Theorem 2sqlem9 15211

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.

Step	Hyp	Ref	Expression
1		2sqlem9.4	. . 3 |- ( ph -> N e. Y )
2		eqeq1 2200	. . . . . . . 8 |- ( z = N -> ( z = ( ( x ^ 2 ) + ( y ^ 2 ) ) <-> N = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) )
3	2	anbi2d 464	. . . . . . 7 |- ( z = N -> ( ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> ( ( x gcd y ) = 1 /\ N = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) )
4	3	2rexbidv 2519	. . . . . 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 5925	. . . . . . . . 9 |- ( x = u -> ( x gcd y ) = ( u gcd y ) )
6	5	eqeq1d 2202	. . . . . . . 8 |- ( x = u -> ( ( x gcd y ) = 1 <-> ( u gcd y ) = 1 ) )
7		oveq1 5925	. . . . . . . . . 10 |- ( x = u -> ( x ^ 2 ) = ( u ^ 2 ) )
8	7	oveq1d 5933	. . . . . . . . 9 |- ( x = u -> ( ( x ^ 2 ) + ( y ^ 2 ) ) = ( ( u ^ 2 ) + ( y ^ 2 ) ) )
9	8	eqeq2d 2205	. . . . . . . 8 |- ( x = u -> ( N = ( ( x ^ 2 ) + ( y ^ 2 ) ) <-> N = ( ( u ^ 2 ) + ( y ^ 2 ) ) ) )
10	6, 9	anbi12d 473	. . . . . . 7 |- ( x = u -> ( ( ( x gcd y ) = 1 /\ N = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> ( ( u gcd y ) = 1 /\ N = ( ( u ^ 2 ) + ( y ^ 2 ) ) ) ) )
11		oveq2 5926	. . . . . . . . 9 |- ( y = v -> ( u gcd y ) = ( u gcd v ) )
12	11	eqeq1d 2202	. . . . . . . 8 |- ( y = v -> ( ( u gcd y ) = 1 <-> ( u gcd v ) = 1 ) )
13		oveq1 5925	. . . . . . . . . 10 |- ( y = v -> ( y ^ 2 ) = ( v ^ 2 ) )
14	13	oveq2d 5934	. . . . . . . . 9 |- ( y = v -> ( ( u ^ 2 ) + ( y ^ 2 ) ) = ( ( u ^ 2 ) + ( v ^ 2 ) ) )
15	14	eqeq2d 2205	. . . . . . . 8 |- ( y = v -> ( N = ( ( u ^ 2 ) + ( y ^ 2 ) ) <-> N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) )
16	12, 15	anbi12d 473	. . . . . . 7 |- ( y = v -> ( ( ( u gcd y ) = 1 /\ N = ( ( u ^ 2 ) + ( y ^ 2 ) ) ) <-> ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) )
17	10, 16	cbvrex2vw 2738	. . . . . 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 ) ) ) )
18	4, 17	bitrdi 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 ) ) ) }
20	18, 19	elab2g 2907	. . . 4 |- ( N e. Y -> ( N e. Y <-> E. u e. ZZ E. v e. ZZ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) )
21	20	ibi 176	. . 3 |- ( N e. Y -> E. u e. ZZ E. v e. ZZ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) )
22	1, 21	syl 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 9343	. . . . . . . . 9 |- 1 e. ZZ
25		zgz 12511	. . . . . . . . 9 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
26	24, 25	ax-mp 5	. . . . . . . 8 |- 1 e. ZZ[_i]
27		sq1 10704	. . . . . . . . 9 |- ( 1 ^ 2 ) = 1
28	27	eqcomi 2197	. . . . . . . 8 |- 1 = ( 1 ^ 2 )
29		fveq2 5554	. . . . . . . . . . 11 |- ( x = 1 -> ( abs ` x ) = ( abs ` 1 ) )
30		abs1 11216	. . . . . . . . . . 11 |- ( abs ` 1 ) = 1
31	29, 30	eqtrdi 2242	. . . . . . . . . 10 |- ( x = 1 -> ( abs ` x ) = 1 )
32	31	oveq1d 5933	. . . . . . . . 9 |- ( x = 1 -> ( ( abs ` x ) ^ 2 ) = ( 1 ^ 2 ) )
33	32	rspceeqv 2882	. . . . . . . 8 |- ( ( 1 e. ZZ[_i] /\ 1 = ( 1 ^ 2 ) ) -> E. x e. ZZ[_i] 1 = ( ( abs ` x ) ^ 2 ) )
34	26, 28, 33	mp2an 426	. . . . . . 7 |- E. x e. ZZ[_i] 1 = ( ( abs ` x ) ^ 2 )
35		2sq.1	. . . . . . . 8 |- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
36	35	2sqlem1 15201	. . . . . . 7 |- ( 1 e. S <-> E. x e. ZZ[_i] 1 = ( ( abs ` x ) ^ 2 ) )
37	34, 36	mpbir 146	. . . . . 6 |- 1 e. S
38	23, 37	eqeltrdi 2284	. . . . 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 ) )
40	39	ad2antrr 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 )
42	41	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> M || N )
43	35, 19	2sqlem7 15208	. . . . . . . . . 10 |- Y C_ ( S i^i NN )
44		inss2 3380	. . . . . . . . . 10 |- ( S i^i NN ) C_ NN
45	43, 44	sstri 3188	. . . . . . . . 9 |- Y C_ NN
46	45, 1	sselid 3177	. . . . . . . 8 |- ( ph -> N e. NN )
47	46	ad2antrr 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 )
49	48	ad2antrr 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 9669	. . . . . . . 8 |- ( M e. ( ZZ>= ` 2 ) <-> ( M e. NN /\ M =/= 1 ) )
52	49, 50, 51	sylanbrc 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 2193	. . . . . . 7 |- ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) ) = ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) )
58		eqid 2193	. . . . . . 7 |- ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) ) = ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) )
59		eqid 2193	. . . . . . 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 2193	. . . . . . 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 ) ) ) )
61	35, 19, 40, 42, 47, 52, 53, 54, 55, 56, 57, 58, 59, 60	2sqlem8 15210	. . . . . 6 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) /\ M =/= 1 ) ) -> M e. S )
62	61	anassrs 400	. . . . 5 |- ( ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) /\ M =/= 1 ) -> M e. S )
63	48	nnzd 9438	. . . . . . . 8 |- ( ph -> M e. ZZ )
64	63	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) -> M e. ZZ )
65		zdceq 9392	. . . . . . 7 |- ( ( M e. ZZ /\ 1 e. ZZ ) -> DECID M = 1 )
66	64, 24, 65	sylancl 413	. . . . . 6 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) -> DECID M = 1 )
67		dcne 2375	. . . . . 6 |- (DECID M = 1 <-> ( M = 1 \/ M =/= 1 ) )
68	66, 67	sylib 122	. . . . 5 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) -> ( M = 1 \/ M =/= 1 ) )
69	38, 62, 68	mpjaodan 799	. . . 4 |- ( ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) /\ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) ) -> M e. S )
70	69	ex 115	. . 3 |- ( ( ph /\ ( u e. ZZ /\ v e. ZZ ) ) -> ( ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) -> M e. S ) )
71	70	rexlimdvva 2619	. 2 |- ( ph -> ( E. u e. ZZ E. v e. ZZ ( ( u gcd v ) = 1 /\ N = ( ( u ^ 2 ) + ( v ^ 2 ) ) ) -> M e. S ) )
72	22, 71	mpd 13	1 |- ( ph -> M e. S )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2164   {cab 2179   =/= wne 2364   A.wral 2472   E.wrex 2473   i^i cin 3152   class class class wbr 4029   |-> cmpt 4090   ran crn 4660   ` cfv 5254  (class class class)co 5918   1c1 7873   + caddc 7875   - cmin 8190   / cdiv 8691   NNcn 8982   2c2 9033   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074   mod cmo 10393   ^cexp 10609   abscabs 11141   || cdvds 11930   gcd cgcd 12079   ZZ[_i]cgz 12507

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-er 6587  df-en 6795  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080  df-prm 12246  df-gz 12508

This theorem is referenced by:  2sqlem10  15212