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Theorem 2sqlem4 22838
Description: Lemma for 2sqlem5 22839. (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 22837 . 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 10859 . . . 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 10858 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
28 sqneg 12042 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u A ^ 2 )  =  ( A ^
2 ) )
2927, 28syl 16 . . . . . 6  |-  ( ph  ->  ( -u A ^
2 )  =  ( A ^ 2 ) )
3029oveq1d 6214 . . . . 5  |-  ( ph  ->  ( ( -u A ^ 2 )  +  ( B ^ 2 ) )  =  ( ( A ^ 2 )  +  ( B ^ 2 ) ) )
3114, 30eqtr4d 2498 . . . 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 10858 . . . . . . . 8  |-  ( ph  ->  D  e.  CC )
3527, 34mulneg1d 9907 . . . . . . 7  |-  ( ph  ->  ( -u A  x.  D )  =  -u ( A  x.  D
) )
3635oveq2d 6215 . . . . . 6  |-  ( ph  ->  ( ( C  x.  B )  +  (
-u A  x.  D
) )  =  ( ( C  x.  B
)  +  -u ( A  x.  D )
) )
3710, 8zmulcld 10863 . . . . . . . 8  |-  ( ph  ->  ( C  x.  B
)  e.  ZZ )
3837zcnd 10858 . . . . . . 7  |-  ( ph  ->  ( C  x.  B
)  e.  CC )
396, 12zmulcld 10863 . . . . . . . 8  |-  ( ph  ->  ( A  x.  D
)  e.  ZZ )
4039zcnd 10858 . . . . . . 7  |-  ( ph  ->  ( A  x.  D
)  e.  CC )
4138, 40negsubd 9835 . . . . . 6  |-  ( ph  ->  ( ( C  x.  B )  +  -u ( A  x.  D
) )  =  ( ( C  x.  B
)  -  ( A  x.  D ) ) )
4236, 41eqtrd 2495 . . . . 5  |-  ( ph  ->  ( ( C  x.  B )  +  (
-u A  x.  D
) )  =  ( ( C  x.  B
)  -  ( A  x.  D ) ) )
4342breq2d 4411 . . . 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 22837 . 2  |-  ( (
ph  /\  P  ||  (
( C  x.  B
)  -  ( A  x.  D ) ) )  ->  N  e.  S )
46 prmz 13884 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
474, 46syl 16 . . . . 5  |-  ( ph  ->  P  e.  ZZ )
48 zsqcl 12052 . . . . . . . 8  |-  ( C  e.  ZZ  ->  ( C ^ 2 )  e.  ZZ )
4910, 48syl 16 . . . . . . 7  |-  ( ph  ->  ( C ^ 2 )  e.  ZZ )
502nnzd 10856 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
5149, 50zmulcld 10863 . . . . . 6  |-  ( ph  ->  ( ( C ^
2 )  x.  N
)  e.  ZZ )
52 zsqcl 12052 . . . . . . 7  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e.  ZZ )
536, 52syl 16 . . . . . 6  |-  ( ph  ->  ( A ^ 2 )  e.  ZZ )
5451, 53zsubcld 10862 . . . . 5  |-  ( ph  ->  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) )  e.  ZZ )
55 dvdsmul1 13671 . . . . 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 10863 . . . . . . . . 9  |-  ( ph  ->  ( C  x.  A
)  e.  ZZ )
5857zcnd 10858 . . . . . . . 8  |-  ( ph  ->  ( C  x.  A
)  e.  CC )
5958sqcld 12122 . . . . . . 7  |-  ( ph  ->  ( ( C  x.  A ) ^ 2 )  e.  CC )
6038sqcld 12122 . . . . . . 7  |-  ( ph  ->  ( ( C  x.  B ) ^ 2 )  e.  CC )
6140sqcld 12122 . . . . . . 7  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  e.  CC )
6259, 60, 61pnpcand 9866 . . . . . 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 10858 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  CC )
6463, 27sqmuld 12136 . . . . . . . . . . 11  |-  ( ph  ->  ( ( C  x.  A ) ^ 2 )  =  ( ( C ^ 2 )  x.  ( A ^
2 ) ) )
658zcnd 10858 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  CC )
6663, 65sqmuld 12136 . . . . . . . . . . 11  |-  ( ph  ->  ( ( C  x.  B ) ^ 2 )  =  ( ( C ^ 2 )  x.  ( B ^
2 ) ) )
6764, 66oveq12d 6217 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( C ^ 2 )  x.  ( B ^ 2 ) ) ) )
6863sqcld 12122 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  e.  CC )
6953zcnd 10858 . . . . . . . . . . 11  |-  ( ph  ->  ( A ^ 2 )  e.  CC )
7065sqcld 12122 . . . . . . . . . . 11  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
7168, 69, 70adddid 9520 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( C ^ 2 )  x.  ( B ^ 2 ) ) ) )
7267, 71eqtr4d 2498 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( ( C ^ 2 )  x.  ( ( A ^ 2 )  +  ( B ^ 2 ) ) ) )
732nncnd 10448 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
7447zcnd 10858 . . . . . . . . . . . . 13  |-  ( ph  ->  P  e.  CC )
7573, 74mulcomd 9517 . . . . . . . . . . . 12  |-  ( ph  ->  ( N  x.  P
)  =  ( P  x.  N ) )
7614, 75eqtr3d 2497 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( P  x.  N ) )
7776oveq2d 6215 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( ( C ^ 2 )  x.  ( P  x.  N ) ) )
7868, 74, 73mul12d 9688 . . . . . . . . . 10  |-  ( ph  ->  ( ( C ^
2 )  x.  ( P  x.  N )
)  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
7977, 78eqtrd 2495 . . . . . . . . 9  |-  ( ph  ->  ( ( C ^
2 )  x.  (
( A ^ 2 )  +  ( B ^ 2 ) ) )  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
8072, 79eqtrd 2495 . . . . . . . 8  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( C  x.  B
) ^ 2 ) )  =  ( P  x.  ( ( C ^ 2 )  x.  N ) ) )
8127, 34sqmuld 12136 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  =  ( ( A ^ 2 )  x.  ( D ^
2 ) ) )
8234sqcld 12122 . . . . . . . . . . . . 13  |-  ( ph  ->  ( D ^ 2 )  e.  CC )
8369, 82mulcomd 9517 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A ^
2 )  x.  ( D ^ 2 ) )  =  ( ( D ^ 2 )  x.  ( A ^ 2 ) ) )
8481, 83eqtrd 2495 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A  x.  D ) ^ 2 )  =  ( ( D ^ 2 )  x.  ( A ^
2 ) ) )
8564, 84oveq12d 6217 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^ 2 ) )  +  ( ( D ^ 2 )  x.  ( A ^ 2 ) ) ) )
8649zcnd 10858 . . . . . . . . . . 11  |-  ( ph  ->  ( C ^ 2 )  e.  CC )
8786, 82, 69adddird 9521 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( C ^ 2 )  +  ( D ^ 2 ) )  x.  ( A ^ 2 ) )  =  ( ( ( C ^ 2 )  x.  ( A ^
2 ) )  +  ( ( D ^
2 )  x.  ( A ^ 2 ) ) ) )
8885, 87eqtr4d 2498 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( ( ( C ^ 2 )  +  ( D ^ 2 ) )  x.  ( A ^
2 ) ) )
8916oveq1d 6214 . . . . . . . . 9  |-  ( ph  ->  ( P  x.  ( A ^ 2 ) )  =  ( ( ( C ^ 2 )  +  ( D ^
2 ) )  x.  ( A ^ 2 ) ) )
9088, 89eqtr4d 2498 . . . . . . . 8  |-  ( ph  ->  ( ( ( C  x.  A ) ^
2 )  +  ( ( A  x.  D
) ^ 2 ) )  =  ( P  x.  ( A ^
2 ) ) )
9180, 90oveq12d 6217 . . . . . . 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 10858 . . . . . . . 8  |-  ( ph  ->  ( ( C ^
2 )  x.  N
)  e.  CC )
9374, 92, 69subdid 9910 . . . . . . 7  |-  ( ph  ->  ( P  x.  (
( ( C ^
2 )  x.  N
)  -  ( A ^ 2 ) ) )  =  ( ( P  x.  ( ( C ^ 2 )  x.  N ) )  -  ( P  x.  ( A ^ 2 ) ) ) )
9491, 93eqtr4d 2498 . . . . . 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 2497 . . . . 5  |-  ( ph  ->  ( ( ( C  x.  B ) ^
2 )  -  (
( A  x.  D
) ^ 2 ) )  =  ( P  x.  ( ( ( C ^ 2 )  x.  N )  -  ( A ^ 2 ) ) ) )
96 subsq 12089 . . . . . 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 2497 . . . 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 4423 . . 3  |-  ( ph  ->  P  ||  ( ( ( C  x.  B
)  +  ( A  x.  D ) )  x.  ( ( C  x.  B )  -  ( A  x.  D
) ) ) )
10037, 39zaddcld 10861 . . . 4  |-  ( ph  ->  ( ( C  x.  B )  +  ( A  x.  D ) )  e.  ZZ )
10137, 39zsubcld 10862 . . . 4  |-  ( ph  ->  ( ( C  x.  B )  -  ( A  x.  D )
)  e.  ZZ )
102 euclemma 13911 . . . 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 1219 . . 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 1370    e. wcel 1758   class class class wbr 4399    |-> cmpt 4457   ran crn 4948   ` cfv 5525  (class class class)co 6199   CCcc 9390    + caddc 9395    x. cmul 9397    - cmin 9705   -ucneg 9706   NNcn 10432   2c2 10481   ZZcz 10756   ^cexp 11981   abscabs 12840    || cdivides 13652   Primecprime 13880   ZZ[_i]cgz 14107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-n0 10690  df-z 10757  df-uz 10972  df-rp 11102  df-fl 11758  df-mod 11825  df-seq 11923  df-exp 11982  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-dvds 13653  df-gcd 13808  df-prm 13881  df-gz 14108
This theorem is referenced by:  2sqlem5  22839
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