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Theorem 2sqlem7 23510
Description: Lemma for 2sq 23516. (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 461 . . . . . . 7  |-  ( ( ( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  ->  z  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )
32reximi 2909 . . . . . 6  |-  ( E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  ->  E. y  e.  ZZ  z  =  ( (
x ^ 2 )  +  ( y ^
2 ) ) )
43reximi 2909 . . . . 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 23504 . . . . 5  |-  ( z  e.  S  <->  E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )
74, 6sylibr 212 . . . 4  |-  ( E. x  e.  ZZ  E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  ->  z  e.  S
)
8 ax-1ne0 9559 . . . . . . . . . 10  |-  1  =/=  0
9 gcdeq0 14031 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( x  gcd  y )  =  0  <-> 
( x  =  0  /\  y  =  0 ) ) )
109adantr 465 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
x  gcd  y )  =  0  <->  ( x  =  0  /\  y  =  0 ) ) )
11 simpr 461 . . . . . . . . . . . . 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 255 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
x  =  0  /\  y  =  0 )  <->  1  =  0 ) )
1413necon3bbid 2688 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( -.  ( x  =  0  /\  y  =  0
)  <->  1  =/=  0
) )
158, 14mpbiri 233 . . . . . . . . 9  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  -.  (
x  =  0  /\  y  =  0 ) )
16 zsqcl2 12219 . . . . . . . . . . . . 13  |-  ( x  e.  ZZ  ->  (
x ^ 2 )  e.  NN0 )
1716ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( x ^ 2 )  e. 
NN0 )
1817nn0red 10854 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( x ^ 2 )  e.  RR )
1917nn0ge0d 10856 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  0  <_  ( x ^ 2 ) )
20 zsqcl2 12219 . . . . . . . . . . . . 13  |-  ( y  e.  ZZ  ->  (
y ^ 2 )  e.  NN0 )
2120ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( y ^ 2 )  e. 
NN0 )
2221nn0red 10854 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( y ^ 2 )  e.  RR )
2321nn0ge0d 10856 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  0  <_  ( y ^ 2 ) )
24 add20 10065 . . . . . . . . . . 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 1228 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
( x ^ 2 )  +  ( y ^ 2 ) )  =  0  <->  ( (
x ^ 2 )  =  0  /\  (
y ^ 2 )  =  0 ) ) )
26 zcn 10870 . . . . . . . . . . . 12  |-  ( x  e.  ZZ  ->  x  e.  CC )
2726ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  x  e.  CC )
28 zcn 10870 . . . . . . . . . . . 12  |-  ( y  e.  ZZ  ->  y  e.  CC )
2928ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  y  e.  CC )
30 sqeq0 12206 . . . . . . . . . . . 12  |-  ( x  e.  CC  ->  (
( x ^ 2 )  =  0  <->  x  =  0 ) )
31 sqeq0 12206 . . . . . . . . . . . 12  |-  ( y  e.  CC  ->  (
( y ^ 2 )  =  0  <->  y  =  0 ) )
3230, 31bi2anan9 871 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( ( x ^ 2 )  =  0  /\  ( y ^ 2 )  =  0 )  <->  ( x  =  0  /\  y  =  0 ) ) )
3327, 29, 32syl2anc 661 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
( x ^ 2 )  =  0  /\  ( y ^ 2 )  =  0 )  <-> 
( x  =  0  /\  y  =  0 ) ) )
3425, 33bitrd 253 . . . . . . . . 9  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
( x ^ 2 )  +  ( y ^ 2 ) )  =  0  <->  ( x  =  0  /\  y  =  0 ) ) )
3515, 34mtbird 301 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  -.  (
( x ^ 2 )  +  ( y ^ 2 ) )  =  0 )
36 nn0addcl 10832 . . . . . . . . . . . 12  |-  ( ( ( x ^ 2 )  e.  NN0  /\  ( y ^ 2 )  e.  NN0 )  ->  ( ( x ^
2 )  +  ( y ^ 2 ) )  e.  NN0 )
3716, 20, 36syl2an 477 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( x ^
2 )  +  ( y ^ 2 ) )  e.  NN0 )
3837adantr 465 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
x ^ 2 )  +  ( y ^
2 ) )  e. 
NN0 )
39 elnn0 10798 . . . . . . . . . 10  |-  ( ( ( x ^ 2 )  +  ( y ^ 2 ) )  e.  NN0  <->  ( ( ( x ^ 2 )  +  ( y ^
2 ) )  e.  NN  \/  ( ( x ^ 2 )  +  ( y ^
2 ) )  =  0 ) )
4038, 39sylib 196 . . . . . . . . 9  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
( x ^ 2 )  +  ( y ^ 2 ) )  e.  NN  \/  (
( x ^ 2 )  +  ( y ^ 2 ) )  =  0 ) )
4140ord 377 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( -.  ( ( x ^
2 )  +  ( y ^ 2 ) )  e.  NN  ->  ( ( x ^ 2 )  +  ( y ^ 2 ) )  =  0 ) )
4235, 41mt3d 125 . . . . . . 7  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( (
x ^ 2 )  +  ( y ^
2 ) )  e.  NN )
43 eleq1 2513 . . . . . . 7  |-  ( z  =  ( ( x ^ 2 )  +  ( y ^ 2 ) )  ->  (
z  e.  NN  <->  ( (
x ^ 2 )  +  ( y ^
2 ) )  e.  NN ) )
4442, 43syl5ibrcom 222 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( x  gcd  y )  =  1 )  ->  ( z  =  ( ( x ^ 2 )  +  ( y ^ 2 ) )  ->  z  e.  NN ) )
4544expimpd 603 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ( x  gcd  y )  =  1  /\  z  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) )  ->  z  e.  NN ) )
4645rexlimivv 2938 . . . 4  |-  ( E. x  e.  ZZ  E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  ->  z  e.  NN )
477, 46elind 3670 . . 3  |-  ( E. x  e.  ZZ  E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  ->  z  e.  ( S  i^i  NN ) )
4847abssi 3557 . 2  |-  { z  |  E. x  e.  ZZ  E. y  e.  ZZ  ( ( x  gcd  y )  =  1  /\  z  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) }  C_  ( S  i^i  NN )
491, 48eqsstri 3516 1  |-  Y  C_  ( S  i^i  NN )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1381    e. wcel 1802   {cab 2426    =/= wne 2636   E.wrex 2792    i^i cin 3457    C_ wss 3458   class class class wbr 4433    |-> cmpt 4491   ran crn 4986   ` cfv 5574  (class class class)co 6277   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    <_ cle 9627   NNcn 10537   2c2 10586   NN0cn0 10796   ZZcz 10865   ^cexp 12140   abscabs 13041    gcd cgcd 14016   ZZ[_i]cgz 14319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-seq 12082  df-exp 12141  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-dvds 13859  df-gcd 14017  df-gz 14320
This theorem is referenced by:  2sqlem8  23512  2sqlem9  23513
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