Theorem 2sqlem4 14923

Description: Lemma for 2sqlem5 14924. (Contributed by Mario Carneiro, 20-Jun-2015.)

Hypotheses

Ref	Expression
2sq.1	|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
2sqlem5.1	|- ( ph -> N e. NN )
2sqlem5.2	|- ( ph -> P e. Prime )
2sqlem4.3	|- ( ph -> A e. ZZ )
2sqlem4.4	|- ( ph -> B e. ZZ )
2sqlem4.5	|- ( ph -> C e. ZZ )
2sqlem4.6	|- ( ph -> D e. ZZ )
2sqlem4.7	|- ( ph -> ( N x. P ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
2sqlem4.8	|- ( ph -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )

Assertion

Ref	Expression
2sqlem4	|- ( ph -> N e. S )

Proof of Theorem 2sqlem4

Step	Hyp	Ref	Expression
1		2sq.1	. . 3 |- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
2		2sqlem5.1	. . . 4 |- ( ph -> N e. NN )
3	2	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> N e. NN )
4		2sqlem5.2	. . . 4 |- ( ph -> P e. Prime )
5	4	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> P e. Prime )
6		2sqlem4.3	. . . 4 |- ( ph -> A e. ZZ )
7	6	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> A e. ZZ )
8		2sqlem4.4	. . . 4 |- ( ph -> B e. ZZ )
9	8	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> B e. ZZ )
10		2sqlem4.5	. . . 4 |- ( ph -> C e. ZZ )
11	10	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> C e. ZZ )
12		2sqlem4.6	. . . 4 |- ( ph -> D e. ZZ )
13	12	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> D e. ZZ )
14		2sqlem4.7	. . . 4 |- ( ph -> ( N x. P ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
15	14	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> ( N x. P ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
16		2sqlem4.8	. . . 4 |- ( ph -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )
17	16	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )
18		simpr 110	. . 3 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> P || ( ( C x. B ) + ( A x. D ) ) )
19	1, 3, 5, 7, 9, 11, 13, 15, 17, 18	2sqlem3 14922	. 2 |- ( ( ph /\ P || ( ( C x. B ) + ( A x. D ) ) ) -> N e. S )
20	2	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> N e. NN )
21	4	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> P e. Prime )
22	6	znegcld 9407	. . . 4 |- ( ph -> -u A e. ZZ )
23	22	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> -u A e. ZZ )
24	8	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> B e. ZZ )
25	10	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> C e. ZZ )
26	12	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> D e. ZZ )
27	6	zcnd 9406	. . . . . . 7 |- ( ph -> A e. CC )
28		sqneg 10610	. . . . . . 7 |- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
29	27, 28	syl 14	. . . . . 6 |- ( ph -> ( -u A ^ 2 ) = ( A ^ 2 ) )
30	29	oveq1d 5911	. . . . 5 |- ( ph -> ( ( -u A ^ 2 ) + ( B ^ 2 ) ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
31	14, 30	eqtr4d 2225	. . . 4 |- ( ph -> ( N x. P ) = ( ( -u A ^ 2 ) + ( B ^ 2 ) ) )
32	31	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> ( N x. P ) = ( ( -u A ^ 2 ) + ( B ^ 2 ) ) )
33	16	adantr 276	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )
34	12	zcnd 9406	. . . . . . . 8 |- ( ph -> D e. CC )
35	27, 34	mulneg1d 8398	. . . . . . 7 |- ( ph -> ( -u A x. D ) = -u ( A x. D ) )
36	35	oveq2d 5912	. . . . . 6 |- ( ph -> ( ( C x. B ) + ( -u A x. D ) ) = ( ( C x. B ) + -u ( A x. D ) ) )
37	10, 8	zmulcld 9411	. . . . . . . 8 |- ( ph -> ( C x. B ) e. ZZ )
38	37	zcnd 9406	. . . . . . 7 |- ( ph -> ( C x. B ) e. CC )
39	6, 12	zmulcld 9411	. . . . . . . 8 |- ( ph -> ( A x. D ) e. ZZ )
40	39	zcnd 9406	. . . . . . 7 |- ( ph -> ( A x. D ) e. CC )
41	38, 40	negsubd 8304	. . . . . 6 |- ( ph -> ( ( C x. B ) + -u ( A x. D ) ) = ( ( C x. B ) - ( A x. D ) ) )
42	36, 41	eqtrd 2222	. . . . 5 |- ( ph -> ( ( C x. B ) + ( -u A x. D ) ) = ( ( C x. B ) - ( A x. D ) ) )
43	42	breq2d 4030	. . . 4 |- ( ph -> ( P || ( ( C x. B ) + ( -u A x. D ) ) <-> P || ( ( C x. B ) - ( A x. D ) ) ) )
44	43	biimpar 297	. . 3 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> P || ( ( C x. B ) + ( -u A x. D ) ) )
45	1, 20, 21, 23, 24, 25, 26, 32, 33, 44	2sqlem3 14922	. 2 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> N e. S )
46		prmz 12143	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
47	4, 46	syl 14	. . . . 5 |- ( ph -> P e. ZZ )
48		zsqcl 10622	. . . . . . . 8 |- ( C e. ZZ -> ( C ^ 2 ) e. ZZ )
49	10, 48	syl 14	. . . . . . 7 |- ( ph -> ( C ^ 2 ) e. ZZ )
50	2	nnzd 9404	. . . . . . 7 |- ( ph -> N e. ZZ )
51	49, 50	zmulcld 9411	. . . . . 6 |- ( ph -> ( ( C ^ 2 ) x. N ) e. ZZ )
52		zsqcl 10622	. . . . . . 7 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
53	6, 52	syl 14	. . . . . 6 |- ( ph -> ( A ^ 2 ) e. ZZ )
54	51, 53	zsubcld 9410	. . . . 5 |- ( ph -> ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) e. ZZ )
55		dvdsmul1 11852	. . . . 5 |- ( ( P e. ZZ /\ ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) e. ZZ ) -> P || ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) )
56	47, 54, 55	syl2anc 411	. . . 4 |- ( ph -> P || ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) )
57	10, 6	zmulcld 9411	. . . . . . . . 9 |- ( ph -> ( C x. A ) e. ZZ )
58	57	zcnd 9406	. . . . . . . 8 |- ( ph -> ( C x. A ) e. CC )
59	58	sqcld 10683	. . . . . . 7 |- ( ph -> ( ( C x. A ) ^ 2 ) e. CC )
60	38	sqcld 10683	. . . . . . 7 |- ( ph -> ( ( C x. B ) ^ 2 ) e. CC )
61	40	sqcld 10683	. . . . . . 7 |- ( ph -> ( ( A x. D ) ^ 2 ) e. CC )
62	59, 60, 61	pnpcand 8335	. . . . . 6 |- ( ph -> ( ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) - ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) ) = ( ( ( C x. B ) ^ 2 ) - ( ( A x. D ) ^ 2 ) ) )
63	10	zcnd 9406	. . . . . . . . . . . 12 |- ( ph -> C e. CC )
64	63, 27	sqmuld 10697	. . . . . . . . . . 11 |- ( ph -> ( ( C x. A ) ^ 2 ) = ( ( C ^ 2 ) x. ( A ^ 2 ) ) )
65	8	zcnd 9406	. . . . . . . . . . . 12 |- ( ph -> B e. CC )
66	63, 65	sqmuld 10697	. . . . . . . . . . 11 |- ( ph -> ( ( C x. B ) ^ 2 ) = ( ( C ^ 2 ) x. ( B ^ 2 ) ) )
67	64, 66	oveq12d 5914	. . . . . . . . . 10 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( C ^ 2 ) x. ( B ^ 2 ) ) ) )
68	63	sqcld 10683	. . . . . . . . . . 11 |- ( ph -> ( C ^ 2 ) e. CC )
69	53	zcnd 9406	. . . . . . . . . . 11 |- ( ph -> ( A ^ 2 ) e. CC )
70	65	sqcld 10683	. . . . . . . . . . 11 |- ( ph -> ( B ^ 2 ) e. CC )
71	68, 69, 70	adddid 8012	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( C ^ 2 ) x. ( B ^ 2 ) ) ) )
72	67, 71	eqtr4d 2225	. . . . . . . . 9 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) )
73	2	nncnd 8963	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
74	47	zcnd 9406	. . . . . . . . . . . . 13 |- ( ph -> P e. CC )
75	73, 74	mulcomd 8009	. . . . . . . . . . . 12 |- ( ph -> ( N x. P ) = ( P x. N ) )
76	14, 75	eqtr3d 2224	. . . . . . . . . . 11 |- ( ph -> ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( P x. N ) )
77	76	oveq2d 5912	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( C ^ 2 ) x. ( P x. N ) ) )
78	68, 74, 73	mul12d 8139	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( P x. N ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
79	77, 78	eqtrd 2222	. . . . . . . . 9 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
80	72, 79	eqtrd 2222	. . . . . . . 8 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
81	27, 34	sqmuld 10697	. . . . . . . . . . . 12 |- ( ph -> ( ( A x. D ) ^ 2 ) = ( ( A ^ 2 ) x. ( D ^ 2 ) ) )
82	34	sqcld 10683	. . . . . . . . . . . . 13 |- ( ph -> ( D ^ 2 ) e. CC )
83	69, 82	mulcomd 8009	. . . . . . . . . . . 12 |- ( ph -> ( ( A ^ 2 ) x. ( D ^ 2 ) ) = ( ( D ^ 2 ) x. ( A ^ 2 ) ) )
84	81, 83	eqtrd 2222	. . . . . . . . . . 11 |- ( ph -> ( ( A x. D ) ^ 2 ) = ( ( D ^ 2 ) x. ( A ^ 2 ) ) )
85	64, 84	oveq12d 5914	. . . . . . . . . 10 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( D ^ 2 ) x. ( A ^ 2 ) ) ) )
86	49	zcnd 9406	. . . . . . . . . . 11 |- ( ph -> ( C ^ 2 ) e. CC )
87	86, 82, 69	adddird 8013	. . . . . . . . . 10 |- ( ph -> ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( D ^ 2 ) x. ( A ^ 2 ) ) ) )
88	85, 87	eqtr4d 2225	. . . . . . . . 9 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) )
89	16	oveq1d 5911	. . . . . . . . 9 |- ( ph -> ( P x. ( A ^ 2 ) ) = ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) )
90	88, 89	eqtr4d 2225	. . . . . . . 8 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( P x. ( A ^ 2 ) ) )
91	80, 90	oveq12d 5914	. . . . . . 7 |- ( ph -> ( ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) - ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) ) = ( ( P x. ( ( C ^ 2 ) x. N ) ) - ( P x. ( A ^ 2 ) ) ) )
92	51	zcnd 9406	. . . . . . . 8 |- ( ph -> ( ( C ^ 2 ) x. N ) e. CC )
93	74, 92, 69	subdid 8401	. . . . . . 7 |- ( ph -> ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) = ( ( P x. ( ( C ^ 2 ) x. N ) ) - ( P x. ( A ^ 2 ) ) ) )
94	91, 93	eqtr4d 2225	. . . . . 6 |- ( ph -> ( ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) - ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) ) = ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) )
95	62, 94	eqtr3d 2224	. . . . 5 |- ( ph -> ( ( ( C x. B ) ^ 2 ) - ( ( A x. D ) ^ 2 ) ) = ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) )
96		subsq 10658	. . . . . 6 |- ( ( ( C x. B ) e. CC /\ ( A x. D ) e. CC ) -> ( ( ( C x. B ) ^ 2 ) - ( ( A x. D ) ^ 2 ) ) = ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) )
97	38, 40, 96	syl2anc 411	. . . . 5 |- ( ph -> ( ( ( C x. B ) ^ 2 ) - ( ( A x. D ) ^ 2 ) ) = ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) )
98	95, 97	eqtr3d 2224	. . . 4 |- ( ph -> ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) = ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) )
99	56, 98	breqtrd 4044	. . 3 |- ( ph -> P || ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) )
100	37, 39	zaddcld 9409	. . . 4 |- ( ph -> ( ( C x. B ) + ( A x. D ) ) e. ZZ )
101	37, 39	zsubcld 9410	. . . 4 |- ( ph -> ( ( C x. B ) - ( A x. D ) ) e. ZZ )
102		euclemma 12178	. . . 4 |- ( ( P e. Prime /\ ( ( C x. B ) + ( A x. D ) ) e. ZZ /\ ( ( C x. B ) - ( A x. D ) ) e. ZZ ) -> ( P || ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) <-> ( P || ( ( C x. B ) + ( A x. D ) ) \/ P || ( ( C x. B ) - ( A x. D ) ) ) ) )
103	4, 100, 101, 102	syl3anc 1249	. . 3 |- ( ph -> ( P || ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) <-> ( P || ( ( C x. B ) + ( A x. D ) ) \/ P || ( ( C x. B ) - ( A x. D ) ) ) ) )
104	99, 103	mpbid 147	. 2 |- ( ph -> ( P || ( ( C x. B ) + ( A x. D ) ) \/ P || ( ( C x. B ) - ( A x. D ) ) ) )
105	19, 45, 104	mpjaodan 799	1 |- ( ph -> N e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2160   class class class wbr 4018   |-> cmpt 4079   ran crn 4645   ` cfv 5235  (class class class)co 5896   CCcc 7839   + caddc 7844   x. cmul 7846   - cmin 8158   -ucneg 8159   NNcn 8949   2c2 9000   ZZcz 9283   ^cexp 10550   abscabs 11038   || cdvds 11826   Primecprime 12139   ZZ[_i]cgz 12401

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-1o 6441  df-2o 6442  df-er 6559  df-en 6767  df-sup 7013  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-q 9650  df-rp 9684  df-fz 10039  df-fzo 10173  df-fl 10301  df-mod 10354  df-seqfrec 10477  df-exp 10551  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-dvds 11827  df-gcd 11976  df-prm 12140  df-gz 12402

This theorem is referenced by:  2sqlem5  14924