Theorem 2sqlem8a 23367

Description: Lemma for 2sqlem8 23368. (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 11144	. . . . . 6 |- ( M e. ( ZZ>= ` 2 ) <-> ( M e. NN /\ M =/= 1 ) )
4	2, 3	sylib 196	. . . . 5 |- ( ph -> ( M e. NN /\ M =/= 1 ) )
5	4	simpld 459	. . . 4 |- ( ph -> M e. NN )
6		2sqlem8.c	. . . 4 |- C = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )
7	1, 5, 6	4sqlem5 14308	. . 3 |- ( ph -> ( C e. ZZ /\ ( ( A - C ) / M ) e. ZZ ) )
8	7	simpld 459	. 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 14308	. . 3 |- ( ph -> ( D e. ZZ /\ ( ( B - D ) / M ) e. ZZ ) )
12	11	simpld 459	. 2 |- ( ph -> D e. ZZ )
13	4	simprd 463	. . . 4 |- ( ph -> M =/= 1 )
14		simpr 461	. . . . . . . . . 10 |- ( ( ph /\ ( C ^ 2 ) = 0 ) -> ( C ^ 2 ) = 0 )
15	1, 5, 6, 14	4sqlem9 14312	. . . . . . . . 9 |- ( ( ph /\ ( C ^ 2 ) = 0 ) -> ( M ^ 2 ) || ( A ^ 2 ) )
16	15	ex 434	. . . . . . . 8 |- ( ph -> ( ( C ^ 2 ) = 0 -> ( M ^ 2 ) || ( A ^ 2 ) ) )
17		eluzelz 11080	. . . . . . . . . 10 |- ( M e. ( ZZ>= ` 2 ) -> M e. ZZ )
18	2, 17	syl 16	. . . . . . . . 9 |- ( ph -> M e. ZZ )
19		dvdssq 14046	. . . . . . . . 9 |- ( ( M e. ZZ /\ A e. ZZ ) -> ( M || A <-> ( M ^ 2 ) || ( A ^ 2 ) ) )
20	18, 1, 19	syl2anc 661	. . . . . . . 8 |- ( ph -> ( M || A <-> ( M ^ 2 ) || ( A ^ 2 ) ) )
21	16, 20	sylibrd 234	. . . . . . 7 |- ( ph -> ( ( C ^ 2 ) = 0 -> M || A ) )
22		simpr 461	. . . . . . . . . 10 |- ( ( ph /\ ( D ^ 2 ) = 0 ) -> ( D ^ 2 ) = 0 )
23	9, 5, 10, 22	4sqlem9 14312	. . . . . . . . 9 |- ( ( ph /\ ( D ^ 2 ) = 0 ) -> ( M ^ 2 ) || ( B ^ 2 ) )
24	23	ex 434	. . . . . . . 8 |- ( ph -> ( ( D ^ 2 ) = 0 -> ( M ^ 2 ) || ( B ^ 2 ) ) )
25		dvdssq 14046	. . . . . . . . 9 |- ( ( M e. ZZ /\ B e. ZZ ) -> ( M || B <-> ( M ^ 2 ) || ( B ^ 2 ) ) )
26	18, 9, 25	syl2anc 661	. . . . . . . 8 |- ( ph -> ( M || B <-> ( M ^ 2 ) || ( B ^ 2 ) ) )
27	24, 26	sylibrd 234	. . . . . . 7 |- ( ph -> ( ( D ^ 2 ) = 0 -> M || B ) )
28		2sqlem8.3	. . . . . . . . . . 11 |- ( ph -> ( A gcd B ) = 1 )
29		ax-1ne0 9550	. . . . . . . . . . . 12 |- 1 =/= 0
30	29	a1i 11	. . . . . . . . . . 11 |- ( ph -> 1 =/= 0 )
31	28, 30	eqnetrd 2753	. . . . . . . . . 10 |- ( ph -> ( A gcd B ) =/= 0 )
32	31	neneqd 2662	. . . . . . . . 9 |- ( ph -> -. ( A gcd B ) = 0 )
33		gcdeq0 14007	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
34	1, 9, 33	syl2anc 661	. . . . . . . . 9 |- ( ph -> ( ( A gcd B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
35	32, 34	mtbid 300	. . . . . . . 8 |- ( ph -> -. ( A = 0 /\ B = 0 ) )
36		dvdslegcd 14002	. . . . . . . 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 1226	. . . . . . 7 |- ( ph -> ( ( M || A /\ M || B ) -> M <_ ( A gcd B ) ) )
38	21, 27, 37	syl2and 483	. . . . . 6 |- ( ph -> ( ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) -> M <_ ( A gcd B ) ) )
39	28	breq2d 4452	. . . . . . 7 |- ( ph -> ( M <_ ( A gcd B ) <-> M <_ 1 ) )
40		nnle1eq1 10553	. . . . . . . 8 |- ( M e. NN -> ( M <_ 1 <-> M = 1 ) )
41	5, 40	syl 16	. . . . . . 7 |- ( ph -> ( M <_ 1 <-> M = 1 ) )
42	39, 41	bitrd 253	. . . . . 6 |- ( ph -> ( M <_ ( A gcd B ) <-> M = 1 ) )
43	38, 42	sylibd 214	. . . . 5 |- ( ph -> ( ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) -> M = 1 ) )
44	43	necon3ad 2670	. . . 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 10956	. . . . 5 |- ( ph -> C e. CC )
47		sqeq0 12187	. . . . 5 |- ( C e. CC -> ( ( C ^ 2 ) = 0 <-> C = 0 ) )
48	46, 47	syl 16	. . . 4 |- ( ph -> ( ( C ^ 2 ) = 0 <-> C = 0 ) )
49	12	zcnd 10956	. . . . 5 |- ( ph -> D e. CC )
50		sqeq0 12187	. . . . 5 |- ( D e. CC -> ( ( D ^ 2 ) = 0 <-> D = 0 ) )
51	49, 50	syl 16	. . . 4 |- ( ph -> ( ( D ^ 2 ) = 0 <-> D = 0 ) )
52	48, 51	anbi12d 710	. . 3 |- ( ph -> ( ( ( C ^ 2 ) = 0 /\ ( D ^ 2 ) = 0 ) <-> ( C = 0 /\ D = 0 ) ) )
53	45, 52	mtbid 300	. 2 |- ( ph -> -. ( C = 0 /\ D = 0 ) )
54		gcdn0cl 14000	. 2 |- ( ( ( C e. ZZ /\ D e. ZZ ) /\ -. ( C = 0 /\ D = 0 ) ) -> ( C gcd D ) e. NN )
55	8, 12, 53, 54	syl21anc 1222	1 |- ( ph -> ( C gcd D ) e. NN )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1374   e. wcel 1762   {cab 2445   =/= wne 2655   A.wral 2807   E.wrex 2808   class class class wbr 4440   |-> cmpt 4498   ran crn 4993   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482   + caddc 9484   <_ cle 9618   - cmin 9794   / cdiv 10195   NNcn 10525   2c2 10574   ZZcz 10853   ZZ>=cuz 11071   ...cfz 11661   mod cmo 11952   ^cexp 12122   abscabs 13017   || cdivides 13836   gcd cgcd 13992   ZZ[_i]cgz 14295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-dvds 13837  df-gcd 13993

This theorem is referenced by:  2sqlem8  23368