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Theorem 2sqlem9 14029
Description: Lemma for 2sq . (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 ) ) ) }
2sqlem9.5  |-  ( ph  ->  A. b  e.  ( 1 ... ( M  -  1 ) ) A. a  e.  Y  ( b  ||  a  ->  b  e.  S ) )
2sqlem9.7  |-  ( ph  ->  M  ||  N )
2sqlem9.6  |-  ( ph  ->  M  e.  NN )
2sqlem9.4  |-  ( ph  ->  N  e.  Y )
Assertion
Ref Expression
2sqlem9  |-  ( ph  ->  M  e.  S )
Distinct variable groups:    a, b, w, x, y, z    ph, x, y    M, a, b, x, y, z    S, a, b, x, y, z   
x, N, y, z    Y, a, b, x, y
Allowed substitution hints:    ph( z, w, a, b)    S( w)    M( w)    N( w, a, b)    Y( z, w)

Proof of Theorem 2sqlem9
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2sqlem9.4 . . 3  |-  ( ph  ->  N  e.  Y )
2 eqeq1 2182 . . . . . . . 8  |-  ( z  =  N  ->  (
z  =  ( ( x ^ 2 )  +  ( y ^
2 ) )  <->  N  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) )
32anbi2d 464 . . . . . . 7  |-  ( z  =  N  ->  (
( ( x  gcd  y )  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )  <->  ( ( x  gcd  y )  =  1  /\  N  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) ) )
432rexbidv 2500 . . . . . 6  |-  ( z  =  N  ->  ( 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  (
( x  gcd  y
)  =  1  /\  N  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) ) ) )
5 oveq1 5872 . . . . . . . . 9  |-  ( x  =  u  ->  (
x  gcd  y )  =  ( u  gcd  y ) )
65eqeq1d 2184 . . . . . . . 8  |-  ( x  =  u  ->  (
( x  gcd  y
)  =  1  <->  (
u  gcd  y )  =  1 ) )
7 oveq1 5872 . . . . . . . . . 10  |-  ( x  =  u  ->  (
x ^ 2 )  =  ( u ^
2 ) )
87oveq1d 5880 . . . . . . . . 9  |-  ( x  =  u  ->  (
( x ^ 2 )  +  ( y ^ 2 ) )  =  ( ( u ^ 2 )  +  ( y ^ 2 ) ) )
98eqeq2d 2187 . . . . . . . 8  |-  ( x  =  u  ->  ( N  =  ( (
x ^ 2 )  +  ( y ^
2 ) )  <->  N  =  ( ( u ^
2 )  +  ( y ^ 2 ) ) ) )
106, 9anbi12d 473 . . . . . . 7  |-  ( x  =  u  ->  (
( ( x  gcd  y )  =  1  /\  N  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )  <->  ( ( u  gcd  y )  =  1  /\  N  =  ( ( u ^
2 )  +  ( y ^ 2 ) ) ) ) )
11 oveq2 5873 . . . . . . . . 9  |-  ( y  =  v  ->  (
u  gcd  y )  =  ( u  gcd  v ) )
1211eqeq1d 2184 . . . . . . . 8  |-  ( y  =  v  ->  (
( u  gcd  y
)  =  1  <->  (
u  gcd  v )  =  1 ) )
13 oveq1 5872 . . . . . . . . . 10  |-  ( y  =  v  ->  (
y ^ 2 )  =  ( v ^
2 ) )
1413oveq2d 5881 . . . . . . . . 9  |-  ( y  =  v  ->  (
( u ^ 2 )  +  ( y ^ 2 ) )  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) )
1514eqeq2d 2187 . . . . . . . 8  |-  ( y  =  v  ->  ( N  =  ( (
u ^ 2 )  +  ( y ^
2 ) )  <->  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) ) )
1612, 15anbi12d 473 . . . . . . 7  |-  ( y  =  v  ->  (
( ( u  gcd  y )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( y ^ 2 ) ) )  <->  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) ) ) )
1710, 16cbvrex2vw 2713 . . . . . 6  |-  ( E. x  e.  ZZ  E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  N  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  <->  E. u  e.  ZZ  E. v  e.  ZZ  (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) ) )
184, 17bitrdi 196 . . . . 5  |-  ( z  =  N  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  <->  E. u  e.  ZZ  E. v  e.  ZZ  (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) ) ) )
19 2sqlem7.2 . . . . 5  |-  Y  =  { z  |  E. x  e.  ZZ  E. y  e.  ZZ  ( ( x  gcd  y )  =  1  /\  z  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) }
2018, 19elab2g 2882 . . . 4  |-  ( N  e.  Y  ->  ( N  e.  Y  <->  E. u  e.  ZZ  E. v  e.  ZZ  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) ) ) )
2120ibi 176 . . 3  |-  ( N  e.  Y  ->  E. u  e.  ZZ  E. v  e.  ZZ  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) ) )
221, 21syl 14 . 2  |-  ( ph  ->  E. u  e.  ZZ  E. v  e.  ZZ  (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) ) )
23 simpr 110 . . . . . 6  |-  ( ( ( ( ph  /\  ( u  e.  ZZ  /\  v  e.  ZZ ) )  /\  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) ) )  /\  M  =  1 )  ->  M  = 
1 )
24 1z 9250 . . . . . . . . 9  |-  1  e.  ZZ
25 zgz 12336 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  1  e.  ZZ[_i]
)
2624, 25ax-mp 5 . . . . . . . 8  |-  1  e.  ZZ[_i]
27 sq1 10581 . . . . . . . . 9  |-  ( 1 ^ 2 )  =  1
2827eqcomi 2179 . . . . . . . 8  |-  1  =  ( 1 ^ 2 )
29 fveq2 5507 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( abs `  x )  =  ( abs `  1
) )
30 abs1 11047 . . . . . . . . . . 11  |-  ( abs `  1 )  =  1
3129, 30eqtrdi 2224 . . . . . . . . . 10  |-  ( x  =  1  ->  ( abs `  x )  =  1 )
3231oveq1d 5880 . . . . . . . . 9  |-  ( x  =  1  ->  (
( abs `  x
) ^ 2 )  =  ( 1 ^ 2 ) )
3332rspceeqv 2857 . . . . . . . 8  |-  ( ( 1  e.  ZZ[_i]  /\  1  =  ( 1 ^ 2 ) )  ->  E. x  e.  ZZ[_i]  1  =  ( ( abs `  x ) ^ 2 ) )
3426, 28, 33mp2an 426 . . . . . . 7  |-  E. x  e.  ZZ[_i] 
1  =  ( ( abs `  x ) ^ 2 )
35 2sq.1 . . . . . . . 8  |-  S  =  ran  ( w  e.  ZZ[_i]  |->  ( ( abs `  w
) ^ 2 ) )
36352sqlem1 14019 . . . . . . 7  |-  ( 1  e.  S  <->  E. x  e.  ZZ[_i] 
1  =  ( ( abs `  x ) ^ 2 ) )
3734, 36mpbir 146 . . . . . 6  |-  1  e.  S
3823, 37eqeltrdi 2266 . . . . 5  |-  ( ( ( ( ph  /\  ( u  e.  ZZ  /\  v  e.  ZZ ) )  /\  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) ) )  /\  M  =  1 )  ->  M  e.  S )
39 2sqlem9.5 . . . . . . . 8  |-  ( ph  ->  A. b  e.  ( 1 ... ( M  -  1 ) ) A. a  e.  Y  ( b  ||  a  ->  b  e.  S ) )
4039ad2antrr 488 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  A. b  e.  ( 1 ... ( M  -  1 ) ) A. a  e.  Y  ( b  ||  a  ->  b  e.  S
) )
41 2sqlem9.7 . . . . . . . 8  |-  ( ph  ->  M  ||  N )
4241ad2antrr 488 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  ||  N )
4335, 192sqlem7 14026 . . . . . . . . . 10  |-  Y  C_  ( S  i^i  NN )
44 inss2 3354 . . . . . . . . . 10  |-  ( S  i^i  NN )  C_  NN
4543, 44sstri 3162 . . . . . . . . 9  |-  Y  C_  NN
4645, 1sselid 3151 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
4746ad2antrr 488 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  N  e.  NN )
48 2sqlem9.6 . . . . . . . . 9  |-  ( ph  ->  M  e.  NN )
4948ad2antrr 488 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  e.  NN )
50 simprr 531 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  =/=  1 )
51 eluz2b3 9575 . . . . . . . 8  |-  ( M  e.  ( ZZ>= `  2
)  <->  ( M  e.  NN  /\  M  =/=  1 ) )
5249, 50, 51sylanbrc 417 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  e.  ( ZZ>= `  2 )
)
53 simplrl 535 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  u  e.  ZZ )
54 simplrr 536 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  v  e.  ZZ )
55 simprll 537 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  (
u  gcd  v )  =  1 )
56 simprlr 538 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) )
57 eqid 2175 . . . . . . 7  |-  ( ( ( u  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  =  ( ( ( u  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
58 eqid 2175 . . . . . . 7  |-  ( ( ( v  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  =  ( ( ( v  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
59 eqid 2175 . . . . . . 7  |-  ( ( ( ( u  +  ( M  /  2
) )  mod  M
)  -  ( M  /  2 ) )  /  ( ( ( ( u  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  gcd  ( ( ( v  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) ) ) )  =  ( ( ( ( u  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )  /  ( ( ( ( u  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  gcd  ( ( ( v  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) ) ) )
60 eqid 2175 . . . . . . 7  |-  ( ( ( ( v  +  ( M  /  2
) )  mod  M
)  -  ( M  /  2 ) )  /  ( ( ( ( u  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  gcd  ( ( ( v  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) ) ) )  =  ( ( ( ( v  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )  /  ( ( ( ( u  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  gcd  ( ( ( v  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) ) ) )
6135, 19, 40, 42, 47, 52, 53, 54, 55, 56, 57, 58, 59, 602sqlem8 14028 . . . . . 6  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  e.  S )
6261anassrs 400 . . . . 5  |-  ( ( ( ( ph  /\  ( u  e.  ZZ  /\  v  e.  ZZ ) )  /\  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) ) )  /\  M  =/=  1
)  ->  M  e.  S )
6348nnzd 9345 . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
6463ad2antrr 488 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
u  gcd  v )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) ) )  ->  M  e.  ZZ )
65 zdceq 9299 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  1  e.  ZZ )  -> DECID  M  =  1 )
6664, 24, 65sylancl 413 . . . . . 6  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
u  gcd  v )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) ) )  -> DECID 
M  =  1 )
67 dcne 2356 . . . . . 6  |-  (DECID  M  =  1  <->  ( M  =  1  \/  M  =/=  1 ) )
6866, 67sylib 122 . . . . 5  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
u  gcd  v )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) ) )  ->  ( M  =  1  \/  M  =/=  1 ) )
6938, 62, 68mpjaodan 798 . . . 4  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
u  gcd  v )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) ) )  ->  M  e.  S
)
7069ex 115 . . 3  |-  ( (
ph  /\  ( u  e.  ZZ  /\  v  e.  ZZ ) )  -> 
( ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) )  ->  M  e.  S ) )
7170rexlimdvva 2600 . 2  |-  ( ph  ->  ( E. u  e.  ZZ  E. v  e.  ZZ  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) )  ->  M  e.  S ) )
7222, 71mpd 13 1  |-  ( ph  ->  M  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 708  DECID wdc 834    = wceq 1353    e. wcel 2146   {cab 2161    =/= wne 2345   A.wral 2453   E.wrex 2454    i^i cin 3126   class class class wbr 3998    |-> cmpt 4059   ran crn 4621   ` cfv 5208  (class class class)co 5865   1c1 7787    + caddc 7789    - cmin 8102    / cdiv 8601   NNcn 8890   2c2 8941   ZZcz 9224   ZZ>=cuz 9499   ...cfz 9977    mod cmo 10290   ^cexp 10487   abscabs 10972    || cdvds 11760    gcd cgcd 11908   ZZ[_i]cgz 12332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-1o 6407  df-2o 6408  df-er 6525  df-en 6731  df-sup 6973  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fzo 10111  df-fl 10238  df-mod 10291  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-dvds 11761  df-gcd 11909  df-prm 12073  df-gz 12333
This theorem is referenced by:  2sqlem10  14030
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