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Theorem 2sqlem8a 14027
Description: Lemma for 2sqlem8 14028. (Contributed by Mario Carneiro, 4-Jun-2016.)
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 )
2sqlem8.n  |-  ( ph  ->  N  e.  NN )
2sqlem8.m  |-  ( ph  ->  M  e.  ( ZZ>= ` 
2 ) )
2sqlem8.1  |-  ( ph  ->  A  e.  ZZ )
2sqlem8.2  |-  ( ph  ->  B  e.  ZZ )
2sqlem8.3  |-  ( ph  ->  ( A  gcd  B
)  =  1 )
2sqlem8.4  |-  ( ph  ->  N  =  ( ( A ^ 2 )  +  ( B ^
2 ) ) )
2sqlem8.c  |-  C  =  ( ( ( A  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
2sqlem8.d  |-  D  =  ( ( ( B  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
Assertion
Ref Expression
2sqlem8a  |-  ( ph  ->  ( C  gcd  D
)  e.  NN )
Distinct variable groups:    a, b, w, x, y, z    A, a, x, y, z    x, C    ph, x, y    B, a, b, x, y    M, a, b, x, y, z    S, a, b, x, y, z    x, D    x, N, y, z    Y, a, b, x, y
Allowed substitution hints:    ph( z, w, a, b)    A( w, b)    B( z, w)    C( y, z, w, a, b)    D( y, z, w, a, b)    S( w)    M( w)    N( w, a, b)    Y( z, w)

Proof of Theorem 2sqlem8a
StepHypRef Expression
1 2sqlem8.1 . . . 4  |-  ( ph  ->  A  e.  ZZ )
2 2sqlem8.m . . . . . 6  |-  ( ph  ->  M  e.  ( ZZ>= ` 
2 ) )
3 eluz2b3 9575 . . . . . 6  |-  ( M  e.  ( ZZ>= `  2
)  <->  ( M  e.  NN  /\  M  =/=  1 ) )
42, 3sylib 122 . . . . 5  |-  ( ph  ->  ( M  e.  NN  /\  M  =/=  1 ) )
54simpld 112 . . . 4  |-  ( ph  ->  M  e.  NN )
6 2sqlem8.c . . . 4  |-  C  =  ( ( ( A  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
71, 5, 64sqlem5 12345 . . 3  |-  ( ph  ->  ( C  e.  ZZ  /\  ( ( A  -  C )  /  M
)  e.  ZZ ) )
87simpld 112 . 2  |-  ( ph  ->  C  e.  ZZ )
9 2sqlem8.2 . . . 4  |-  ( ph  ->  B  e.  ZZ )
10 2sqlem8.d . . . 4  |-  D  =  ( ( ( B  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
119, 5, 104sqlem5 12345 . . 3  |-  ( ph  ->  ( D  e.  ZZ  /\  ( ( B  -  D )  /  M
)  e.  ZZ ) )
1211simpld 112 . 2  |-  ( ph  ->  D  e.  ZZ )
134simprd 114 . . . 4  |-  ( ph  ->  M  =/=  1 )
14 simpr 110 . . . . . . . . . 10  |-  ( (
ph  /\  ( C ^ 2 )  =  0 )  ->  ( C ^ 2 )  =  0 )
151, 5, 6, 144sqlem9 12349 . . . . . . . . 9  |-  ( (
ph  /\  ( C ^ 2 )  =  0 )  ->  ( M ^ 2 )  ||  ( A ^ 2 ) )
1615ex 115 . . . . . . . 8  |-  ( ph  ->  ( ( C ^
2 )  =  0  ->  ( M ^
2 )  ||  ( A ^ 2 ) ) )
17 eluzelz 9508 . . . . . . . . . 10  |-  ( M  e.  ( ZZ>= `  2
)  ->  M  e.  ZZ )
182, 17syl 14 . . . . . . . . 9  |-  ( ph  ->  M  e.  ZZ )
19 dvdssq 11997 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  ZZ )  ->  ( M  ||  A  <->  ( M ^ 2 ) 
||  ( A ^
2 ) ) )
2018, 1, 19syl2anc 411 . . . . . . . 8  |-  ( ph  ->  ( M  ||  A  <->  ( M ^ 2 ) 
||  ( A ^
2 ) ) )
2116, 20sylibrd 169 . . . . . . 7  |-  ( ph  ->  ( ( C ^
2 )  =  0  ->  M  ||  A
) )
22 simpr 110 . . . . . . . . . 10  |-  ( (
ph  /\  ( D ^ 2 )  =  0 )  ->  ( D ^ 2 )  =  0 )
239, 5, 10, 224sqlem9 12349 . . . . . . . . 9  |-  ( (
ph  /\  ( D ^ 2 )  =  0 )  ->  ( M ^ 2 )  ||  ( B ^ 2 ) )
2423ex 115 . . . . . . . 8  |-  ( ph  ->  ( ( D ^
2 )  =  0  ->  ( M ^
2 )  ||  ( B ^ 2 ) ) )
25 dvdssq 11997 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  B  e.  ZZ )  ->  ( M  ||  B  <->  ( M ^ 2 ) 
||  ( B ^
2 ) ) )
2618, 9, 25syl2anc 411 . . . . . . . 8  |-  ( ph  ->  ( M  ||  B  <->  ( M ^ 2 ) 
||  ( B ^
2 ) ) )
2724, 26sylibrd 169 . . . . . . 7  |-  ( ph  ->  ( ( D ^
2 )  =  0  ->  M  ||  B
) )
28 2sqlem8.3 . . . . . . . . . . 11  |-  ( ph  ->  ( A  gcd  B
)  =  1 )
29 1ne0 8958 . . . . . . . . . . . 12  |-  1  =/=  0
3029a1i 9 . . . . . . . . . . 11  |-  ( ph  ->  1  =/=  0 )
3128, 30eqnetrd 2369 . . . . . . . . . 10  |-  ( ph  ->  ( A  gcd  B
)  =/=  0 )
3231neneqd 2366 . . . . . . . . 9  |-  ( ph  ->  -.  ( A  gcd  B )  =  0 )
33 gcdeq0 11943 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
341, 9, 33syl2anc 411 . . . . . . . . 9  |-  ( ph  ->  ( ( A  gcd  B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
3532, 34mtbid 672 . . . . . . . 8  |-  ( ph  ->  -.  ( A  =  0  /\  B  =  0 ) )
36 dvdslegcd 11930 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( M  ||  A  /\  M  ||  B
)  ->  M  <_  ( A  gcd  B ) ) )
3718, 1, 9, 35, 36syl31anc 1241 . . . . . . 7  |-  ( ph  ->  ( ( M  ||  A  /\  M  ||  B
)  ->  M  <_  ( A  gcd  B ) ) )
3821, 27, 37syl2and 295 . . . . . 6  |-  ( ph  ->  ( ( ( C ^ 2 )  =  0  /\  ( D ^ 2 )  =  0 )  ->  M  <_  ( A  gcd  B
) ) )
3928breq2d 4010 . . . . . . 7  |-  ( ph  ->  ( M  <_  ( A  gcd  B )  <->  M  <_  1 ) )
40 nnle1eq1 8914 . . . . . . . 8  |-  ( M  e.  NN  ->  ( M  <_  1  <->  M  = 
1 ) )
415, 40syl 14 . . . . . . 7  |-  ( ph  ->  ( M  <_  1  <->  M  =  1 ) )
4239, 41bitrd 188 . . . . . 6  |-  ( ph  ->  ( M  <_  ( A  gcd  B )  <->  M  = 
1 ) )
4338, 42sylibd 149 . . . . 5  |-  ( ph  ->  ( ( ( C ^ 2 )  =  0  /\  ( D ^ 2 )  =  0 )  ->  M  =  1 ) )
4443necon3ad 2387 . . . 4  |-  ( ph  ->  ( M  =/=  1  ->  -.  ( ( C ^ 2 )  =  0  /\  ( D ^ 2 )  =  0 ) ) )
4513, 44mpd 13 . . 3  |-  ( ph  ->  -.  ( ( C ^ 2 )  =  0  /\  ( D ^ 2 )  =  0 ) )
468zcnd 9347 . . . . 5  |-  ( ph  ->  C  e.  CC )
47 sqeq0 10551 . . . . 5  |-  ( C  e.  CC  ->  (
( C ^ 2 )  =  0  <->  C  =  0 ) )
4846, 47syl 14 . . . 4  |-  ( ph  ->  ( ( C ^
2 )  =  0  <-> 
C  =  0 ) )
4912zcnd 9347 . . . . 5  |-  ( ph  ->  D  e.  CC )
50 sqeq0 10551 . . . . 5  |-  ( D  e.  CC  ->  (
( D ^ 2 )  =  0  <->  D  =  0 ) )
5149, 50syl 14 . . . 4  |-  ( ph  ->  ( ( D ^
2 )  =  0  <-> 
D  =  0 ) )
5248, 51anbi12d 473 . . 3  |-  ( ph  ->  ( ( ( C ^ 2 )  =  0  /\  ( D ^ 2 )  =  0 )  <->  ( C  =  0  /\  D  =  0 ) ) )
5345, 52mtbid 672 . 2  |-  ( ph  ->  -.  ( C  =  0  /\  D  =  0 ) )
54 gcdn0cl 11928 . 2  |-  ( ( ( C  e.  ZZ  /\  D  e.  ZZ )  /\  -.  ( C  =  0  /\  D  =  0 ) )  ->  ( C  gcd  D )  e.  NN )
558, 12, 53, 54syl21anc 1237 1  |-  ( ph  ->  ( C  gcd  D
)  e.  NN )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2146   {cab 2161    =/= wne 2345   A.wral 2453   E.wrex 2454   class class class wbr 3998    |-> cmpt 4059   ran crn 4621   ` cfv 5208  (class class class)co 5865   CCcc 7784   0cc0 7786   1c1 7787    + caddc 7789    <_ cle 7967    - 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-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
This theorem is referenced by:  2sqlem8  14028
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