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Theorem expdioph 27219
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 6808 . . . . . . . . . . . . 13  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  a : ( 1 ... 3 ) --> NN0 )
4 df-2 9820 . . . . . . . . . . . . . . 15  |-  2  =  ( 1  +  1 )
5 df-3 9821 . . . . . . . . . . . . . . . 16  |-  3  =  ( 2  +  1 )
6 ssid 3210 . . . . . . . . . . . . . . . 16  |-  ( 1 ... 3 )  C_  ( 1 ... 3
)
75, 6jm2.27dlem5 27209 . . . . . . . . . . . . . . 15  |-  ( 1 ... 2 )  C_  ( 1 ... 3
)
84, 7jm2.27dlem5 27209 . . . . . . . . . . . . . 14  |-  ( 1 ... 1 )  C_  ( 1 ... 3
)
9 1nn 9773 . . . . . . . . . . . . . . 15  |-  1  e.  NN
109jm2.27dlem3 27207 . . . . . . . . . . . . . 14  |-  1  e.  ( 1 ... 1
)
118, 10sselii 3190 . . . . . . . . . . . . 13  |-  1  e.  ( 1 ... 3
)
12 ffvelrn 5679 . . . . . . . . . . . . 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 9983 . . . . . . . . . . 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 10310 . . . . . . . . . . . 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 10061 . . . . . . . . . . . . . . . 16  |-  ( ( a `  2 )  e.  NN  ->  (
a `  2 )  e.  ZZ )
27 1exp 11147 . . . . . . . . . . . . . . . 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 2307 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( a `
 3 )  =  ( 1 ^ (
a `  2 )
)  <->  ( a ` 
3 )  =  1 ) )
31 oveq1 5881 . . . . . . . . . . . . . . 15  |-  ( ( a `  1 )  =  1  ->  (
( a `  1
) ^ ( a `
 2 ) )  =  ( 1 ^ ( a `  2
) ) )
3231eqeq2d 2307 . . . . . . . . . . . . . 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 11153 . . . . . . . . . . . . . 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 2307 . . . . . . . . . . . 12  |-  ( ( a  e.  ( NN0 
^m  ( 1 ... 3 ) )  /\  ( a `  2
)  e.  NN )  ->  ( ( a `
 3 )  =  ( 0 ^ (
a `  2 )
)  <->  ( a ` 
3 )  =  0 ) )
43 oveq1 5881 . . . . . . . . . . . . . 14  |-  ( ( a `  1 )  =  0  ->  (
( a `  1
) ^ ( a `
 2 ) )  =  ( 0 ^ ( a `  2
) ) )
4443eqeq2d 2307 . . . . . . . . . . . . 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 9893 . . . . . . . . . . . 12  |-  2  e.  NN
5554jm2.27dlem3 27207 . . . . . . . . . . 11  |-  2  e.  ( 1 ... 2
)
567, 55sselii 3190 . . . . . . . . . 10  |-  2  e.  ( 1 ... 3
)
57 ffvelrn 5679 . . . . . . . . . 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 9983 . . . . . . . . . . 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 9793 . . . . . . . . . . 11  |-  -.  0  e.  NN
63 eleq1 2356 . . . . . . . . . . 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 10036 . . . . . . . . . . 11  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
a `  1 )  e.  CC )
6968exp0d 11255 . . . . . . . . . 10  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
( a `  1
) ^ 0 )  =  1 )
7069eqeq2d 2307 . . . . . . . . 9  |-  ( a  e.  ( NN0  ^m  ( 1 ... 3
) )  ->  (
( a `  3
)  =  ( ( a `  1 ) ^ 0 )  <->  ( a `  3 )  =  1 ) )
71 oveq2 5882 . . . . . . . . . . 11  |-  ( ( a `  2 )  =  0  ->  (
( a `  1
) ^ ( a `
 2 ) )  =  ( ( a `
 1 ) ^
0 ) )
7271eqeq2d 2307 . . . . . . . . . 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 2791 . 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 9999 . . . . 5  |-  3  e.  NN0
82 ovex 5899 . . . . . 6  |-  ( 1 ... 3 )  e. 
_V
83 mzpproj 26918 . . . . . 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 26991 . . . . 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 26918 . . . . . . . . 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 10069 . . . . . . . . 9  |-  1  e.  ZZ
90 mzpconstmpt 26921 . . . . . . . . 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 26960 . . . . . . . 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 9894 . . . . . . . . . 10  |-  3  e.  NN
9594jm2.27dlem3 27207 . . . . . . . . 9  |-  3  e.  ( 1 ... 3
)
96 mzpproj 26918 . . . . . . . . 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 26960 . . . . . . . 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 26963 . . . . . . 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 27218 . . . . . 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 26964 . . . . . 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 26959 . . . . . . 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 26959 . . . . . . 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 26963 . . . . . 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 26964 . . . . 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 26963 . . . 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 26959 . . . . 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 26963 . . . 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 26964 . . 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 2366 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 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   0cc0 8753   1c1 8754   NNcn 9762   2c2 9811   3c3 9812   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   ^cexp 11120  mzPolycmzp 26903  Diophcdioph 26937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-gcd 12702  df-prm 12775  df-numer 12822  df-denom 12823  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-mzpcl 26904  df-mzp 26905  df-dioph 26938  df-squarenn 27029  df-pell1qr 27030  df-pell14qr 27031  df-pell1234qr 27032  df-pellfund 27033  df-rmx 27090  df-rmy 27091
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