MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2sqlem4 Unicode version

Theorem 2sqlem4 21011
Description: Lemma for 2sqlem5 21012. (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 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  N  e.  NN )
4 2sqlem5.2 . . . 4  |-  ( ph  ->  P  e.  Prime )
54adantr 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  P  e.  Prime )
6 2sqlem4.3 . . . 4  |-  ( ph  ->  A  e.  ZZ )
76adantr 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  A  e.  ZZ )
8 2sqlem4.4 . . . 4  |-  ( ph  ->  B  e.  ZZ )
98adantr 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  B  e.  ZZ )
10 2sqlem4.5 . . . 4  |-  ( ph  ->  C  e.  ZZ )
1110adantr 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  C  e.  ZZ )
12 2sqlem4.6 . . . 4  |-  ( ph  ->  D  e.  ZZ )
1312adantr 452 . . 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 452 . . 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 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  P  =  ( ( C ^
2 )  +  ( D ^ 2 ) ) )
18 simpr 448 . . 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 21010 . 2  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  N  e.  S )
202adantr 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  N  e.  NN )
214adantr 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  P  e.  Prime )
226znegcld 10302 . . . 4  |-  ( ph  -> 
-u A  e.  ZZ )
2322adantr 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  -u A  e.  ZZ )
248adantr 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  B  e.  ZZ )
2510adantr 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  C  e.  ZZ )
2612adantr 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  D  e.  ZZ )
276zcnd 10301 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
28 sqneg 11362 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u A ^ 2 )  =  ( A ^
2 ) )
2927, 28syl 16 . . . . . 6  |-  ( ph  ->  ( -u A ^
2 )  =  ( A ^ 2 ) )
3029oveq1d 6028 . . . . 5  |-  ( ph  ->  ( ( -u A ^ 2 )  +  ( B ^ 2 ) )  =  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )
3114, 30eqtr4d 2415 . . . 4  |-  ( ph  ->  ( N  x.  P
)  =  ( (
-u A ^ 2 )  +  ( B ^ 2 ) ) )
3231adantr 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  ( N  x.  P )  =  ( ( -u A ^
2 )  +  ( B ^ 2 ) ) )
3316adantr 452 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  P  =  ( ( C ^
2 )  +  ( D ^ 2 ) ) )
3412zcnd 10301 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
3527, 34mulneg1d 9411 . . . . . . 7  |-  ( ph  ->  ( -u A  x.  D )  =  -u ( A  x.  D
) )
3635oveq2d 6029 . . . . . 6  |-  ( ph  ->  ( ( C  x.  B )  +  (
-u A  x.  D
) )  =  ( ( C  x.  B
)  +  -u ( A  x.  D )
) )
3710, 8zmulcld 10306 . . . . . . . 8  |-  ( ph  ->  ( C  x.  B
)  e.  ZZ )
3837zcnd 10301 . . . . . . 7  |-  ( ph  ->  ( C  x.  B
)  e.  CC )
396, 12zmulcld 10306 . . . . . . . 8  |-  ( ph  ->  ( A  x.  D
)  e.  ZZ )
4039zcnd 10301 . . . . . . 7  |-  ( ph  ->  ( A  x.  D
)  e.  CC )
4138, 40negsubd 9342 . . . . . 6  |-  ( ph  ->  ( ( C  x.  B )  +  -u ( A  x.  D
) )  =  ( ( C  x.  B
)  -  ( A  x.  D ) ) )
4236, 41eqtrd 2412 . . . . 5  |-  ( ph  ->  ( ( C  x.  B )  +  (
-u A  x.  D
) )  =  ( ( C  x.  B
)  -  ( A  x.  D ) ) )
4342breq2d 4158 . . . 4  |-  ( ph  ->  ( P  ||  (
( C  x.  B
)  +  ( -u A  x.  D )
)  <->  P  ||  ( ( C  x.  B )  -  ( A  x.  D ) ) ) )
4443biimpar 472 . . 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 21010 . 2  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  N  e.  S )
46 prmz 13003 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
474, 46syl 16 . . . . 5  |-  ( ph  ->  P  e.  ZZ )
48 zsqcl 11372 . . . . . . . 8  |-  ( C  e.  ZZ  ->  ( C ^ 2 )  e.  ZZ )
4910, 48syl 16 . . . . . . 7  |-  ( ph  ->  ( C ^ 2 )  e.  ZZ )
502nnzd 10299 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
5149, 50zmulcld 10306 . . . . . 6  |-  ( ph  ->  ( ( C ^
2 )  x.  N
)  e.  ZZ )
52 zsqcl 11372 . . . . . . 7  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e.  ZZ )
536, 52syl 16 . . . . . 6  |-  ( ph  ->  ( A ^ 2 )  e.  ZZ )
5451, 53zsubcld 10305 . . . . 5  |-  ( ph  ->  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) )  e.  ZZ )
55 dvdsmul1 12791 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) )  e.  ZZ )  ->  P  ||  ( P  x.  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) ) ) )
5647, 54, 55syl2anc 643 . . . 4  |-  ( ph  ->  P  ||  ( P  x.  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) ) ) )
5710, 6zmulcld 10306 . . . . . . . . 9  |-  ( ph  ->  ( C  x.  A
)  e.  ZZ )
5857zcnd 10301 . . . . . . . 8  |-  ( ph  ->  ( C  x.  A
)  e.  CC )
5958sqcld 11441 . . . . . . 7  |-  ( ph  ->  ( ( C  x.  A ) ^ 2 )  e.  CC )
6038sqcld 11441 . . . . . . 7  |-  ( ph  ->  ( ( C  x.  B ) ^ 2 )  e.  CC )
6140sqcld 11441 . . . . . . 7  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  e.  CC )
6259, 60, 61pnpcand 9373 . . . . . 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 10301 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  CC )
6463, 27sqmuld 11455 . . . . . . . . . . 11  |-  ( ph  ->  ( ( C  x.  A ) ^ 2 )  =  ( ( C ^ 2 )  x.  ( A ^
2 ) ) )
658zcnd 10301 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  CC )
6663, 65sqmuld 11455 . . . . . . . . . . 11  |-  ( ph  ->  ( ( C  x.  B ) ^ 2 )  =  ( ( C ^ 2 )  x.  ( B ^
2 ) ) )
6764, 66oveq12d 6031 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( C ^ 2 )  x.  ( B ^ 2 ) ) ) )
6863sqcld 11441 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  e.  CC )
6953zcnd 10301 . . . . . . . . . . 11  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
7065sqcld 11441 . . . . . . . . . . 11  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
7168, 69, 70adddid 9038 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( C ^ 2 )  x.  ( B ^ 2 ) ) ) )
7267, 71eqtr4d 2415 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( ( C ^ 2 )  x.  ( ( A ^ 2 )  +  ( B ^ 2 ) ) ) )
732nncnd 9941 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
7447zcnd 10301 . . . . . . . . . . . . 13  |-  ( ph  ->  P  e.  CC )
7573, 74mulcomd 9035 . . . . . . . . . . . 12  |-  ( ph  ->  ( N  x.  P
)  =  ( P  x.  N ) )
7614, 75eqtr3d 2414 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( P  x.  N ) )
7776oveq2d 6029 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( C ^ 2 )  x.  ( P  x.  N ) ) )
7868, 74, 73mul12d 9200 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  ( P  x.  N )
)  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
7977, 78eqtrd 2412 . . . . . . . . 9  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
8072, 79eqtrd 2412 . . . . . . . 8  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
8127, 34sqmuld 11455 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  =  ( ( A ^ 2 )  x.  ( D ^
2 ) ) )
8234sqcld 11441 . . . . . . . . . . . . 13  |-  ( ph  ->  ( D ^ 2 )  e.  CC )
8369, 82mulcomd 9035 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A ^
2 )  x.  ( D ^ 2 ) )  =  ( ( D ^ 2 )  x.  ( A ^ 2 ) ) )
8481, 83eqtrd 2412 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  =  ( ( D ^ 2 )  x.  ( A ^
2 ) ) )
8564, 84oveq12d 6031 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( D ^ 2 )  x.  ( A ^ 2 ) ) ) )
8649zcnd 10301 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  e.  CC )
8786, 82, 69adddird 9039 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C ^ 2 )  +  ( D ^ 2 ) )  x.  ( A ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^
2 ) )  +  ( ( D ^
2 )  x.  ( A ^ 2 ) ) ) )
8885, 87eqtr4d 2415 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( ( ( C ^ 2 )  +  ( D ^ 2 ) )  x.  ( A ^
2 ) ) )
8916oveq1d 6028 . . . . . . . . 9  |-  ( ph  ->  ( P  x.  ( A ^ 2 ) )  =  ( ( ( C ^ 2 )  +  ( D ^
2 ) )  x.  ( A ^ 2 ) ) )
9088, 89eqtr4d 2415 . . . . . . . 8  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( P  x.  ( A ^
2 ) ) )
9180, 90oveq12d 6031 . . . . . . 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 10301 . . . . . . . 8  |-  ( ph  ->  ( ( C ^
2 )  x.  N
)  e.  CC )
9374, 92, 69subdid 9414 . . . . . . 7  |-  ( ph  ->  ( P  x.  (
( ( C ^
2 )  x.  N
)  -  ( A ^ 2 ) ) )  =  ( ( P  x.  ( ( C ^ 2 )  x.  N ) )  -  ( P  x.  ( A ^ 2 ) ) ) )
9491, 93eqtr4d 2415 . . . . . 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 2414 . . . . 5  |-  ( ph  ->  ( ( ( C  x.  B ) ^
2 )  -  (
( A  x.  D
) ^ 2 ) )  =  ( P  x.  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) ) ) )
96 subsq 11408 . . . . . 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 643 . . . . 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 2414 . . . 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 4170 . . 3  |-  ( ph  ->  P  ||  ( ( ( C  x.  B
)  +  ( A  x.  D ) )  x.  ( ( C  x.  B )  -  ( A  x.  D
) ) ) )
10037, 39zaddcld 10304 . . . 4  |-  ( ph  ->  ( ( C  x.  B )  +  ( A  x.  D ) )  e.  ZZ )
10137, 39zsubcld 10305 . . . 4  |-  ( ph  ->  ( ( C  x.  B )  -  ( A  x.  D )
)  e.  ZZ )
102 euclemma 13028 . . . 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 1184 . . 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 202 . 2  |-  ( ph  ->  ( P  ||  (
( C  x.  B
)  +  ( A  x.  D ) )  \/  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) ) )
10519, 45, 104mpjaodan 762 1  |-  ( ph  ->  N  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4146    e. cmpt 4200   ran crn 4812   ` cfv 5387  (class class class)co 6013   CCcc 8914    + caddc 8919    x. cmul 8921    - cmin 9216   -ucneg 9217   NNcn 9925   2c2 9974   ZZcz 10207   ^cexp 11302   abscabs 11959    || cdivides 12772   Primecprime 12999   ZZ [ _i ]cgz 13217
This theorem is referenced by:  2sqlem5  21012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fl 11122  df-mod 11171  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-dvds 12773  df-gcd 12927  df-prm 13000  df-gz 13218
  Copyright terms: Public domain W3C validator