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Theorem 2sqlem4 21108
Description: Lemma for 2sqlem5 21109. (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 21107 . 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 10337 . . . 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 10336 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
28 sqneg 11401 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u A ^ 2 )  =  ( A ^
2 ) )
2927, 28syl 16 . . . . . 6  |-  ( ph  ->  ( -u A ^
2 )  =  ( A ^ 2 ) )
3029oveq1d 6059 . . . . 5  |-  ( ph  ->  ( ( -u A ^ 2 )  +  ( B ^ 2 ) )  =  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )
3114, 30eqtr4d 2443 . . . 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 10336 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
3527, 34mulneg1d 9446 . . . . . . 7  |-  ( ph  ->  ( -u A  x.  D )  =  -u ( A  x.  D
) )
3635oveq2d 6060 . . . . . 6  |-  ( ph  ->  ( ( C  x.  B )  +  (
-u A  x.  D
) )  =  ( ( C  x.  B
)  +  -u ( A  x.  D )
) )
3710, 8zmulcld 10341 . . . . . . . 8  |-  ( ph  ->  ( C  x.  B
)  e.  ZZ )
3837zcnd 10336 . . . . . . 7  |-  ( ph  ->  ( C  x.  B
)  e.  CC )
396, 12zmulcld 10341 . . . . . . . 8  |-  ( ph  ->  ( A  x.  D
)  e.  ZZ )
4039zcnd 10336 . . . . . . 7  |-  ( ph  ->  ( A  x.  D
)  e.  CC )
4138, 40negsubd 9377 . . . . . 6  |-  ( ph  ->  ( ( C  x.  B )  +  -u ( A  x.  D
) )  =  ( ( C  x.  B
)  -  ( A  x.  D ) ) )
4236, 41eqtrd 2440 . . . . 5  |-  ( ph  ->  ( ( C  x.  B )  +  (
-u A  x.  D
) )  =  ( ( C  x.  B
)  -  ( A  x.  D ) ) )
4342breq2d 4188 . . . 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 21107 . 2  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  N  e.  S )
46 prmz 13042 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
474, 46syl 16 . . . . 5  |-  ( ph  ->  P  e.  ZZ )
48 zsqcl 11411 . . . . . . . 8  |-  ( C  e.  ZZ  ->  ( C ^ 2 )  e.  ZZ )
4910, 48syl 16 . . . . . . 7  |-  ( ph  ->  ( C ^ 2 )  e.  ZZ )
502nnzd 10334 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
5149, 50zmulcld 10341 . . . . . 6  |-  ( ph  ->  ( ( C ^
2 )  x.  N
)  e.  ZZ )
52 zsqcl 11411 . . . . . . 7  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e.  ZZ )
536, 52syl 16 . . . . . 6  |-  ( ph  ->  ( A ^ 2 )  e.  ZZ )
5451, 53zsubcld 10340 . . . . 5  |-  ( ph  ->  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) )  e.  ZZ )
55 dvdsmul1 12830 . . . . 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 10341 . . . . . . . . 9  |-  ( ph  ->  ( C  x.  A
)  e.  ZZ )
5857zcnd 10336 . . . . . . . 8  |-  ( ph  ->  ( C  x.  A
)  e.  CC )
5958sqcld 11480 . . . . . . 7  |-  ( ph  ->  ( ( C  x.  A ) ^ 2 )  e.  CC )
6038sqcld 11480 . . . . . . 7  |-  ( ph  ->  ( ( C  x.  B ) ^ 2 )  e.  CC )
6140sqcld 11480 . . . . . . 7  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  e.  CC )
6259, 60, 61pnpcand 9408 . . . . . 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 10336 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  CC )
6463, 27sqmuld 11494 . . . . . . . . . . 11  |-  ( ph  ->  ( ( C  x.  A ) ^ 2 )  =  ( ( C ^ 2 )  x.  ( A ^
2 ) ) )
658zcnd 10336 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  CC )
6663, 65sqmuld 11494 . . . . . . . . . . 11  |-  ( ph  ->  ( ( C  x.  B ) ^ 2 )  =  ( ( C ^ 2 )  x.  ( B ^
2 ) ) )
6764, 66oveq12d 6062 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( C ^ 2 )  x.  ( B ^ 2 ) ) ) )
6863sqcld 11480 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  e.  CC )
6953zcnd 10336 . . . . . . . . . . 11  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
7065sqcld 11480 . . . . . . . . . . 11  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
7168, 69, 70adddid 9072 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( C ^ 2 )  x.  ( B ^ 2 ) ) ) )
7267, 71eqtr4d 2443 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( ( C ^ 2 )  x.  ( ( A ^ 2 )  +  ( B ^ 2 ) ) ) )
732nncnd 9976 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
7447zcnd 10336 . . . . . . . . . . . . 13  |-  ( ph  ->  P  e.  CC )
7573, 74mulcomd 9069 . . . . . . . . . . . 12  |-  ( ph  ->  ( N  x.  P
)  =  ( P  x.  N ) )
7614, 75eqtr3d 2442 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( P  x.  N ) )
7776oveq2d 6060 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( C ^ 2 )  x.  ( P  x.  N ) ) )
7868, 74, 73mul12d 9235 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  ( P  x.  N )
)  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
7977, 78eqtrd 2440 . . . . . . . . 9  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
8072, 79eqtrd 2440 . . . . . . . 8  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
8127, 34sqmuld 11494 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  =  ( ( A ^ 2 )  x.  ( D ^
2 ) ) )
8234sqcld 11480 . . . . . . . . . . . . 13  |-  ( ph  ->  ( D ^ 2 )  e.  CC )
8369, 82mulcomd 9069 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A ^
2 )  x.  ( D ^ 2 ) )  =  ( ( D ^ 2 )  x.  ( A ^ 2 ) ) )
8481, 83eqtrd 2440 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  =  ( ( D ^ 2 )  x.  ( A ^
2 ) ) )
8564, 84oveq12d 6062 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( D ^ 2 )  x.  ( A ^ 2 ) ) ) )
8649zcnd 10336 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  e.  CC )
8786, 82, 69adddird 9073 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C ^ 2 )  +  ( D ^ 2 ) )  x.  ( A ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^
2 ) )  +  ( ( D ^
2 )  x.  ( A ^ 2 ) ) ) )
8885, 87eqtr4d 2443 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( ( ( C ^ 2 )  +  ( D ^ 2 ) )  x.  ( A ^
2 ) ) )
8916oveq1d 6059 . . . . . . . . 9  |-  ( ph  ->  ( P  x.  ( A ^ 2 ) )  =  ( ( ( C ^ 2 )  +  ( D ^
2 ) )  x.  ( A ^ 2 ) ) )
9088, 89eqtr4d 2443 . . . . . . . 8  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( P  x.  ( A ^
2 ) ) )
9180, 90oveq12d 6062 . . . . . . 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 10336 . . . . . . . 8  |-  ( ph  ->  ( ( C ^
2 )  x.  N
)  e.  CC )
9374, 92, 69subdid 9449 . . . . . . 7  |-  ( ph  ->  ( P  x.  (
( ( C ^
2 )  x.  N
)  -  ( A ^ 2 ) ) )  =  ( ( P  x.  ( ( C ^ 2 )  x.  N ) )  -  ( P  x.  ( A ^ 2 ) ) ) )
9491, 93eqtr4d 2443 . . . . . 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 2442 . . . . 5  |-  ( ph  ->  ( ( ( C  x.  B ) ^
2 )  -  (
( A  x.  D
) ^ 2 ) )  =  ( P  x.  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) ) ) )
96 subsq 11447 . . . . . 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 2442 . . . 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 4200 . . 3  |-  ( ph  ->  P  ||  ( ( ( C  x.  B
)  +  ( A  x.  D ) )  x.  ( ( C  x.  B )  -  ( A  x.  D
) ) ) )
10037, 39zaddcld 10339 . . . 4  |-  ( ph  ->  ( ( C  x.  B )  +  ( A  x.  D ) )  e.  ZZ )
10137, 39zsubcld 10340 . . . 4  |-  ( ph  ->  ( ( C  x.  B )  -  ( A  x.  D )
)  e.  ZZ )
102 euclemma 13067 . . . 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 1721   class class class wbr 4176    e. cmpt 4230   ran crn 4842   ` cfv 5417  (class class class)co 6044   CCcc 8948    + caddc 8953    x. cmul 8955    - cmin 9251   -ucneg 9252   NNcn 9960   2c2 10009   ZZcz 10242   ^cexp 11341   abscabs 11998    || cdivides 12811   Primecprime 13038   ZZ [ _i ]cgz 13256
This theorem is referenced by:  2sqlem5  21109
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-fl 11161  df-mod 11210  df-seq 11283  df-exp 11342  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-dvds 12812  df-gcd 12966  df-prm 13039  df-gz 13257
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