Theorem 2sqlem8a 16241

Description: Lemma for 2sqlem8 16242. (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

Step	Hyp	Ref	Expression
1		2sqlem8.1	. . . 4 |- ( ph -> A e. ZZ )
2		2sqlem8.m	. . . . . 6 |- ( ph -> M e. ( ZZ>= ` 2 ) )
3		eluz2b3 10004	. . . . . 6 |- ( M e. ( ZZ>= ` 2 ) <-> ( M e. NN /\ M =/= 1 ) )
4	2, 3	sylib 122	. . . . 5 |- ( ph -> ( M e. NN /\ M =/= 1 ) )
5	4	simpld 112	. . . 4 |- ( ph -> M e. NN )
6		2sqlem8.c	. . . 4 |- C = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )
7	1, 5, 6	4sqlem5 13161	. . 3 |- ( ph -> ( C e. ZZ /\ ( ( A - C ) / M ) e. ZZ ) )
8	7	simpld 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 ) )
11	9, 5, 10	4sqlem5 13161	. . 3 |- ( ph -> ( D e. ZZ /\ ( ( B - D ) / M ) e. ZZ ) )
12	11	simpld 112	. 2 |- ( ph -> D e. ZZ )
13	4	simprd 114	. . . 4 |- ( ph -> M =/= 1 )
14		simpr 110	. . . . . . . . . 10 |- ( ( ph /\ ( C ^ 2 ) = 0 ) -> ( C ^ 2 ) = 0 )
15	1, 5, 6, 14	4sqlem9 13165	. . . . . . . . 9 |- ( ( ph /\ ( C ^ 2 ) = 0 ) -> ( M ^ 2 ) || ( A ^ 2 ) )
16	15	ex 115	. . . . . . . 8 |- ( ph -> ( ( C ^ 2 ) = 0 -> ( M ^ 2 ) || ( A ^ 2 ) ) )
17		eluzelz 9931	. . . . . . . . . 10 |- ( M e. ( ZZ>= ` 2 ) -> M e. ZZ )
18	2, 17	syl 14	. . . . . . . . 9 |- ( ph -> M e. ZZ )
19		dvdssq 12808	. . . . . . . . 9 |- ( ( M e. ZZ /\ A e. ZZ ) -> ( M || A <-> ( M ^ 2 ) || ( A ^ 2 ) ) )
20	18, 1, 19	syl2anc 415	. . . . . . . 8 |- ( ph -> ( M || A <-> ( M ^ 2 ) || ( A ^ 2 ) ) )
21	16, 20	sylibrd 169	. . . . . . 7 |- ( ph -> ( ( C ^ 2 ) = 0 -> M || A ) )
22		simpr 110	. . . . . . . . . 10 |- ( ( ph /\ ( D ^ 2 ) = 0 ) -> ( D ^ 2 ) = 0 )
23	9, 5, 10, 22	4sqlem9 13165	. . . . . . . . 9 |- ( ( ph /\ ( D ^ 2 ) = 0 ) -> ( M ^ 2 ) || ( B ^ 2 ) )
24	23	ex 115	. . . . . . . 8 |- ( ph -> ( ( D ^ 2 ) = 0 -> ( M ^ 2 ) || ( B ^ 2 ) ) )
25		dvdssq 12808	. . . . . . . . 9 |- ( ( M e. ZZ /\ B e. ZZ ) -> ( M || B <-> ( M ^ 2 ) || ( B ^ 2 ) ) )
26	18, 9, 25	syl2anc 415	. . . . . . . 8 |- ( ph -> ( M || B <-> ( M ^ 2 ) || ( B ^ 2 ) ) )
27	24, 26	sylibrd 169	. . . . . . 7 |- ( ph -> ( ( D ^ 2 ) = 0 -> M || B ) )
28		2sqlem8.3	. . . . . . . . . . 11 |- ( ph -> ( A gcd B ) = 1 )
29		1ne0 9372	. . . . . . . . . . . 12 |- 1 =/= 0
30	29	a1i 9	. . . . . . . . . . 11 |- ( ph -> 1 =/= 0 )
31	28, 30	eqnetrd 2444	. . . . . . . . . 10 |- ( ph -> ( A gcd B ) =/= 0 )
32	31	neneqd 2441	. . . . . . . . 9 |- ( ph -> -. ( A gcd B ) = 0 )
33		gcdeq0 12754	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
34	1, 9, 33	syl2anc 415	. . . . . . . . 9 |- ( ph -> ( ( A gcd B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
35	32, 34	mtbid 683	. . . . . . . 8 |- ( ph -> -. ( A = 0 /\ B = 0 ) )
36		dvdslegcd 12741	. . . . . . . 8 |- ( ( ( M e. ZZ /\ A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( ( M || A /\ M || B ) -> M <_ ( A gcd B ) ) )
37	18, 1, 9, 35, 36	syl31anc 1281	. . . . . . 7 |- ( ph -> ( ( M || A /\ M || B ) -> M <_ ( A gcd B ) ) )
38	21, 27, 37	syl2and 295	. . . . . 6 |- ( ph -> ( ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) -> M <_ ( A gcd B ) ) )
39	28	breq2d 4142	. . . . . . 7 |- ( ph -> ( M <_ ( A gcd B ) <-> M <_ 1 ) )
40		nnle1eq1 9328	. . . . . . . 8 |- ( M e. NN -> ( M <_ 1 <-> M = 1 ) )
41	5, 40	syl 14	. . . . . . 7 |- ( ph -> ( M <_ 1 <-> M = 1 ) )
42	39, 41	bitrd 188	. . . . . 6 |- ( ph -> ( M <_ ( A gcd B ) <-> M = 1 ) )
43	38, 42	sylibd 149	. . . . 5 |- ( ph -> ( ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) -> M = 1 ) )
44	43	necon3ad 2462	. . . 4 |- ( ph -> ( M =/= 1 -> -. ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) ) )
45	13, 44	mpd 13	. . 3 |- ( ph -> -. ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) )
46	8	zcnd 9769	. . . . 5 |- ( ph -> C e. CC )
47		sqeq0 11039	. . . . 5 |- ( C e. CC -> ( ( C ^ 2 ) = 0 <-> C = 0 ) )
48	46, 47	syl 14	. . . 4 |- ( ph -> ( ( C ^ 2 ) = 0 <-> C = 0 ) )
49	12	zcnd 9769	. . . . 5 |- ( ph -> D e. CC )
50		sqeq0 11039	. . . . 5 |- ( D e. CC -> ( ( D ^ 2 ) = 0 <-> D = 0 ) )
51	49, 50	syl 14	. . . 4 |- ( ph -> ( ( D ^ 2 ) = 0 <-> D = 0 ) )
52	48, 51	anbi12d 477	. . 3 |- ( ph -> ( ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) <-> ( C = 0 /\ D = 0 ) ) )
53	45, 52	mtbid 683	. 2 |- ( ph -> -. ( C = 0 /\ D = 0 ) )
54		gcdn0cl 12739	. 2 |- ( ( ( C e. ZZ /\ D e. ZZ ) /\ -. ( C = 0 /\ D = 0 ) ) -> ( C gcd D ) e. NN )
55	8, 12, 53, 54	syl21anc 1277	1 |- ( ph -> ( C gcd D ) e. NN )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   =/= wne 2420   A.wral 2528   E.wrex 2529   class class class wbr 4130   |-> cmpt 4192   ran crn 4775   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   <_ cle 8361   - cmin 8497   / cdiv 9002   NNcn 9304   2c2 9355   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411   mod cmo 10759   ^cexp 10975   abscabs 11763   || cdvds 12554   gcd cgcd 12730   ZZ[_i]cgz 13148

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731

This theorem is used by:  2sqlem8  16242