Theorem 4sqlem19 12932

Description: Lemma for 4sq 12933. The proof is by strong induction - we show that if all the integers less than k are in S, then k is as well. In this part of the proof we do the induction argument and dispense with all the cases except the odd prime case, which is sent to 4sqlem18 12931. If k is 0 , 1 , 2, we show k e. S directly; otherwise if k is composite, k is the product of two numbers less than it (and hence in S by assumption), so by mul4sq 12917 k e. S. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)

Hypothesis

Ref	Expression
4sqlem11.1	|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }

Assertion

Ref	Expression
4sqlem19	|- NN0 = S

Distinct variable groups:   S, n   w, n, x, y, z

Allowed substitution hints:   S( x, y, z, w)

Proof of Theorem 4sqlem19

Dummy variables i j k m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn0 9371	. . . 4 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
2		eleq1 2292	. . . . . 6 |- ( j = 1 -> ( j e. S <-> 1 e. S ) )
3		eleq1 2292	. . . . . 6 |- ( j = m -> ( j e. S <-> m e. S ) )
4		eleq1 2292	. . . . . 6 |- ( j = i -> ( j e. S <-> i e. S ) )
5		eleq1 2292	. . . . . 6 |- ( j = ( m x. i ) -> ( j e. S <-> ( m x. i ) e. S ) )
6		eleq1 2292	. . . . . 6 |- ( j = k -> ( j e. S <-> k e. S ) )
7		abs1 11583	. . . . . . . . . . 11 |- ( abs ` 1 ) = 1
8	7	oveq1i 6011	. . . . . . . . . 10 |- ( ( abs ` 1 ) ^ 2 ) = ( 1 ^ 2 )
9		sq1 10855	. . . . . . . . . 10 |- ( 1 ^ 2 ) = 1
10	8, 9	eqtri 2250	. . . . . . . . 9 |- ( ( abs ` 1 ) ^ 2 ) = 1
11		abs0 11569	. . . . . . . . . . 11 |- ( abs ` 0 ) = 0
12	11	oveq1i 6011	. . . . . . . . . 10 |- ( ( abs ` 0 ) ^ 2 ) = ( 0 ^ 2 )
13		sq0 10852	. . . . . . . . . 10 |- ( 0 ^ 2 ) = 0
14	12, 13	eqtri 2250	. . . . . . . . 9 |- ( ( abs ` 0 ) ^ 2 ) = 0
15	10, 14	oveq12i 6013	. . . . . . . 8 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = ( 1 + 0 )
16		1p0e1 9226	. . . . . . . 8 |- ( 1 + 0 ) = 1
17	15, 16	eqtri 2250	. . . . . . 7 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = 1
18		1z 9472	. . . . . . . . 9 |- 1 e. ZZ
19		zgz 12896	. . . . . . . . 9 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
20	18, 19	ax-mp 5	. . . . . . . 8 |- 1 e. ZZ[_i]
21		0z 9457	. . . . . . . . 9 |- 0 e. ZZ
22		zgz 12896	. . . . . . . . 9 |- ( 0 e. ZZ -> 0 e. ZZ[_i] )
23	21, 22	ax-mp 5	. . . . . . . 8 |- 0 e. ZZ[_i]
24		4sqlem11.1	. . . . . . . . 9 |- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }
25	24	4sqlem4a 12914	. . . . . . . 8 |- ( ( 1 e. ZZ[_i] /\ 0 e. ZZ[_i] ) -> ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S )
26	20, 23, 25	mp2an 426	. . . . . . 7 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S
27	17, 26	eqeltrri 2303	. . . . . 6 |- 1 e. S
28	10, 10	oveq12i 6013	. . . . . . . . . 10 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) = ( 1 + 1 )
29		df-2 9169	. . . . . . . . . 10 |- 2 = ( 1 + 1 )
30	28, 29	eqtr4i 2253	. . . . . . . . 9 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) = 2
31	24	4sqlem4a 12914	. . . . . . . . . 10 |- ( ( 1 e. ZZ[_i] /\ 1 e. ZZ[_i] ) -> ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) e. S )
32	20, 20, 31	mp2an 426	. . . . . . . . 9 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) e. S
33	30, 32	eqeltrri 2303	. . . . . . . 8 |- 2 e. S
34		eleq1 2292	. . . . . . . . 9 |- ( j = 2 -> ( j e. S <-> 2 e. S ) )
35	34	adantl 277	. . . . . . . 8 |- ( ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) /\ j = 2 ) -> ( j e. S <-> 2 e. S ) )
36	33, 35	mpbiri 168	. . . . . . 7 |- ( ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) /\ j = 2 ) -> j e. S )
37		eldifsn 3795	. . . . . . . . 9 |- ( j e. ( Prime \ { 2 } ) <-> ( j e. Prime /\ j =/= 2 ) )
38		oddprm 12782	. . . . . . . . . . 11 |- ( j e. ( Prime \ { 2 } ) -> ( ( j - 1 ) / 2 ) e. NN )
39	38	adantr 276	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( j - 1 ) / 2 ) e. NN )
40		eldifi 3326	. . . . . . . . . . . . . . . 16 |- ( j e. ( Prime \ { 2 } ) -> j e. Prime )
41	40	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. Prime )
42		prmnn 12632	. . . . . . . . . . . . . . 15 |- ( j e. Prime -> j e. NN )
43		nncn 9118	. . . . . . . . . . . . . . 15 |- ( j e. NN -> j e. CC )
44	41, 42, 43	3syl 17	. . . . . . . . . . . . . 14 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. CC )
45		ax-1cn 8092	. . . . . . . . . . . . . 14 |- 1 e. CC
46		subcl 8345	. . . . . . . . . . . . . 14 |- ( ( j e. CC /\ 1 e. CC ) -> ( j - 1 ) e. CC )
47	44, 45, 46	sylancl 413	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( j - 1 ) e. CC )
48		2cnd 9183	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> 2 e. CC )
49		2ap0 9203	. . . . . . . . . . . . . 14 |- 2 # 0
50	49	a1i 9	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> 2 # 0 )
51	47, 48, 50	divcanap2d 8939	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 2 x. ( ( j - 1 ) / 2 ) ) = ( j - 1 ) )
52	51	oveq1d 6016	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( 2 x. ( ( j - 1 ) / 2 ) ) + 1 ) = ( ( j - 1 ) + 1 ) )
53		npcan 8355	. . . . . . . . . . . 12 |- ( ( j e. CC /\ 1 e. CC ) -> ( ( j - 1 ) + 1 ) = j )
54	44, 45, 53	sylancl 413	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( j - 1 ) + 1 ) = j )
55	52, 54	eqtr2d 2263	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j = ( ( 2 x. ( ( j - 1 ) / 2 ) ) + 1 ) )
56	51	oveq2d 6017	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( 2 x. ( ( j - 1 ) / 2 ) ) ) = ( 0 ... ( j - 1 ) ) )
57		nnm1nn0 9410	. . . . . . . . . . . . . . 15 |- ( j e. NN -> ( j - 1 ) e. NN0 )
58	41, 42, 57	3syl 17	. . . . . . . . . . . . . 14 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( j - 1 ) e. NN0 )
59		elnn0uz 9760	. . . . . . . . . . . . . 14 |- ( ( j - 1 ) e. NN0 <-> ( j - 1 ) e. ( ZZ>= ` 0 ) )
60	58, 59	sylib 122	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( j - 1 ) e. ( ZZ>= ` 0 ) )
61		eluzfz1 10227	. . . . . . . . . . . . 13 |- ( ( j - 1 ) e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... ( j - 1 ) ) )
62		fzsplit 10247	. . . . . . . . . . . . 13 |- ( 0 e. ( 0 ... ( j - 1 ) ) -> ( 0 ... ( j - 1 ) ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) )
63	60, 61, 62	3syl 17	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( j - 1 ) ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) )
64	56, 63	eqtrd 2262	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( 2 x. ( ( j - 1 ) / 2 ) ) ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) )
65		fz0sn 10317	. . . . . . . . . . . . . 14 |- ( 0 ... 0 ) = { 0 }
66	14, 14	oveq12i 6013	. . . . . . . . . . . . . . . . 17 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = ( 0 + 0 )
67		00id 8287	. . . . . . . . . . . . . . . . 17 |- ( 0 + 0 ) = 0
68	66, 67	eqtri 2250	. . . . . . . . . . . . . . . 16 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = 0
69	24	4sqlem4a 12914	. . . . . . . . . . . . . . . . 17 |- ( ( 0 e. ZZ[_i] /\ 0 e. ZZ[_i] ) -> ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S )
70	23, 23, 69	mp2an 426	. . . . . . . . . . . . . . . 16 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S
71	68, 70	eqeltrri 2303	. . . . . . . . . . . . . . 15 |- 0 e. S
72		snssi 3812	. . . . . . . . . . . . . . 15 |- ( 0 e. S -> { 0 } C_ S )
73	71, 72	ax-mp 5	. . . . . . . . . . . . . 14 |- { 0 } C_ S
74	65, 73	eqsstri 3256	. . . . . . . . . . . . 13 |- ( 0 ... 0 ) C_ S
75	74	a1i 9	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... 0 ) C_ S )
76		0p1e1 9224	. . . . . . . . . . . . . 14 |- ( 0 + 1 ) = 1
77	76	oveq1i 6011	. . . . . . . . . . . . 13 |- ( ( 0 + 1 ) ... ( j - 1 ) ) = ( 1 ... ( j - 1 ) )
78		simpr 110	. . . . . . . . . . . . . 14 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> A. m e. ( 1 ... ( j - 1 ) ) m e. S )
79		dfss3 3213	. . . . . . . . . . . . . 14 |- ( ( 1 ... ( j - 1 ) ) C_ S <-> A. m e. ( 1 ... ( j - 1 ) ) m e. S )
80	78, 79	sylibr 134	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 1 ... ( j - 1 ) ) C_ S )
81	77, 80	eqsstrid 3270	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( 0 + 1 ) ... ( j - 1 ) ) C_ S )
82	75, 81	unssd 3380	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) C_ S )
83	64, 82	eqsstrd 3260	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( 2 x. ( ( j - 1 ) / 2 ) ) ) C_ S )
84		oveq1 6008	. . . . . . . . . . . 12 |- ( k = i -> ( k x. j ) = ( i x. j ) )
85	84	eleq1d 2298	. . . . . . . . . . 11 |- ( k = i -> ( ( k x. j ) e. S <-> ( i x. j ) e. S ) )
86	85	cbvrabv 2798	. . . . . . . . . 10 |- { k e. NN | ( k x. j ) e. S } = { i e. NN | ( i x. j ) e. S }
87		eqid 2229	. . . . . . . . . 10 |- inf ( { k e. NN | ( k x. j ) e. S } , RR , < ) = inf ( { k e. NN | ( k x. j ) e. S } , RR , < )
88	24, 39, 55, 41, 83, 86, 87	4sqlem18 12931	. . . . . . . . 9 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. S )
89	37, 88	sylanbr 285	. . . . . . . 8 |- ( ( ( j e. Prime /\ j =/= 2 ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. S )
90	89	an32s 568	. . . . . . 7 |- ( ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) /\ j =/= 2 ) -> j e. S )
91		prmz 12633	. . . . . . . . . 10 |- ( j e. Prime -> j e. ZZ )
92	91	adantr 276	. . . . . . . . 9 |- ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. ZZ )
93		2z 9474	. . . . . . . . 9 |- 2 e. ZZ
94		zdceq 9522	. . . . . . . . 9 |- ( ( j e. ZZ /\ 2 e. ZZ ) -> DECID j = 2 )
95	92, 93, 94	sylancl 413	. . . . . . . 8 |- ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> DECID j = 2 )
96		dcne 2411	. . . . . . . 8 |- (DECID j = 2 <-> ( j = 2 \/ j =/= 2 ) )
97	95, 96	sylib 122	. . . . . . 7 |- ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( j = 2 \/ j =/= 2 ) )
98	36, 90, 97	mpjaodan 803	. . . . . 6 |- ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. S )
99	24	mul4sq 12917	. . . . . . 7 |- ( ( m e. S /\ i e. S ) -> ( m x. i ) e. S )
100	99	a1i 9	. . . . . 6 |- ( ( m e. ( ZZ>= ` 2 ) /\ i e. ( ZZ>= ` 2 ) ) -> ( ( m e. S /\ i e. S ) -> ( m x. i ) e. S ) )
101	2, 3, 4, 5, 6, 27, 98, 100	prmind2 12642	. . . . 5 |- ( k e. NN -> k e. S )
102		id 19	. . . . . 6 |- ( k = 0 -> k = 0 )
103	102, 71	eqeltrdi 2320	. . . . 5 |- ( k = 0 -> k e. S )
104	101, 103	jaoi 721	. . . 4 |- ( ( k e. NN \/ k = 0 ) -> k e. S )
105	1, 104	sylbi 121	. . 3 |- ( k e. NN0 -> k e. S )
106	105	ssriv 3228	. 2 |- NN0 C_ S
107	24	4sqlem1 12911	. 2 |- S C_ NN0
108	106, 107	eqssi 3240	1 |- NN0 = S

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   {cab 2215   =/= wne 2400   A.wral 2508   E.wrex 2509   {crab 2512   \ cdif 3194   u. cun 3195   C_ wss 3197   {csn 3666   class class class wbr 4083   ` cfv 5318  (class class class)co 6001  infcinf 7150   CCcc 7997   RRcr 7998   0cc0 7999   1c1 8000   + caddc 8002   x. cmul 8004   < clt 8181   - cmin 8317   # cap 8728   / cdiv 8819   NNcn 9110   2c2 9161   NN0cn0 9369   ZZcz 9446   ZZ>=cuz 9722   ...cfz 10204   ^cexp 10760   abscabs 11508   Primecprime 12629   ZZ[_i]cgz 12892

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-2o 6563  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-sup 7151  df-inf 7152  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-dvds 12299  df-gcd 12475  df-prm 12630  df-gz 12893

This theorem is referenced by:  4sq  12933