Theorem 4sqlem19 12765

Description: Lemma for 4sq 12766. 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 12764. 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 12750 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 9299	. . . 4 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
2		eleq1 2268	. . . . . 6 |- ( j = 1 -> ( j e. S <-> 1 e. S ) )
3		eleq1 2268	. . . . . 6 |- ( j = m -> ( j e. S <-> m e. S ) )
4		eleq1 2268	. . . . . 6 |- ( j = i -> ( j e. S <-> i e. S ) )
5		eleq1 2268	. . . . . 6 |- ( j = ( m x. i ) -> ( j e. S <-> ( m x. i ) e. S ) )
6		eleq1 2268	. . . . . 6 |- ( j = k -> ( j e. S <-> k e. S ) )
7		abs1 11416	. . . . . . . . . . 11 |- ( abs ` 1 ) = 1
8	7	oveq1i 5956	. . . . . . . . . 10 |- ( ( abs ` 1 ) ^ 2 ) = ( 1 ^ 2 )
9		sq1 10780	. . . . . . . . . 10 |- ( 1 ^ 2 ) = 1
10	8, 9	eqtri 2226	. . . . . . . . 9 |- ( ( abs ` 1 ) ^ 2 ) = 1
11		abs0 11402	. . . . . . . . . . 11 |- ( abs ` 0 ) = 0
12	11	oveq1i 5956	. . . . . . . . . 10 |- ( ( abs ` 0 ) ^ 2 ) = ( 0 ^ 2 )
13		sq0 10777	. . . . . . . . . 10 |- ( 0 ^ 2 ) = 0
14	12, 13	eqtri 2226	. . . . . . . . 9 |- ( ( abs ` 0 ) ^ 2 ) = 0
15	10, 14	oveq12i 5958	. . . . . . . 8 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = ( 1 + 0 )
16		1p0e1 9154	. . . . . . . 8 |- ( 1 + 0 ) = 1
17	15, 16	eqtri 2226	. . . . . . 7 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = 1
18		1z 9400	. . . . . . . . 9 |- 1 e. ZZ
19		zgz 12729	. . . . . . . . 9 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
20	18, 19	ax-mp 5	. . . . . . . 8 |- 1 e. ZZ[_i]
21		0z 9385	. . . . . . . . 9 |- 0 e. ZZ
22		zgz 12729	. . . . . . . . 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 12747	. . . . . . . 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 2279	. . . . . 6 |- 1 e. S
28	10, 10	oveq12i 5958	. . . . . . . . . 10 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) = ( 1 + 1 )
29		df-2 9097	. . . . . . . . . 10 |- 2 = ( 1 + 1 )
30	28, 29	eqtr4i 2229	. . . . . . . . 9 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) = 2
31	24	4sqlem4a 12747	. . . . . . . . . 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 2279	. . . . . . . 8 |- 2 e. S
34		eleq1 2268	. . . . . . . . 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 3760	. . . . . . . . 9 |- ( j e. ( Prime \ { 2 } ) <-> ( j e. Prime /\ j =/= 2 ) )
38		oddprm 12615	. . . . . . . . . . 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 3295	. . . . . . . . . . . . . . . 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 12465	. . . . . . . . . . . . . . 15 |- ( j e. Prime -> j e. NN )
43		nncn 9046	. . . . . . . . . . . . . . 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 8020	. . . . . . . . . . . . . 14 |- 1 e. CC
46		subcl 8273	. . . . . . . . . . . . . 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 9111	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> 2 e. CC )
49		2ap0 9131	. . . . . . . . . . . . . 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 8867	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 2 x. ( ( j - 1 ) / 2 ) ) = ( j - 1 ) )
52	51	oveq1d 5961	. . . . . . . . . . 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 8283	. . . . . . . . . . . 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 2239	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j = ( ( 2 x. ( ( j - 1 ) / 2 ) ) + 1 ) )
56	51	oveq2d 5962	. . . . . . . . . . . 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 9338	. . . . . . . . . . . . . . 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 9688	. . . . . . . . . . . . . 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 10155	. . . . . . . . . . . . 13 |- ( ( j - 1 ) e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... ( j - 1 ) ) )
62		fzsplit 10175	. . . . . . . . . . . . 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 2238	. . . . . . . . . . 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 10245	. . . . . . . . . . . . . 14 |- ( 0 ... 0 ) = { 0 }
66	14, 14	oveq12i 5958	. . . . . . . . . . . . . . . . 17 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = ( 0 + 0 )
67		00id 8215	. . . . . . . . . . . . . . . . 17 |- ( 0 + 0 ) = 0
68	66, 67	eqtri 2226	. . . . . . . . . . . . . . . 16 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = 0
69	24	4sqlem4a 12747	. . . . . . . . . . . . . . . . 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 2279	. . . . . . . . . . . . . . 15 |- 0 e. S
72		snssi 3777	. . . . . . . . . . . . . . 15 |- ( 0 e. S -> { 0 } C_ S )
73	71, 72	ax-mp 5	. . . . . . . . . . . . . 14 |- { 0 } C_ S
74	65, 73	eqsstri 3225	. . . . . . . . . . . . 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 9152	. . . . . . . . . . . . . 14 |- ( 0 + 1 ) = 1
77	76	oveq1i 5956	. . . . . . . . . . . . 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 3182	. . . . . . . . . . . . . 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 3239	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( 0 + 1 ) ... ( j - 1 ) ) C_ S )
82	75, 81	unssd 3349	. . . . . . . . . . 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 3229	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( 2 x. ( ( j - 1 ) / 2 ) ) ) C_ S )
84		oveq1 5953	. . . . . . . . . . . 12 |- ( k = i -> ( k x. j ) = ( i x. j ) )
85	84	eleq1d 2274	. . . . . . . . . . 11 |- ( k = i -> ( ( k x. j ) e. S <-> ( i x. j ) e. S ) )
86	85	cbvrabv 2771	. . . . . . . . . 10 |- { k e. NN | ( k x. j ) e. S } = { i e. NN | ( i x. j ) e. S }
87		eqid 2205	. . . . . . . . . 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 12764	. . . . . . . . 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 12466	. . . . . . . . . 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 9402	. . . . . . . . 9 |- 2 e. ZZ
94		zdceq 9450	. . . . . . . . 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 2387	. . . . . . . 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 800	. . . . . 6 |- ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. S )
99	24	mul4sq 12750	. . . . . . 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 12475	. . . . 5 |- ( k e. NN -> k e. S )
102		id 19	. . . . . 6 |- ( k = 0 -> k = 0 )
103	102, 71	eqeltrdi 2296	. . . . 5 |- ( k = 0 -> k e. S )
104	101, 103	jaoi 718	. . . 4 |- ( ( k e. NN \/ k = 0 ) -> k e. S )
105	1, 104	sylbi 121	. . 3 |- ( k e. NN0 -> k e. S )
106	105	ssriv 3197	. 2 |- NN0 C_ S
107	24	4sqlem1 12744	. 2 |- S C_ NN0
108	106, 107	eqssi 3209	1 |- NN0 = S

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710  DECID wdc 836   = wceq 1373   e. wcel 2176   {cab 2191   =/= wne 2376   A.wral 2484   E.wrex 2485   {crab 2488   \ cdif 3163   u. cun 3164   C_ wss 3166   {csn 3633   class class class wbr 4045   ` cfv 5272  (class class class)co 5946  infcinf 7087   CCcc 7925   RRcr 7926   0cc0 7927   1c1 7928   + caddc 7930   x. cmul 7932   < clt 8109   - cmin 8245   # cap 8656   / cdiv 8747   NNcn 9038   2c2 9089   NN0cn0 9297   ZZcz 9374   ZZ>=cuz 9650   ...cfz 10132   ^cexp 10685   abscabs 11341   Primecprime 12462   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-xor 1396  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-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-frec 6479  df-1o 6504  df-2o 6505  df-oadd 6508  df-er 6622  df-en 6830  df-dom 6831  df-fin 6832  df-sup 7088  df-inf 7089  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-ihash 10923  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:  4sq  12766