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Theorem 2sqlem4 22681
Description: Lemma for 2sqlem5 22682. (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 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  N  e.  NN )
4 2sqlem5.2 . . . 4  |-  ( ph  ->  P  e.  Prime )
54adantr 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  P  e.  Prime )
6 2sqlem4.3 . . . 4  |-  ( ph  ->  A  e.  ZZ )
76adantr 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  A  e.  ZZ )
8 2sqlem4.4 . . . 4  |-  ( ph  ->  B  e.  ZZ )
98adantr 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  B  e.  ZZ )
10 2sqlem4.5 . . . 4  |-  ( ph  ->  C  e.  ZZ )
1110adantr 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  C  e.  ZZ )
12 2sqlem4.6 . . . 4  |-  ( ph  ->  D  e.  ZZ )
1312adantr 465 . . 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 465 . . 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 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  P  =  ( ( C ^
2 )  +  ( D ^ 2 ) ) )
18 simpr 461 . . 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 22680 . 2  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  +  ( A  x.  D ) ) )  ->  N  e.  S )
202adantr 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  N  e.  NN )
214adantr 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  P  e.  Prime )
226znegcld 10741 . . . 4  |-  ( ph  -> 
-u A  e.  ZZ )
2322adantr 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  -u A  e.  ZZ )
248adantr 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  B  e.  ZZ )
2510adantr 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  C  e.  ZZ )
2612adantr 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  D  e.  ZZ )
276zcnd 10740 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
28 sqneg 11918 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u A ^ 2 )  =  ( A ^
2 ) )
2927, 28syl 16 . . . . . 6  |-  ( ph  ->  ( -u A ^
2 )  =  ( A ^ 2 ) )
3029oveq1d 6101 . . . . 5  |-  ( ph  ->  ( ( -u A ^ 2 )  +  ( B ^ 2 ) )  =  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )
3114, 30eqtr4d 2473 . . . 4  |-  ( ph  ->  ( N  x.  P
)  =  ( (
-u A ^ 2 )  +  ( B ^ 2 ) ) )
3231adantr 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  ( N  x.  P )  =  ( ( -u A ^
2 )  +  ( B ^ 2 ) ) )
3316adantr 465 . . 3  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  P  =  ( ( C ^
2 )  +  ( D ^ 2 ) ) )
3412zcnd 10740 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
3527, 34mulneg1d 9789 . . . . . . 7  |-  ( ph  ->  ( -u A  x.  D )  =  -u ( A  x.  D
) )
3635oveq2d 6102 . . . . . 6  |-  ( ph  ->  ( ( C  x.  B )  +  (
-u A  x.  D
) )  =  ( ( C  x.  B
)  +  -u ( A  x.  D )
) )
3710, 8zmulcld 10745 . . . . . . . 8  |-  ( ph  ->  ( C  x.  B
)  e.  ZZ )
3837zcnd 10740 . . . . . . 7  |-  ( ph  ->  ( C  x.  B
)  e.  CC )
396, 12zmulcld 10745 . . . . . . . 8  |-  ( ph  ->  ( A  x.  D
)  e.  ZZ )
4039zcnd 10740 . . . . . . 7  |-  ( ph  ->  ( A  x.  D
)  e.  CC )
4138, 40negsubd 9717 . . . . . 6  |-  ( ph  ->  ( ( C  x.  B )  +  -u ( A  x.  D
) )  =  ( ( C  x.  B
)  -  ( A  x.  D ) ) )
4236, 41eqtrd 2470 . . . . 5  |-  ( ph  ->  ( ( C  x.  B )  +  (
-u A  x.  D
) )  =  ( ( C  x.  B
)  -  ( A  x.  D ) ) )
4342breq2d 4299 . . . 4  |-  ( ph  ->  ( P  ||  (
( C  x.  B
)  +  ( -u A  x.  D )
)  <->  P  ||  ( ( C  x.  B )  -  ( A  x.  D ) ) ) )
4443biimpar 485 . . 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 22680 . 2  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  N  e.  S )
46 prmz 13759 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
474, 46syl 16 . . . . 5  |-  ( ph  ->  P  e.  ZZ )
48 zsqcl 11928 . . . . . . . 8  |-  ( C  e.  ZZ  ->  ( C ^ 2 )  e.  ZZ )
4910, 48syl 16 . . . . . . 7  |-  ( ph  ->  ( C ^ 2 )  e.  ZZ )
502nnzd 10738 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
5149, 50zmulcld 10745 . . . . . 6  |-  ( ph  ->  ( ( C ^
2 )  x.  N
)  e.  ZZ )
52 zsqcl 11928 . . . . . . 7  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e.  ZZ )
536, 52syl 16 . . . . . 6  |-  ( ph  ->  ( A ^ 2 )  e.  ZZ )
5451, 53zsubcld 10744 . . . . 5  |-  ( ph  ->  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) )  e.  ZZ )
55 dvdsmul1 13546 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) )  e.  ZZ )  ->  P  ||  ( P  x.  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) ) ) )
5647, 54, 55syl2anc 661 . . . 4  |-  ( ph  ->  P  ||  ( P  x.  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) ) ) )
5710, 6zmulcld 10745 . . . . . . . . 9  |-  ( ph  ->  ( C  x.  A
)  e.  ZZ )
5857zcnd 10740 . . . . . . . 8  |-  ( ph  ->  ( C  x.  A
)  e.  CC )
5958sqcld 11998 . . . . . . 7  |-  ( ph  ->  ( ( C  x.  A ) ^ 2 )  e.  CC )
6038sqcld 11998 . . . . . . 7  |-  ( ph  ->  ( ( C  x.  B ) ^ 2 )  e.  CC )
6140sqcld 11998 . . . . . . 7  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  e.  CC )
6259, 60, 61pnpcand 9748 . . . . . 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 10740 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  CC )
6463, 27sqmuld 12012 . . . . . . . . . . 11  |-  ( ph  ->  ( ( C  x.  A ) ^ 2 )  =  ( ( C ^ 2 )  x.  ( A ^
2 ) ) )
658zcnd 10740 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  CC )
6663, 65sqmuld 12012 . . . . . . . . . . 11  |-  ( ph  ->  ( ( C  x.  B ) ^ 2 )  =  ( ( C ^ 2 )  x.  ( B ^
2 ) ) )
6764, 66oveq12d 6104 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( C ^ 2 )  x.  ( B ^ 2 ) ) ) )
6863sqcld 11998 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  e.  CC )
6953zcnd 10740 . . . . . . . . . . 11  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
7065sqcld 11998 . . . . . . . . . . 11  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
7168, 69, 70adddid 9402 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( C ^ 2 )  x.  ( B ^ 2 ) ) ) )
7267, 71eqtr4d 2473 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( ( C ^ 2 )  x.  ( ( A ^ 2 )  +  ( B ^ 2 ) ) ) )
732nncnd 10330 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
7447zcnd 10740 . . . . . . . . . . . . 13  |-  ( ph  ->  P  e.  CC )
7573, 74mulcomd 9399 . . . . . . . . . . . 12  |-  ( ph  ->  ( N  x.  P
)  =  ( P  x.  N ) )
7614, 75eqtr3d 2472 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( P  x.  N ) )
7776oveq2d 6102 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( C ^ 2 )  x.  ( P  x.  N ) ) )
7868, 74, 73mul12d 9570 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  ( P  x.  N )
)  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
7977, 78eqtrd 2470 . . . . . . . . 9  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
8072, 79eqtrd 2470 . . . . . . . 8  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
8127, 34sqmuld 12012 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  =  ( ( A ^ 2 )  x.  ( D ^
2 ) ) )
8234sqcld 11998 . . . . . . . . . . . . 13  |-  ( ph  ->  ( D ^ 2 )  e.  CC )
8369, 82mulcomd 9399 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A ^
2 )  x.  ( D ^ 2 ) )  =  ( ( D ^ 2 )  x.  ( A ^ 2 ) ) )
8481, 83eqtrd 2470 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  =  ( ( D ^ 2 )  x.  ( A ^
2 ) ) )
8564, 84oveq12d 6104 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( D ^ 2 )  x.  ( A ^ 2 ) ) ) )
8649zcnd 10740 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  e.  CC )
8786, 82, 69adddird 9403 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C ^ 2 )  +  ( D ^ 2 ) )  x.  ( A ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^
2 ) )  +  ( ( D ^
2 )  x.  ( A ^ 2 ) ) ) )
8885, 87eqtr4d 2473 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( ( ( C ^ 2 )  +  ( D ^ 2 ) )  x.  ( A ^
2 ) ) )
8916oveq1d 6101 . . . . . . . . 9  |-  ( ph  ->  ( P  x.  ( A ^ 2 ) )  =  ( ( ( C ^ 2 )  +  ( D ^
2 ) )  x.  ( A ^ 2 ) ) )
9088, 89eqtr4d 2473 . . . . . . . 8  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( P  x.  ( A ^
2 ) ) )
9180, 90oveq12d 6104 . . . . . . 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 10740 . . . . . . . 8  |-  ( ph  ->  ( ( C ^
2 )  x.  N
)  e.  CC )
9374, 92, 69subdid 9792 . . . . . . 7  |-  ( ph  ->  ( P  x.  (
( ( C ^
2 )  x.  N
)  -  ( A ^ 2 ) ) )  =  ( ( P  x.  ( ( C ^ 2 )  x.  N ) )  -  ( P  x.  ( A ^ 2 ) ) ) )
9491, 93eqtr4d 2473 . . . . . 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 2472 . . . . 5  |-  ( ph  ->  ( ( ( C  x.  B ) ^
2 )  -  (
( A  x.  D
) ^ 2 ) )  =  ( P  x.  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) ) ) )
96 subsq 11965 . . . . . 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 661 . . . . 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 2472 . . . 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 4311 . . 3  |-  ( ph  ->  P  ||  ( ( ( C  x.  B
)  +  ( A  x.  D ) )  x.  ( ( C  x.  B )  -  ( A  x.  D
) ) ) )
10037, 39zaddcld 10743 . . . 4  |-  ( ph  ->  ( ( C  x.  B )  +  ( A  x.  D ) )  e.  ZZ )
10137, 39zsubcld 10744 . . . 4  |-  ( ph  ->  ( ( C  x.  B )  -  ( A  x.  D )
)  e.  ZZ )
102 euclemma 13786 . . . 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 1218 . . 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 210 . 2  |-  ( ph  ->  ( P  ||  (
( C  x.  B
)  +  ( A  x.  D ) )  \/  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) ) )
10519, 45, 104mpjaodan 784 1  |-  ( ph  ->  N  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4287    e. cmpt 4345   ran crn 4836   ` cfv 5413  (class class class)co 6086   CCcc 9272    + caddc 9277    x. cmul 9279    - cmin 9587   -ucneg 9588   NNcn 10314   2c2 10363   ZZcz 10638   ^cexp 11857   abscabs 12715    || cdivides 13527   Primecprime 13755   ZZ[_i]cgz 13982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-dvds 13528  df-gcd 13683  df-prm 13756  df-gz 13983
This theorem is referenced by:  2sqlem5  22682
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