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Theorem 2sqlem7 21146
Description: Lemma for 2sq 21152. (Contributed by Mario Carneiro, 19-Jun-2015.)
Hypotheses
Ref Expression
2sq.1  |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w
) ^ 2 ) )
2sqlem7.2  |-  Y  =  { z  |  E. x  e.  ZZ  E. y  e.  ZZ  ( ( x  gcd  y )  =  1  /\  z  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) }
Assertion
Ref Expression
2sqlem7  |-  Y  C_  ( S  i^i  NN )
Distinct variable groups:    x, w, y, z    x, S, y, z    x, Y, y
Allowed substitution hints:    S( w)    Y( z, w)

Proof of Theorem 2sqlem7
StepHypRef Expression
1 2sqlem7.2 . 2  |-  Y  =  { z  |  E. x  e.  ZZ  E. y  e.  ZZ  ( ( x  gcd  y )  =  1  /\  z  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) }
2 simpr 448 . . . . . . 7  |-  ( ( ( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  ->  z  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )
32reximi 2805 . . . . . 6  |-  ( E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  ->  E. y  e.  ZZ  z  =  ( (
x ^ 2 )  +  ( y ^
2 ) ) )
43reximi 2805 . . . . 5  |-  ( E. x  e.  ZZ  E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  ->  E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )
5 2sq.1 . . . . . 6  |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w
) ^ 2 ) )
652sqlem2 21140 . . . . 5  |-  ( z  e.  S  <->  E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )
74, 6sylibr 204 . . . 4  |-  ( E. x  e.  ZZ  E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  ->  z  e.  S
)
8 ax-1ne0 9051 . . . . . . . . . 10  |-  1  =/=  0
9 gcdeq0 13013 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( x  gcd  y )  =  0  <-> 
( x  =  0  /\  y  =  0 ) ) )
109adantr 452 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
x  gcd  y )  =  0  <->  ( x  =  0  /\  y  =  0 ) ) )
11 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( x  gcd  y )  =  1 )
1211eqeq1d 2443 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
x  gcd  y )  =  0  <->  1  = 
0 ) )
1310, 12bitr3d 247 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
x  =  0  /\  y  =  0 )  <->  1  =  0 ) )
1413necon3bbid 2632 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( -.  ( x  =  0  /\  y  =  0
)  <->  1  =/=  0
) )
158, 14mpbiri 225 . . . . . . . . 9  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  -.  (
x  =  0  /\  y  =  0 ) )
16 zsqcl2 11451 . . . . . . . . . . . . 13  |-  ( x  e.  ZZ  ->  (
x ^ 2 )  e.  NN0 )
1716ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( x ^ 2 )  e. 
NN0 )
1817nn0red 10267 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( x ^ 2 )  e.  RR )
1917nn0ge0d 10269 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  0  <_  ( x ^ 2 ) )
20 zsqcl2 11451 . . . . . . . . . . . . 13  |-  ( y  e.  ZZ  ->  (
y ^ 2 )  e.  NN0 )
2120ad2antlr 708 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( y ^ 2 )  e. 
NN0 )
2221nn0red 10267 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( y ^ 2 )  e.  RR )
2321nn0ge0d 10269 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  0  <_  ( y ^ 2 ) )
24 add20 9532 . . . . . . . . . . 11  |-  ( ( ( ( x ^
2 )  e.  RR  /\  0  <_  ( x ^ 2 ) )  /\  ( ( y ^ 2 )  e.  RR  /\  0  <_ 
( y ^ 2 ) ) )  -> 
( ( ( x ^ 2 )  +  ( y ^ 2 ) )  =  0  <-> 
( ( x ^
2 )  =  0  /\  ( y ^
2 )  =  0 ) ) )
2518, 19, 22, 23, 24syl22anc 1185 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
( x ^ 2 )  +  ( y ^ 2 ) )  =  0  <->  ( (
x ^ 2 )  =  0  /\  (
y ^ 2 )  =  0 ) ) )
26 zcn 10279 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  x  e.  CC )
2726ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  x  e.  CC )
28 zcn 10279 . . . . . . . . . . . 12  |-  ( y  e.  ZZ  ->  y  e.  CC )
2928ad2antlr 708 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  y  e.  CC )
30 sqeq0 11438 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
( x ^ 2 )  =  0  <->  x  =  0 ) )
31 sqeq0 11438 . . . . . . . . . . . 12  |-  ( y  e.  CC  ->  (
( y ^ 2 )  =  0  <->  y  =  0 ) )
3230, 31bi2anan9 844 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( x ^ 2 )  =  0  /\  ( y ^ 2 )  =  0 )  <->  ( x  =  0  /\  y  =  0 ) ) )
3327, 29, 32syl2anc 643 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
( x ^ 2 )  =  0  /\  ( y ^ 2 )  =  0 )  <-> 
( x  =  0  /\  y  =  0 ) ) )
3425, 33bitrd 245 . . . . . . . . 9  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
( x ^ 2 )  +  ( y ^ 2 ) )  =  0  <->  ( x  =  0  /\  y  =  0 ) ) )
3515, 34mtbird 293 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  -.  (
( x ^ 2 )  +  ( y ^ 2 ) )  =  0 )
36 nn0addcl 10247 . . . . . . . . . . . 12  |-  ( ( ( x ^ 2 )  e.  NN0  /\  ( y ^ 2 )  e.  NN0 )  ->  ( ( x ^
2 )  +  ( y ^ 2 ) )  e.  NN0 )
3716, 20, 36syl2an 464 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( x ^
2 )  +  ( y ^ 2 ) )  e.  NN0 )
3837adantr 452 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
x ^ 2 )  +  ( y ^
2 ) )  e. 
NN0 )
39 elnn0 10215 . . . . . . . . . 10  |-  ( ( ( x ^ 2 )  +  ( y ^ 2 ) )  e.  NN0  <->  ( ( ( x ^ 2 )  +  ( y ^
2 ) )  e.  NN  \/  ( ( x ^ 2 )  +  ( y ^
2 ) )  =  0 ) )
4038, 39sylib 189 . . . . . . . . 9  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
( x ^ 2 )  +  ( y ^ 2 ) )  e.  NN  \/  (
( x ^ 2 )  +  ( y ^ 2 ) )  =  0 ) )
4140ord 367 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( -.  ( ( x ^
2 )  +  ( y ^ 2 ) )  e.  NN  ->  ( ( x ^ 2 )  +  ( y ^ 2 ) )  =  0 ) )
4235, 41mt3d 119 . . . . . . 7  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
x ^ 2 )  +  ( y ^
2 ) )  e.  NN )
43 eleq1 2495 . . . . . . 7  |-  ( z  =  ( ( x ^ 2 )  +  ( y ^ 2 ) )  ->  (
z  e.  NN  <->  ( (
x ^ 2 )  +  ( y ^
2 ) )  e.  NN ) )
4442, 43syl5ibrcom 214 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( z  =  ( ( x ^ 2 )  +  ( y ^ 2 ) )  ->  z  e.  NN ) )
4544expimpd 587 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ( x  gcd  y )  =  1  /\  z  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) )  ->  z  e.  NN ) )
4645rexlimivv 2827 . . . 4  |-  ( E. x  e.  ZZ  E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  ->  z  e.  NN )
47 elin 3522 . . . 4  |-  ( z  e.  ( S  i^i  NN )  <->  ( z  e.  S  /\  z  e.  NN ) )
487, 46, 47sylanbrc 646 . . 3  |-  ( E. x  e.  ZZ  E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  ->  z  e.  ( S  i^i  NN ) )
4948abssi 3410 . 2  |-  { z  |  E. x  e.  ZZ  E. y  e.  ZZ  ( ( x  gcd  y )  =  1  /\  z  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) }  C_  ( S  i^i  NN )
501, 49eqsstri 3370 1  |-  Y  C_  ( S  i^i  NN )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   E.wrex 2698    i^i cin 3311    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   ran crn 4871   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    <_ cle 9113   NNcn 9992   2c2 10041   NN0cn0 10213   ZZcz 10274   ^cexp 11374   abscabs 12031    gcd cgcd 12998   ZZ [ _i ]cgz 13289
This theorem is referenced by:  2sqlem8  21148  2sqlem9  21149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-gz 13290
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