Theorem 2sqlem4 15710

Description: Lemma for 2sqlem5 15711. (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 15709	. 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 9532	. . . 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 9531	. . . . . . 7 |- ( ph -> A e. CC )
28		sqneg 10780	. . . . . . 7 |- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
29	27, 28	syl 14	. . . . . 6 |- ( ph -> ( -u A ^ 2 ) = ( A ^ 2 ) )
30	29	oveq1d 5982	. . . . 5 |- ( ph -> ( ( -u A ^ 2 ) + ( B ^ 2 ) ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )
31	14, 30	eqtr4d 2243	. . . 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 9531	. . . . . . . 8 |- ( ph -> D e. CC )
35	27, 34	mulneg1d 8518	. . . . . . 7 |- ( ph -> ( -u A x. D ) = -u ( A x. D ) )
36	35	oveq2d 5983	. . . . . 6 |- ( ph -> ( ( C x. B ) + ( -u A x. D ) ) = ( ( C x. B ) + -u ( A x. D ) ) )
37	10, 8	zmulcld 9536	. . . . . . . 8 |- ( ph -> ( C x. B ) e. ZZ )
38	37	zcnd 9531	. . . . . . 7 |- ( ph -> ( C x. B ) e. CC )
39	6, 12	zmulcld 9536	. . . . . . . 8 |- ( ph -> ( A x. D ) e. ZZ )
40	39	zcnd 9531	. . . . . . 7 |- ( ph -> ( A x. D ) e. CC )
41	38, 40	negsubd 8424	. . . . . 6 |- ( ph -> ( ( C x. B ) + -u ( A x. D ) ) = ( ( C x. B ) - ( A x. D ) ) )
42	36, 41	eqtrd 2240	. . . . 5 |- ( ph -> ( ( C x. B ) + ( -u A x. D ) ) = ( ( C x. B ) - ( A x. D ) ) )
43	42	breq2d 4071	. . . 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 15709	. 2 |- ( ( ph /\ P || ( ( C x. B ) - ( A x. D ) ) ) -> N e. S )
46		prmz 12548	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
47	4, 46	syl 14	. . . . 5 |- ( ph -> P e. ZZ )
48		zsqcl 10792	. . . . . . . 8 |- ( C e. ZZ -> ( C ^ 2 ) e. ZZ )
49	10, 48	syl 14	. . . . . . 7 |- ( ph -> ( C ^ 2 ) e. ZZ )
50	2	nnzd 9529	. . . . . . 7 |- ( ph -> N e. ZZ )
51	49, 50	zmulcld 9536	. . . . . 6 |- ( ph -> ( ( C ^ 2 ) x. N ) e. ZZ )
52		zsqcl 10792	. . . . . . 7 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
53	6, 52	syl 14	. . . . . 6 |- ( ph -> ( A ^ 2 ) e. ZZ )
54	51, 53	zsubcld 9535	. . . . 5 |- ( ph -> ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) e. ZZ )
55		dvdsmul1 12239	. . . . 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 9536	. . . . . . . . 9 |- ( ph -> ( C x. A ) e. ZZ )
58	57	zcnd 9531	. . . . . . . 8 |- ( ph -> ( C x. A ) e. CC )
59	58	sqcld 10853	. . . . . . 7 |- ( ph -> ( ( C x. A ) ^ 2 ) e. CC )
60	38	sqcld 10853	. . . . . . 7 |- ( ph -> ( ( C x. B ) ^ 2 ) e. CC )
61	40	sqcld 10853	. . . . . . 7 |- ( ph -> ( ( A x. D ) ^ 2 ) e. CC )
62	59, 60, 61	pnpcand 8455	. . . . . 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 9531	. . . . . . . . . . . 12 |- ( ph -> C e. CC )
64	63, 27	sqmuld 10867	. . . . . . . . . . 11 |- ( ph -> ( ( C x. A ) ^ 2 ) = ( ( C ^ 2 ) x. ( A ^ 2 ) ) )
65	8	zcnd 9531	. . . . . . . . . . . 12 |- ( ph -> B e. CC )
66	63, 65	sqmuld 10867	. . . . . . . . . . 11 |- ( ph -> ( ( C x. B ) ^ 2 ) = ( ( C ^ 2 ) x. ( B ^ 2 ) ) )
67	64, 66	oveq12d 5985	. . . . . . . . . 10 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( C ^ 2 ) x. ( B ^ 2 ) ) ) )
68	63	sqcld 10853	. . . . . . . . . . 11 |- ( ph -> ( C ^ 2 ) e. CC )
69	53	zcnd 9531	. . . . . . . . . . 11 |- ( ph -> ( A ^ 2 ) e. CC )
70	65	sqcld 10853	. . . . . . . . . . 11 |- ( ph -> ( B ^ 2 ) e. CC )
71	68, 69, 70	adddid 8132	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( C ^ 2 ) x. ( B ^ 2 ) ) ) )
72	67, 71	eqtr4d 2243	. . . . . . . . 9 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) )
73	2	nncnd 9085	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
74	47	zcnd 9531	. . . . . . . . . . . . 13 |- ( ph -> P e. CC )
75	73, 74	mulcomd 8129	. . . . . . . . . . . 12 |- ( ph -> ( N x. P ) = ( P x. N ) )
76	14, 75	eqtr3d 2242	. . . . . . . . . . 11 |- ( ph -> ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( P x. N ) )
77	76	oveq2d 5983	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( ( C ^ 2 ) x. ( P x. N ) ) )
78	68, 74, 73	mul12d 8259	. . . . . . . . . 10 |- ( ph -> ( ( C ^ 2 ) x. ( P x. N ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
79	77, 78	eqtrd 2240	. . . . . . . . 9 |- ( ph -> ( ( C ^ 2 ) x. ( ( A ^ 2 ) + ( B ^ 2 ) ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
80	72, 79	eqtrd 2240	. . . . . . . 8 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( C x. B ) ^ 2 ) ) = ( P x. ( ( C ^ 2 ) x. N ) ) )
81	27, 34	sqmuld 10867	. . . . . . . . . . . 12 |- ( ph -> ( ( A x. D ) ^ 2 ) = ( ( A ^ 2 ) x. ( D ^ 2 ) ) )
82	34	sqcld 10853	. . . . . . . . . . . . 13 |- ( ph -> ( D ^ 2 ) e. CC )
83	69, 82	mulcomd 8129	. . . . . . . . . . . 12 |- ( ph -> ( ( A ^ 2 ) x. ( D ^ 2 ) ) = ( ( D ^ 2 ) x. ( A ^ 2 ) ) )
84	81, 83	eqtrd 2240	. . . . . . . . . . 11 |- ( ph -> ( ( A x. D ) ^ 2 ) = ( ( D ^ 2 ) x. ( A ^ 2 ) ) )
85	64, 84	oveq12d 5985	. . . . . . . . . 10 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( D ^ 2 ) x. ( A ^ 2 ) ) ) )
86	49	zcnd 9531	. . . . . . . . . . 11 |- ( ph -> ( C ^ 2 ) e. CC )
87	86, 82, 69	adddird 8133	. . . . . . . . . 10 |- ( ph -> ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) = ( ( ( C ^ 2 ) x. ( A ^ 2 ) ) + ( ( D ^ 2 ) x. ( A ^ 2 ) ) ) )
88	85, 87	eqtr4d 2243	. . . . . . . . 9 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) )
89	16	oveq1d 5982	. . . . . . . . 9 |- ( ph -> ( P x. ( A ^ 2 ) ) = ( ( ( C ^ 2 ) + ( D ^ 2 ) ) x. ( A ^ 2 ) ) )
90	88, 89	eqtr4d 2243	. . . . . . . 8 |- ( ph -> ( ( ( C x. A ) ^ 2 ) + ( ( A x. D ) ^ 2 ) ) = ( P x. ( A ^ 2 ) ) )
91	80, 90	oveq12d 5985	. . . . . . 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 9531	. . . . . . . 8 |- ( ph -> ( ( C ^ 2 ) x. N ) e. CC )
93	74, 92, 69	subdid 8521	. . . . . . 7 |- ( ph -> ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) = ( ( P x. ( ( C ^ 2 ) x. N ) ) - ( P x. ( A ^ 2 ) ) ) )
94	91, 93	eqtr4d 2243	. . . . . 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 2242	. . . . 5 |- ( ph -> ( ( ( C x. B ) ^ 2 ) - ( ( A x. D ) ^ 2 ) ) = ( P x. ( ( ( C ^ 2 ) x. N ) - ( A ^ 2 ) ) ) )
96		subsq 10828	. . . . . 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 2242	. . . 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 4085	. . 3 |- ( ph -> P || ( ( ( C x. B ) + ( A x. D ) ) x. ( ( C x. B ) - ( A x. D ) ) ) )
100	37, 39	zaddcld 9534	. . . 4 |- ( ph -> ( ( C x. B ) + ( A x. D ) ) e. ZZ )
101	37, 39	zsubcld 9535	. . . 4 |- ( ph -> ( ( C x. B ) - ( A x. D ) ) e. ZZ )
102		euclemma 12583	. . . 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 1250	. . 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 800	1 |- ( ph -> N e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2178   class class class wbr 4059   |-> cmpt 4121   ran crn 4694   ` cfv 5290  (class class class)co 5967   CCcc 7958   + caddc 7963   x. cmul 7965   - cmin 8278   -ucneg 8279   NNcn 9071   2c2 9122   ZZcz 9407   ^cexp 10720   abscabs 11423   || cdvds 12213   Primecprime 12544   ZZ[_i]cgz 12807

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-2o 6526  df-er 6643  df-en 6851  df-sup 7112  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-gcd 12390  df-prm 12545  df-gz 12808

This theorem is referenced by:  2sqlem5  15711