( a `  1 ) ^ ( a ` 
2 ) )  <->  ( (
( a `  2
)  e.  NN  /\  ( ( ( ( a `  1 )  =  1  /\  (
a `  3 )  =  1 )  \/  ( ( ( a `
 1 )  e.  ( ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) )  \/  (
( a `  1
)  =  0  /\  ( a `  3
)  =  0 ) ) )  \/  (
( a `  2
)  =  0  /\  ( a `  3
)  =  1 ) ) ) )
8079rabbiia 2778 . 2  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) }  =  { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( ( ( a `  2 )  e.  NN  /\  (
( ( ( a `
 1 )  =  1  /\  ( a `
 3 )  =  1 )  \/  (
( ( a ` 
1 )  e.  (
ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) )  \/  (
( a `  1
)  =  0  /\  ( a `  3
)  =  0 ) ) )  \/  (
( a `  2
)  =  0  /\  ( a `  3
)  =  1 ) ) }
81 3nn0 9983 . . . . 5  |-  3  e.  NN0
82 ovex 5883 . . . . . 6  |-  ( 1 ... 3 )  e. 
_V
83 mzpproj 26815 . . . . . 6  |-  ( ( ( 1 ... 3
)  e.  _V  /\  2  e.  ( 1 ... 3 ) )  ->  ( a  e.  ( ZZ  ^m  (
1 ... 3 ) ) 
|->  ( a `  2
) )  e.  (mzPoly `  ( 1 ... 3
) ) )
8482, 56, 83mp2an 653 . . . . 5  |-  ( a  e.  ( ZZ  ^m  ( 1 ... 3
) )  |->  ( a `
 2 ) )  e.  (mzPoly `  (
1 ... 3 ) )
85 elnnrabdioph 26888 . . . . 5  |-  ( ( 3  e.  NN0  /\  ( a  e.  ( ZZ  ^m  ( 1 ... 3 ) ) 
|->  ( a `  2
) )  e.  (mzPoly `  ( 1 ... 3
) ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( a ` 
2 )  e.  NN }  e.  (Dioph `  3
) )
8681, 84, 85mp2an 653 . . . 4  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( a `  2 )  e.  NN }  e.  (Dioph `  3 )
87 mzpproj 26815 . . . . . . . . 9  |-  ( ( ( 1 ... 3
)  e.  _V  /\  1  e.  ( 1 ... 3 ) )  ->  ( a  e.  ( ZZ  ^m  (
1 ... 3 ) ) 
|->  ( a `  1
) )  e.  (mzPoly `  ( 1 ... 3
) ) )
8882, 11, 87mp2an 653 . . . . . . . 8  |-  ( a  e.  ( ZZ  ^m  ( 1 ... 3
) )  |->  ( a `
 1 ) )  e.  (mzPoly `  (
1 ... 3 ) )
89 1z 10053 . . . . . . . . 9  |-  1  e.  ZZ
90 mzpconstmpt 26818 . . . . . . . . 9  |-  ( ( ( 1 ... 3
)  e.  _V  /\  1  e.  ZZ )  ->  ( a  e.  ( ZZ  ^m  ( 1 ... 3 ) ) 
|->  1 )  e.  (mzPoly `  ( 1 ... 3
) ) )
9182, 89, 90mp2an 653 . . . . . . . 8  |-  ( a  e.  ( ZZ  ^m  ( 1 ... 3
) )  |->  1 )  e.  (mzPoly `  (
1 ... 3 ) )
92 eqrabdioph 26857 . . . . . . . 8  |-  ( ( 3  e.  NN0  /\  ( a  e.  ( ZZ  ^m  ( 1 ... 3 ) ) 
|->  ( a `  1
) )  e.  (mzPoly `  ( 1 ... 3
) )  /\  (
a  e.  ( ZZ 
^m  ( 1 ... 3 ) )  |->  1 )  e.  (mzPoly `  ( 1 ... 3
) ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( a ` 
1 )  =  1 }  e.  (Dioph ` 
3 ) )
9381, 88, 91, 92mp3an 1277 . . . . . . 7  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( a `  1 )  =  1 }  e.  (Dioph `  3 )
94 3nn 9878 . . . . . . . . . 10  |-  3  e.  NN
9594jm2.27dlem3 27104 . . . . . . . . 9  |-  3  e.  ( 1 ... 3
)
96 mzpproj 26815 . . . . . . . . 9  |-  ( ( ( 1 ... 3
)  e.  _V  /\  3  e.  ( 1 ... 3 ) )  ->  ( a  e.  ( ZZ  ^m  (
1 ... 3 ) ) 
|->  ( a `  3
) )  e.  (mzPoly `  ( 1 ... 3
) ) )
9782, 95, 96mp2an 653 . . . . . . . 8  |-  ( a  e.  ( ZZ  ^m  ( 1 ... 3
) )  |->  ( a `
 3 ) )  e.  (mzPoly `  (
1 ... 3 ) )
98 eqrabdioph 26857 . . . . . . . 8  |-  ( ( 3  e.  NN0  /\  ( a  e.  ( ZZ  ^m  ( 1 ... 3 ) ) 
|->  ( a `  3
) )  e.  (mzPoly `  ( 1 ... 3
) )  /\  (
a  e.  ( ZZ 
^m  ( 1 ... 3 ) )  |->  1 )  e.  (mzPoly `  ( 1 ... 3
) ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( a ` 
3 )  =  1 }  e.  (Dioph ` 
3 ) )
9981, 97, 91, 98mp3an 1277 . . . . . . 7  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( a `  3 )  =  1 }  e.  (Dioph `  3 )
100 anrabdioph 26860 . . . . . . 7  |-  ( ( { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( a ` 
1 )  =  1 }  e.  (Dioph ` 
3 )  /\  {
a  e.  ( NN0 
^m  ( 1 ... 3 ) )  |  ( a `  3
)  =  1 }  e.  (Dioph `  3
) )  ->  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( ( a `  1
)  =  1  /\  ( a `  3
)  =  1 ) }  e.  (Dioph ` 
3 ) )
10193, 99, 100mp2an 653 . . . . . 6  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( ( a `  1
)  =  1  /\  ( a `  3
)  =  1 ) }  e.  (Dioph ` 
3 )
102 expdiophlem2 27115 . . . . . 6  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( ( ( a ` 
1 )  e.  (
ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) }  e.  (Dioph `  3 )
103 orrabdioph 26861 . . . . . 6  |-  ( ( { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( ( a `
 1 )  =  1  /\  ( a `
 3 )  =  1 ) }  e.  (Dioph `  3 )  /\  { a  e.  ( NN0 
^m  ( 1 ... 3 ) )  |  ( ( ( a `
 1 )  e.  ( ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) }  e.  (Dioph `  3 ) )  ->  { a  e.  ( NN0  ^m  (
1 ... 3 ) )  |  ( ( ( a `  1 )  =  1  /\  (
a `  3 )  =  1 )  \/  ( ( ( a `
 1 )  e.  ( ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) ) }  e.  (Dioph `  3 ) )
104101, 102, 103mp2an 653 . . . . 5  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( ( ( a ` 
1 )  =  1  /\  ( a ` 
3 )  =  1 )  \/  ( ( ( a `  1
)  e.  ( ZZ>= ` 
2 )  /\  (
a `  2 )  e.  NN )  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) ) ) }  e.  (Dioph ` 
3 )
105 eq0rabdioph 26856 . . . . . . 7  |-  ( ( 3  e.  NN0  /\  ( a  e.  ( ZZ  ^m  ( 1 ... 3 ) ) 
|->  ( a `  1
) )  e.  (mzPoly `  ( 1 ... 3
) ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( a ` 
1 )  =  0 }  e.  (Dioph ` 
3 ) )
10681, 88, 105mp2an 653 . . . . . 6  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( a `  1 )  =  0 }  e.  (Dioph `  3 )
107 eq0rabdioph 26856 . . . . . . 7  |-  ( ( 3  e.  NN0  /\  ( a  e.  ( ZZ  ^m  ( 1 ... 3 ) ) 
|->  ( a `  3
) )  e.  (mzPoly `  ( 1 ... 3
) ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( a ` 
3 )  =  0 }  e.  (Dioph ` 
3 ) )
10881, 97, 107mp2an 653 . . . . . 6  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( a `  3 )  =  0 }  e.  (Dioph `  3 )
109 anrabdioph 26860 . . . . . 6  |-  ( ( { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( a ` 
1 )  =  0 }  e.  (Dioph ` 
3 )  /\  {
a  e.  ( NN0 
^m  ( 1 ... 3 ) )  |  ( a `  3
)  =  0 }  e.  (Dioph `  3
) )  ->  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( ( a `  1
)  =  0  /\  ( a `  3
)  =  0 ) }  e.  (Dioph ` 
3 ) )
110106, 108, 109mp2an 653 . . . . 5  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( ( a `  1
)  =  0  /\  ( a `  3
)  =  0 ) }  e.  (Dioph ` 
3 )
111 orrabdioph 26861 . . . . 5  |-  ( ( { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( ( ( a `  1 )  =  1  /\  (
a `  3 )  =  1 )  \/  ( ( ( a `
 1 )  e.  ( ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) ) }  e.  (Dioph `  3 )  /\  { a  e.  ( NN0 
^m  ( 1 ... 3 ) )  |  ( ( a ` 
1 )  =  0  /\  ( a ` 
3 )  =  0 ) }  e.  (Dioph `  3 ) )  ->  { a  e.  ( NN0  ^m  (
1 ... 3 ) )  |  ( ( ( ( a `  1
)  =  1  /\  ( a `  3
)  =  1 )  \/  ( ( ( a `  1 )  e.  ( ZZ>= `  2
)  /\  ( a `  2 )  e.  NN )  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) ) )  \/  ( ( a `
 1 )  =  0  /\  ( a `
 3 )  =  0 ) ) }  e.  (Dioph `  3
) )
112104, 110, 111mp2an 653 . . . 4  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( ( ( ( a `
 1 )  =  1  /\  ( a `
 3 )  =  1 )  \/  (
( ( a ` 
1 )  e.  (
ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) )  \/  (
( a `  1
)  =  0  /\  ( a `  3
)  =  0 ) ) }  e.  (Dioph `  3 )
113 anrabdioph 26860 . . . 4  |-  ( ( { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( a ` 
2 )  e.  NN }  e.  (Dioph `  3
)  /\  { a  e.  ( NN0  ^m  (
1 ... 3 ) )  |  ( ( ( ( a `  1
)  =  1  /\  ( a `  3
)  =  1 )  \/  ( ( ( a `  1 )  e.  ( ZZ>= `  2
)  /\  ( a `  2 )  e.  NN )  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) ) )  \/  ( ( a `
 1 )  =  0  /\  ( a `
 3 )  =  0 ) ) }  e.  (Dioph `  3
) )  ->  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( ( a `  2
)  e.  NN  /\  ( ( ( ( a `  1 )  =  1  /\  (
a `  3 )  =  1 )  \/  ( ( ( a `
 1 )  e.  ( ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) )  \/  (
( a `  1
)  =  0  /\  ( a `  3
)  =  0 ) ) ) }  e.  (Dioph `  3 ) )
11486, 112, 113mp2an 653 . . 3  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( ( a `  2
)  e.  NN  /\  ( ( ( ( a `  1 )  =  1  /\  (
a `  3 )  =  1 )  \/  ( ( ( a `
 1 )  e.  ( ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) )  \/  (
( a `  1
)  =  0  /\  ( a `  3
)  =  0 ) ) ) }  e.  (Dioph `  3 )
115 eq0rabdioph 26856 . . . . 5  |-  ( ( 3  e.  NN0  /\  ( a  e.  ( ZZ  ^m  ( 1 ... 3 ) ) 
|->  ( a `  2
) )  e.  (mzPoly `  ( 1 ... 3
) ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( a ` 
2 )  =  0 }  e.  (Dioph ` 
3 ) )
11681, 84, 115mp2an 653 . . . 4  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( a `  2 )  =  0 }  e.  (Dioph `  3 )
117 anrabdioph 26860 . . . 4  |-  ( ( { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( a ` 
2 )  =  0 }  e.  (Dioph ` 
3 )  /\  {
a  e.  ( NN0 
^m  ( 1 ... 3 ) )  |  ( a `  3
)  =  1 }  e.  (Dioph `  3
) )  ->  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( ( a `  2
)  =  0  /\  ( a `  3
)  =  1 ) }  e.  (Dioph ` 
3 ) )
118116, 99, 117mp2an 653 . . 3  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( ( a `  2
)  =  0  /\  ( a `  3
)  =  1 ) }  e.  (Dioph ` 
3 )
119 orrabdioph 26861 . . 3  |-  ( ( { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  |  ( ( a `
 2 )  e.  NN  /\  ( ( ( ( a ` 
1 )  =  1  /\  ( a ` 
3 )  =  1 )  \/  ( ( ( a `  1
)  e.  ( ZZ>= ` 
2 )  /\  (
a `  2 )  e.  NN )  /\  (
a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) ) )  \/  ( ( a `
 1 )  =  0  /\  ( a `
 3 )  =  0 ) ) ) }  e.  (Dioph ` 
3 )  /\  {
a  e.  ( NN0 
^m  ( 1 ... 3 ) )  |  ( ( a ` 
2 )  =  0  /\  ( a ` 
3 )  =  1 ) }  e.  (Dioph `  3 ) )  ->  { a  e.  ( NN0  ^m  (
1 ... 3 ) )  |  ( ( ( a `  2 )  e.  NN  /\  (
( ( ( a `
 1 )  =  1  /\  ( a `
 3 )  =  1 )  \/  (
( ( a ` 
1 )  e.  (
ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) )  \/  (
( a `  1
)  =  0  /\  ( a `  3
)  =  0 ) ) )  \/  (
( a `  2
)  =  0  /\  ( a `  3
)  =  1 ) ) }  e.  (Dioph `  3 ) )
120114, 118, 119mp2an 653 . 2  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( ( ( a ` 
2 )  e.  NN  /\  ( ( ( ( a `  1 )  =  1  /\  (
a `  3 )  =  1 )  \/  ( ( ( a `
 1 )  e.  ( ZZ>= `  2 )  /\  ( a `  2
)  e.  NN )  /\  ( a ` 
3 )  =  ( ( a `  1
) ^ ( a `
 2 ) ) ) )  \/  (
( a `  1
)  =  0  /\  ( a `  3
)  =  0 ) ) )  \/  (
( a `  2
)  =  0  /\  ( a `  3
)  =  1 ) ) }  e.  (Dioph `  3 )
12180, 120eqeltri 2353 1  |-  { a  e.  ( NN0  ^m  ( 1 ... 3
) )  |  ( a `  3 )  =  ( ( a `
 1 ) ^
( a `  2
) ) }  e.  (Dioph `  3 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   0cc0 8737   1c1 8738   NNcn 9746   2c2 9795   3c3 9796   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   ^cexp 11104  mzPolycmzp 26800  Diophcdioph 26834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-numer 12806  df-denom 12807  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-mzpcl 26801  df-mzp 26802  df-dioph 26835  df-squarenn 26926  df-pell1qr 26927  df-pell14qr 26928  df-pell1234qr 26929  df-pellfund 26930  df-rmx 26987  df-rmy 26988
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