Theorem 2sqlem9 15634

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 2212	. . . . . . . 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 2531	. . . . . 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 5953	. . . . . . . . 9 |- ( x = u -> ( x gcd y ) = ( u gcd y ) )
6	5	eqeq1d 2214	. . . . . . . 8 |- ( x = u -> ( ( x gcd y ) = 1 <-> ( u gcd y ) = 1 ) )
7		oveq1 5953	. . . . . . . . . 10 |- ( x = u -> ( x ^ 2 ) = ( u ^ 2 ) )
8	7	oveq1d 5961	. . . . . . . . 9 |- ( x = u -> ( ( x ^ 2 ) + ( y ^ 2 ) ) = ( ( u ^ 2 ) + ( y ^ 2 ) ) )
9	8	eqeq2d 2217	. . . . . . . 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 5954	. . . . . . . . 9 |- ( y = v -> ( u gcd y ) = ( u gcd v ) )
12	11	eqeq1d 2214	. . . . . . . 8 |- ( y = v -> ( ( u gcd y ) = 1 <-> ( u gcd v ) = 1 ) )
13		oveq1 5953	. . . . . . . . . 10 |- ( y = v -> ( y ^ 2 ) = ( v ^ 2 ) )
14	13	oveq2d 5962	. . . . . . . . 9 |- ( y = v -> ( ( u ^ 2 ) + ( y ^ 2 ) ) = ( ( u ^ 2 ) + ( v ^ 2 ) ) )
15	14	eqeq2d 2217	. . . . . . . 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 2750	. . . . . 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 2920	. . . 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 9400	. . . . . . . . 9 |- 1 e. ZZ
25		zgz 12729	. . . . . . . . 9 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
26	24, 25	ax-mp 5	. . . . . . . 8 |- 1 e. ZZ[_i]
27		sq1 10780	. . . . . . . . 9 |- ( 1 ^ 2 ) = 1
28	27	eqcomi 2209	. . . . . . . 8 |- 1 = ( 1 ^ 2 )
29		fveq2 5578	. . . . . . . . . . 11 |- ( x = 1 -> ( abs ` x ) = ( abs ` 1 ) )
30		abs1 11416	. . . . . . . . . . 11 |- ( abs ` 1 ) = 1
31	29, 30	eqtrdi 2254	. . . . . . . . . 10 |- ( x = 1 -> ( abs ` x ) = 1 )
32	31	oveq1d 5961	. . . . . . . . 9 |- ( x = 1 -> ( ( abs ` x ) ^ 2 ) = ( 1 ^ 2 ) )
33	32	rspceeqv 2895	. . . . . . . 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 15624	. . . . . . 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 2296	. . . . 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 15631	. . . . . . . . . 10 |- Y C_ ( S i^i NN )
44		inss2 3394	. . . . . . . . . 10 |- ( S i^i NN ) C_ NN
45	43, 44	sstri 3202	. . . . . . . . 9 |- Y C_ NN
46	45, 1	sselid 3191	. . . . . . . 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 9727	. . . . . . . 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 2205	. . . . . . 7 |- ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) ) = ( ( ( u + ( M / 2 ) ) mod M ) - ( M / 2 ) )
58		eqid 2205	. . . . . . 7 |- ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) ) = ( ( ( v + ( M / 2 ) ) mod M ) - ( M / 2 ) )
59		eqid 2205	. . . . . . 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 2205	. . . . . . 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 15633	. . . . . 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 9496	. . . . . . . 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 9450	. . . . . . 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 2387	. . . . . 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 800	. . . 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 2631	. 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 710  DECID wdc 836   = wceq 1373   e. wcel 2176   {cab 2191   =/= wne 2376   A.wral 2484   E.wrex 2485   i^i cin 3165   class class class wbr 4045   |-> cmpt 4106   ran crn 4677   ` cfv 5272  (class class class)co 5946   1c1 7928   + caddc 7930   - cmin 8245   / cdiv 8747   NNcn 9038   2c2 9089   ZZcz 9374   ZZ>=cuz 9650   ...cfz 10132   mod cmo 10469   ^cexp 10685   abscabs 11341   || cdvds 12131   gcd cgcd 12307   ZZ[_i]cgz 12725

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-1o 6504  df-2o 6505  df-er 6622  df-en 6830  df-sup 7088  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-fz 10133  df-fzo 10267  df-fl 10415  df-mod 10470  df-seqfrec 10595  df-exp 10686  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-dvds 12132  df-gcd 12308  df-prm 12463  df-gz 12726

This theorem is referenced by:  2sqlem10  15635