Theorem 2sqlem8a 24291

Description: Lemma for 2sqlem8 24292. (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 11234	. . . . . 6 |- ( M e. ( ZZ>= ` 2 ) <-> ( M e. NN /\ M =/= 1 ) )
4	2, 3	sylib 200	. . . . 5 |- ( ph -> ( M e. NN /\ M =/= 1 ) )
5	4	simpld 461	. . . 4 |- ( ph -> M e. NN )
6		2sqlem8.c	. . . 4 |- C = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )
7	1, 5, 6	4sqlem5 14879	. . 3 |- ( ph -> ( C e. ZZ /\ ( ( A - C ) / M ) e. ZZ ) )
8	7	simpld 461	. 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 14879	. . 3 |- ( ph -> ( D e. ZZ /\ ( ( B - D ) / M ) e. ZZ ) )
12	11	simpld 461	. 2 |- ( ph -> D e. ZZ )
13	4	simprd 465	. . . 4 |- ( ph -> M =/= 1 )
14		simpr 463	. . . . . . . . . 10 |- ( ( ph /\ ( C ^ 2 ) = 0 ) -> ( C ^ 2 ) = 0 )
15	1, 5, 6, 14	4sqlem9 14883	. . . . . . . . 9 |- ( ( ph /\ ( C ^ 2 ) = 0 ) -> ( M ^ 2 ) || ( A ^ 2 ) )
16	15	ex 436	. . . . . . . 8 |- ( ph -> ( ( C ^ 2 ) = 0 -> ( M ^ 2 ) || ( A ^ 2 ) ) )
17		eluzelz 11170	. . . . . . . . . 10 |- ( M e. ( ZZ>= ` 2 ) -> M e. ZZ )
18	2, 17	syl 17	. . . . . . . . 9 |- ( ph -> M e. ZZ )
19		dvdssq 14521	. . . . . . . . 9 |- ( ( M e. ZZ /\ A e. ZZ ) -> ( M || A <-> ( M ^ 2 ) || ( A ^ 2 ) ) )
20	18, 1, 19	syl2anc 666	. . . . . . . 8 |- ( ph -> ( M || A <-> ( M ^ 2 ) || ( A ^ 2 ) ) )
21	16, 20	sylibrd 238	. . . . . . 7 |- ( ph -> ( ( C ^ 2 ) = 0 -> M || A ) )
22		simpr 463	. . . . . . . . . 10 |- ( ( ph /\ ( D ^ 2 ) = 0 ) -> ( D ^ 2 ) = 0 )
23	9, 5, 10, 22	4sqlem9 14883	. . . . . . . . 9 |- ( ( ph /\ ( D ^ 2 ) = 0 ) -> ( M ^ 2 ) || ( B ^ 2 ) )
24	23	ex 436	. . . . . . . 8 |- ( ph -> ( ( D ^ 2 ) = 0 -> ( M ^ 2 ) || ( B ^ 2 ) ) )
25		dvdssq 14521	. . . . . . . . 9 |- ( ( M e. ZZ /\ B e. ZZ ) -> ( M || B <-> ( M ^ 2 ) || ( B ^ 2 ) ) )
26	18, 9, 25	syl2anc 666	. . . . . . . 8 |- ( ph -> ( M || B <-> ( M ^ 2 ) || ( B ^ 2 ) ) )
27	24, 26	sylibrd 238	. . . . . . 7 |- ( ph -> ( ( D ^ 2 ) = 0 -> M || B ) )
28		2sqlem8.3	. . . . . . . . . . 11 |- ( ph -> ( A gcd B ) = 1 )
29		ax-1ne0 9610	. . . . . . . . . . . 12 |- 1 =/= 0
30	29	a1i 11	. . . . . . . . . . 11 |- ( ph -> 1 =/= 0 )
31	28, 30	eqnetrd 2718	. . . . . . . . . 10 |- ( ph -> ( A gcd B ) =/= 0 )
32	31	neneqd 2626	. . . . . . . . 9 |- ( ph -> -. ( A gcd B ) = 0 )
33		gcdeq0 14478	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
34	1, 9, 33	syl2anc 666	. . . . . . . . 9 |- ( ph -> ( ( A gcd B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
35	32, 34	mtbid 302	. . . . . . . 8 |- ( ph -> -. ( A = 0 /\ B = 0 ) )
36		dvdslegcd 14471	. . . . . . . 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 1268	. . . . . . 7 |- ( ph -> ( ( M || A /\ M || B ) -> M <_ ( A gcd B ) ) )
38	21, 27, 37	syl2and 486	. . . . . 6 |- ( ph -> ( ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) -> M <_ ( A gcd B ) ) )
39	28	breq2d 4433	. . . . . . 7 |- ( ph -> ( M <_ ( A gcd B ) <-> M <_ 1 ) )
40		nnle1eq1 10639	. . . . . . . 8 |- ( M e. NN -> ( M <_ 1 <-> M = 1 ) )
41	5, 40	syl 17	. . . . . . 7 |- ( ph -> ( M <_ 1 <-> M = 1 ) )
42	39, 41	bitrd 257	. . . . . 6 |- ( ph -> ( M <_ ( A gcd B ) <-> M = 1 ) )
43	38, 42	sylibd 218	. . . . 5 |- ( ph -> ( ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) -> M = 1 ) )
44	43	necon3ad 2635	. . . 4 |- ( ph -> ( M =/= 1 -> -. ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) ) )
45	13, 44	mpd 15	. . 3 |- ( ph -> -. ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) )
46	8	zcnd 11043	. . . . 5 |- ( ph -> C e. CC )
47		sqeq0 12340	. . . . 5 |- ( C e. CC -> ( ( C ^ 2 ) = 0 <-> C = 0 ) )
48	46, 47	syl 17	. . . 4 |- ( ph -> ( ( C ^ 2 ) = 0 <-> C = 0 ) )
49	12	zcnd 11043	. . . . 5 |- ( ph -> D e. CC )
50		sqeq0 12340	. . . . 5 |- ( D e. CC -> ( ( D ^ 2 ) = 0 <-> D = 0 ) )
51	49, 50	syl 17	. . . 4 |- ( ph -> ( ( D ^ 2 ) = 0 <-> D = 0 ) )
52	48, 51	anbi12d 716	. . 3 |- ( ph -> ( ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) <-> ( C = 0 /\ D = 0 ) ) )
53	45, 52	mtbid 302	. 2 |- ( ph -> -. ( C = 0 /\ D = 0 ) )
54		gcdn0cl 14469	. 2 |- ( ( ( C e. ZZ /\ D e. ZZ ) /\ -. ( C = 0 /\ D = 0 ) ) -> ( C gcd D ) e. NN )
55	8, 12, 53, 54	syl21anc 1264	1 |- ( ph -> ( C gcd D ) e. NN )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1438   e. wcel 1869   {cab 2408   =/= wne 2619   A.wral 2776   E.wrex 2777   class class class wbr 4421   |-> cmpt 4480   ran crn 4852   ` cfv 5599  (class class class)co 6303   CCcc 9539   0cc0 9541   1c1 9542   + caddc 9544   <_ cle 9678   - cmin 9862   / cdiv 10271   NNcn 10611   2c2 10661   ZZcz 10939   ZZ>=cuz 11161   ...cfz 11786   mod cmo 12097   ^cexp 12273   abscabs 13291   || cdvds 14298   gcd cgcd 14461   ZZ[_i]cgz 14866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-sup 7960  df-inf 7961  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-n0 10872  df-z 10940  df-uz 11162  df-rp 11305  df-fl 12029  df-mod 12098  df-seq 12215  df-exp 12274  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-dvds 14299  df-gcd 14462

This theorem is referenced by:  2sqlem8  24292