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Theorem 2sqlem4 20606
Description: Lemma for 2sqlem5 20607. (Contributed by Mario Carneiro, 20-Jun-2015.)
Hypotheses
Ref Expression
2sq.1  |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w
) ^ 2 ) )
2sqlem5.1  |-  ( ph  ->  N  e.  NN )
2sqlem5.2  |-  ( ph  ->  P  e.  Prime )
2sqlem4.3  |-  ( ph  ->  A  e.  ZZ )
2sqlem4.4  |-  ( ph  ->  B  e.  ZZ )
2sqlem4.5  |-  ( ph  ->  C  e.  ZZ )
2sqlem4.6  |-  ( ph  ->  D  e.  ZZ )
2sqlem4.7  |-  ( ph  ->  ( N  x.  P
)  =  ( ( A ^ 2 )  +  ( B ^
2 ) ) )
2sqlem4.8  |-  ( ph  ->  P  =  ( ( C ^ 2 )  +  ( D ^
2 ) ) )
Assertion
Ref Expression
2sqlem4  |-  ( ph  ->  N  e.  S )

Proof of Theorem 2sqlem4
StepHypRef Expression
1 2sq.1 . . 3  |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w
) ^ 2 ) )
2 2sqlem5.1 . . . 4  |-  ( ph  ->  N  e.  NN )
32adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  N  e.  NN )
4 2sqlem5.2 . . . 4  |-  ( ph  ->  P  e.  Prime )
54adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  P  e.  Prime )
6 2sqlem4.3 . . . 4  |-  ( ph  ->  A  e.  ZZ )
76adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  A  e.  ZZ )
8 2sqlem4.4 . . . 4  |-  ( ph  ->  B  e.  ZZ )
98adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  B  e.  ZZ )
10 2sqlem4.5 . . . 4  |-  ( ph  ->  C  e.  ZZ )
1110adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  C  e.  ZZ )
12 2sqlem4.6 . . . 4  |-  ( ph  ->  D  e.  ZZ )
1312adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  D  e.  ZZ )
14 2sqlem4.7 . . . 4  |-  ( ph  ->  ( N  x.  P
)  =  ( ( A ^ 2 )  +  ( B ^
2 ) ) )
1514adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  ( N  x.  P )  =  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )
16 2sqlem4.8 . . . 4  |-  ( ph  ->  P  =  ( ( C ^ 2 )  +  ( D ^
2 ) ) )
1716adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  P  =  ( ( C ^
2 )  +  ( D ^ 2 ) ) )
18 simpr 447 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )
191, 3, 5, 7, 9, 11, 13, 15, 17, 182sqlem3 20605 . 2  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  N  e.  S )
202adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  N  e.  NN )
214adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  P  e.  Prime )
226znegcld 10119 . . . 4  |-  ( ph  -> 
-u A  e.  ZZ )
2322adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  -u A  e.  ZZ )
248adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  B  e.  ZZ )
2510adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  C  e.  ZZ )
2612adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  D  e.  ZZ )
276zcnd 10118 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
28 sqneg 11164 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u A ^ 2 )  =  ( A ^
2 ) )
2927, 28syl 15 . . . . . 6  |-  ( ph  ->  ( -u A ^
2 )  =  ( A ^ 2 ) )
3029oveq1d 5873 . . . . 5  |-  ( ph  ->  ( ( -u A ^ 2 )  +  ( B ^ 2 ) )  =  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )
3114, 30eqtr4d 2318 . . . 4  |-  ( ph  ->  ( N  x.  P
)  =  ( (
-u A ^ 2 )  +  ( B ^ 2 ) ) )
3231adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  ( N  x.  P )  =  ( ( -u A ^
2 )  +  ( B ^ 2 ) ) )
3316adantr 451 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  P  =  ( ( C ^
2 )  +  ( D ^ 2 ) ) )
3412zcnd 10118 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
3527, 34mulneg1d 9232 . . . . . . 7  |-  ( ph  ->  ( -u A  x.  D )  =  -u ( A  x.  D
) )
3635oveq2d 5874 . . . . . 6  |-  ( ph  ->  ( ( C  x.  B )  +  (
-u A  x.  D
) )  =  ( ( C  x.  B
)  +  -u ( A  x.  D )
) )
3710, 8zmulcld 10123 . . . . . . . 8  |-  ( ph  ->  ( C  x.  B
)  e.  ZZ )
3837zcnd 10118 . . . . . . 7  |-  ( ph  ->  ( C  x.  B
)  e.  CC )
396, 12zmulcld 10123 . . . . . . . 8  |-  ( ph  ->  ( A  x.  D
)  e.  ZZ )
4039zcnd 10118 . . . . . . 7  |-  ( ph  ->  ( A  x.  D
)  e.  CC )
4138, 40negsubd 9163 . . . . . 6  |-  ( ph  ->  ( ( C  x.  B )  +  -u ( A  x.  D
) )  =  ( ( C  x.  B
)  -  ( A  x.  D ) ) )
4236, 41eqtrd 2315 . . . . 5  |-  ( ph  ->  ( ( C  x.  B )  +  (
-u A  x.  D
) )  =  ( ( C  x.  B
)  -  ( A  x.  D ) ) )
4342breq2d 4035 . . . 4  |-  ( ph  ->  ( P  ||  (
( C  x.  B
)  +  ( -u A  x.  D )
)  <->  P  ||  ( ( C  x.  B )  -  ( A  x.  D ) ) ) )
4443biimpar 471 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  P  ||  (
( C  x.  B
)  +  ( -u A  x.  D )
) )
451, 20, 21, 23, 24, 25, 26, 32, 33, 442sqlem3 20605 . 2  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  N  e.  S )
46 prmz 12762 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
474, 46syl 15 . . . . 5  |-  ( ph  ->  P  e.  ZZ )
48 zsqcl 11174 . . . . . . . 8  |-  ( C  e.  ZZ  ->  ( C ^ 2 )  e.  ZZ )
4910, 48syl 15 . . . . . . 7  |-  ( ph  ->  ( C ^ 2 )  e.  ZZ )
502nnzd 10116 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
5149, 50zmulcld 10123 . . . . . 6  |-  ( ph  ->  ( ( C ^
2 )  x.  N
)  e.  ZZ )
52 zsqcl 11174 . . . . . . 7  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e.  ZZ )
536, 52syl 15 . . . . . 6  |-  ( ph  ->  ( A ^ 2 )  e.  ZZ )
5451, 53zsubcld 10122 . . . . 5  |-  ( ph  ->  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) )  e.  ZZ )
55 dvdsmul1 12550 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) )  e.  ZZ )  ->  P  ||  ( P  x.  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) ) ) )
5647, 54, 55syl2anc 642 . . . 4  |-  ( ph  ->  P  ||  ( P  x.  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) ) ) )
5710, 6zmulcld 10123 . . . . . . . . 9  |-  ( ph  ->  ( C  x.  A
)  e.  ZZ )
5857zcnd 10118 . . . . . . . 8  |-  ( ph  ->  ( C  x.  A
)  e.  CC )
5958sqcld 11243 . . . . . . 7  |-  ( ph  ->  ( ( C  x.  A ) ^ 2 )  e.  CC )
6038sqcld 11243 . . . . . . 7  |-  ( ph  ->  ( ( C  x.  B ) ^ 2 )  e.  CC )
6140sqcld 11243 . . . . . . 7  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  e.  CC )
6259, 60, 61pnpcand 9194 . . . . . 6  |-  ( ph  ->  ( ( ( ( C  x.  A ) ^ 2 )  +  ( ( C  x.  B ) ^ 2 ) )  -  (
( ( C  x.  A ) ^ 2 )  +  ( ( A  x.  D ) ^ 2 ) ) )  =  ( ( ( C  x.  B
) ^ 2 )  -  ( ( A  x.  D ) ^
2 ) ) )
6310zcnd 10118 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  CC )
6463, 27sqmuld 11257 . . . . . . . . . . 11  |-  ( ph  ->  ( ( C  x.  A ) ^ 2 )  =  ( ( C ^ 2 )  x.  ( A ^
2 ) ) )
658zcnd 10118 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  CC )
6663, 65sqmuld 11257 . . . . . . . . . . 11  |-  ( ph  ->  ( ( C  x.  B ) ^ 2 )  =  ( ( C ^ 2 )  x.  ( B ^
2 ) ) )
6764, 66oveq12d 5876 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( C ^ 2 )  x.  ( B ^ 2 ) ) ) )
6863sqcld 11243 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  e.  CC )
6953zcnd 10118 . . . . . . . . . . 11  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
7065sqcld 11243 . . . . . . . . . . 11  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
7168, 69, 70adddid 8859 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( C ^ 2 )  x.  ( B ^ 2 ) ) ) )
7267, 71eqtr4d 2318 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( ( C ^ 2 )  x.  ( ( A ^ 2 )  +  ( B ^ 2 ) ) ) )
732nncnd 9762 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
7447zcnd 10118 . . . . . . . . . . . . 13  |-  ( ph  ->  P  e.  CC )
7573, 74mulcomd 8856 . . . . . . . . . . . 12  |-  ( ph  ->  ( N  x.  P
)  =  ( P  x.  N ) )
7614, 75eqtr3d 2317 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( P  x.  N ) )
7776oveq2d 5874 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( C ^ 2 )  x.  ( P  x.  N ) ) )
7868, 74, 73mul12d 9021 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  ( P  x.  N )
)  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
7977, 78eqtrd 2315 . . . . . . . . 9  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
8072, 79eqtrd 2315 . . . . . . . 8  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
8127, 34sqmuld 11257 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  =  ( ( A ^ 2 )  x.  ( D ^
2 ) ) )
8234sqcld 11243 . . . . . . . . . . . . 13  |-  ( ph  ->  ( D ^ 2 )  e.  CC )
8369, 82mulcomd 8856 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A ^
2 )  x.  ( D ^ 2 ) )  =  ( ( D ^ 2 )  x.  ( A ^ 2 ) ) )
8481, 83eqtrd 2315 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  =  ( ( D ^ 2 )  x.  ( A ^
2 ) ) )
8564, 84oveq12d 5876 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( D ^ 2 )  x.  ( A ^ 2 ) ) ) )
8649zcnd 10118 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  e.  CC )
8786, 82, 69adddird 8860 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C ^ 2 )  +  ( D ^ 2 ) )  x.  ( A ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^
2 ) )  +  ( ( D ^
2 )  x.  ( A ^ 2 ) ) ) )
8885, 87eqtr4d 2318 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( ( ( C ^ 2 )  +  ( D ^ 2 ) )  x.  ( A ^
2 ) ) )
8916oveq1d 5873 . . . . . . . . 9  |-  ( ph  ->  ( P  x.  ( A ^ 2 ) )  =  ( ( ( C ^ 2 )  +  ( D ^
2 ) )  x.  ( A ^ 2 ) ) )
9088, 89eqtr4d 2318 . . . . . . . 8  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( P  x.  ( A ^
2 ) ) )
9180, 90oveq12d 5876 . . . . . . 7  |-  ( ph  ->  ( ( ( ( C  x.  A ) ^ 2 )  +  ( ( C  x.  B ) ^ 2 ) )  -  (
( ( C  x.  A ) ^ 2 )  +  ( ( A  x.  D ) ^ 2 ) ) )  =  ( ( P  x.  ( ( C ^ 2 )  x.  N ) )  -  ( P  x.  ( A ^ 2 ) ) ) )
9251zcnd 10118 . . . . . . . 8  |-  ( ph  ->  ( ( C ^
2 )  x.  N
)  e.  CC )
9374, 92, 69subdid 9235 . . . . . . 7  |-  ( ph  ->  ( P  x.  (
( ( C ^
2 )  x.  N
)  -  ( A ^ 2 ) ) )  =  ( ( P  x.  ( ( C ^ 2 )  x.  N ) )  -  ( P  x.  ( A ^ 2 ) ) ) )
9491, 93eqtr4d 2318 . . . . . 6  |-  ( ph  ->  ( ( ( ( C  x.  A ) ^ 2 )  +  ( ( C  x.  B ) ^ 2 ) )  -  (
( ( C  x.  A ) ^ 2 )  +  ( ( A  x.  D ) ^ 2 ) ) )  =  ( P  x.  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) ) ) )
9562, 94eqtr3d 2317 . . . . 5  |-  ( ph  ->  ( ( ( C  x.  B ) ^
2 )  -  (
( A  x.  D
) ^ 2 ) )  =  ( P  x.  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) ) ) )
96 subsq 11210 . . . . . 6  |-  ( ( ( C  x.  B
)  e.  CC  /\  ( A  x.  D
)  e.  CC )  ->  ( ( ( C  x.  B ) ^ 2 )  -  ( ( A  x.  D ) ^ 2 ) )  =  ( ( ( C  x.  B )  +  ( A  x.  D ) )  x.  ( ( C  x.  B )  -  ( A  x.  D ) ) ) )
9738, 40, 96syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( ( C  x.  B ) ^
2 )  -  (
( A  x.  D
) ^ 2 ) )  =  ( ( ( C  x.  B
)  +  ( A  x.  D ) )  x.  ( ( C  x.  B )  -  ( A  x.  D
) ) ) )
9895, 97eqtr3d 2317 . . . 4  |-  ( ph  ->  ( P  x.  (
( ( C ^
2 )  x.  N
)  -  ( A ^ 2 ) ) )  =  ( ( ( C  x.  B
)  +  ( A  x.  D ) )  x.  ( ( C  x.  B )  -  ( A  x.  D
) ) ) )
9956, 98breqtrd 4047 . . 3  |-  ( ph  ->  P  ||  ( ( ( C  x.  B
)  +  ( A  x.  D ) )  x.  ( ( C  x.  B )  -  ( A  x.  D
) ) ) )
10037, 39zaddcld 10121 . . . 4  |-  ( ph  ->  ( ( C  x.  B )  +  ( A  x.  D ) )  e.  ZZ )
10137, 39zsubcld 10122 . . . 4  |-  ( ph  ->  ( ( C  x.  B )  -  ( A  x.  D )
)  e.  ZZ )
102 euclemma 12787 . . . 4  |-  ( ( P  e.  Prime  /\  (
( C  x.  B
)  +  ( A  x.  D ) )  e.  ZZ  /\  (
( C  x.  B
)  -  ( A  x.  D ) )  e.  ZZ )  -> 
( P  ||  (
( ( C  x.  B )  +  ( A  x.  D ) )  x.  ( ( C  x.  B )  -  ( A  x.  D ) ) )  <-> 
( P  ||  (
( C  x.  B
)  +  ( A  x.  D ) )  \/  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) ) ) )
1034, 100, 101, 102syl3anc 1182 . . 3  |-  ( ph  ->  ( P  ||  (
( ( C  x.  B )  +  ( A  x.  D ) )  x.  ( ( C  x.  B )  -  ( A  x.  D ) ) )  <-> 
( P  ||  (
( C  x.  B
)  +  ( A  x.  D ) )  \/  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) ) ) )
10499, 103mpbid 201 . 2  |-  ( ph  ->  ( P  ||  (
( C  x.  B
)  +  ( A  x.  D ) )  \/  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) ) )
10519, 45, 104mpjaodan 761 1  |-  ( ph  ->  N  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   CCcc 8735    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038   NNcn 9746   2c2 9795   ZZcz 10024   ^cexp 11104   abscabs 11719    || cdivides 12531   Primecprime 12758   ZZ [ _i ]cgz 12976
This theorem is referenced by:  2sqlem5  20607
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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
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-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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  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-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-gz 12977
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