Theorem 4sqlem19 14918

Description: Lemma for 4sq 14919. 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 14917. 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 14903 k e. S. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)

Hypothesis

Ref	Expression
4sq.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:   w, n, x, y, z   S, n

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

Proof of Theorem 4sqlem19

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

Step	Hyp	Ref	Expression
1		elnn0 10884	. . . 4 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
2		eleq1 2496	. . . . . 6 |- ( j = 1 -> ( j e. S <-> 1 e. S ) )
3		eleq1 2496	. . . . . 6 |- ( j = m -> ( j e. S <-> m e. S ) )
4		eleq1 2496	. . . . . 6 |- ( j = i -> ( j e. S <-> i e. S ) )
5		eleq1 2496	. . . . . 6 |- ( j = ( m x. i ) -> ( j e. S <-> ( m x. i ) e. S ) )
6		eleq1 2496	. . . . . 6 |- ( j = k -> ( j e. S <-> k e. S ) )
7		abs1 13366	. . . . . . . . . . 11 |- ( abs ` 1 ) = 1
8	7	oveq1i 6321	. . . . . . . . . 10 |- ( ( abs ` 1 ) ^ 2 ) = ( 1 ^ 2 )
9		sq1 12381	. . . . . . . . . 10 |- ( 1 ^ 2 ) = 1
10	8, 9	eqtri 2452	. . . . . . . . 9 |- ( ( abs ` 1 ) ^ 2 ) = 1
11		abs0 13354	. . . . . . . . . . 11 |- ( abs ` 0 ) = 0
12	11	oveq1i 6321	. . . . . . . . . 10 |- ( ( abs ` 0 ) ^ 2 ) = ( 0 ^ 2 )
13		sq0 12378	. . . . . . . . . 10 |- ( 0 ^ 2 ) = 0
14	12, 13	eqtri 2452	. . . . . . . . 9 |- ( ( abs ` 0 ) ^ 2 ) = 0
15	10, 14	oveq12i 6323	. . . . . . . 8 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = ( 1 + 0 )
16		1p0e1 10735	. . . . . . . 8 |- ( 1 + 0 ) = 1
17	15, 16	eqtri 2452	. . . . . . 7 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = 1
18		1z 10980	. . . . . . . . 9 |- 1 e. ZZ
19		zgz 14882	. . . . . . . . 9 |- ( 1 e. ZZ -> 1 e. ZZ[_i] )
20	18, 19	ax-mp 5	. . . . . . . 8 |- 1 e. ZZ[_i]
21		0z 10961	. . . . . . . . 9 |- 0 e. ZZ
22		zgz 14882	. . . . . . . . 9 |- ( 0 e. ZZ -> 0 e. ZZ[_i] )
23	21, 22	ax-mp 5	. . . . . . . 8 |- 0 e. ZZ[_i]
24		4sq.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 14900	. . . . . . . 8 |- ( ( 1 e. ZZ[_i] /\ 0 e. ZZ[_i] ) -> ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S )
26	20, 23, 25	mp2an 677	. . . . . . 7 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S
27	17, 26	eqeltrri 2509	. . . . . 6 |- 1 e. S
28		eleq1 2496	. . . . . . 7 |- ( j = 2 -> ( j e. S <-> 2 e. S ) )
29		eldifsn 4131	. . . . . . . . 9 |- ( j e. ( Prime \ { 2 } ) <-> ( j e. Prime /\ j =/= 2 ) )
30		oddprm 14770	. . . . . . . . . . 11 |- ( j e. ( Prime \ { 2 } ) -> ( ( j - 1 ) / 2 ) e. NN )
31	30	adantr 467	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( j - 1 ) / 2 ) e. NN )
32		eldifi 3593	. . . . . . . . . . . . . . . 16 |- ( j e. ( Prime \ { 2 } ) -> j e. Prime )
33	32	adantr 467	. . . . . . . . . . . . . . 15 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. Prime )
34		prmnn 14630	. . . . . . . . . . . . . . 15 |- ( j e. Prime -> j e. NN )
35		nncn 10630	. . . . . . . . . . . . . . 15 |- ( j e. NN -> j e. CC )
36	33, 34, 35	3syl 18	. . . . . . . . . . . . . 14 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. CC )
37		ax-1cn 9610	. . . . . . . . . . . . . 14 |- 1 e. CC
38		subcl 9887	. . . . . . . . . . . . . 14 |- ( ( j e. CC /\ 1 e. CC ) -> ( j - 1 ) e. CC )
39	36, 37, 38	sylancl 667	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( j - 1 ) e. CC )
40		2cnd 10695	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> 2 e. CC )
41		2ne0 10715	. . . . . . . . . . . . . 14 |- 2 =/= 0
42	41	a1i 11	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> 2 =/= 0 )
43	39, 40, 42	divcan2d 10398	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 2 x. ( ( j - 1 ) / 2 ) ) = ( j - 1 ) )
44	43	oveq1d 6326	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( 2 x. ( ( j - 1 ) / 2 ) ) + 1 ) = ( ( j - 1 ) + 1 ) )
45		npcan 9897	. . . . . . . . . . . 12 |- ( ( j e. CC /\ 1 e. CC ) -> ( ( j - 1 ) + 1 ) = j )
46	36, 37, 45	sylancl 667	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( j - 1 ) + 1 ) = j )
47	44, 46	eqtr2d 2465	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j = ( ( 2 x. ( ( j - 1 ) / 2 ) ) + 1 ) )
48	43	oveq2d 6327	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( 2 x. ( ( j - 1 ) / 2 ) ) ) = ( 0 ... ( j - 1 ) ) )
49		nnm1nn0 10924	. . . . . . . . . . . . . . 15 |- ( j e. NN -> ( j - 1 ) e. NN0 )
50	33, 34, 49	3syl 18	. . . . . . . . . . . . . 14 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( j - 1 ) e. NN0 )
51		elnn0uz 11209	. . . . . . . . . . . . . 14 |- ( ( j - 1 ) e. NN0 <-> ( j - 1 ) e. ( ZZ>= ` 0 ) )
52	50, 51	sylib 200	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( j - 1 ) e. ( ZZ>= ` 0 ) )
53		eluzfz1 11819	. . . . . . . . . . . . 13 |- ( ( j - 1 ) e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... ( j - 1 ) ) )
54		fzsplit 11838	. . . . . . . . . . . . 13 |- ( 0 e. ( 0 ... ( j - 1 ) ) -> ( 0 ... ( j - 1 ) ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) )
55	52, 53, 54	3syl 18	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( j - 1 ) ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) )
56	48, 55	eqtrd 2464	. . . . . . . . . . 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 ) ) ) )
57		fzsn 11853	. . . . . . . . . . . . . . 15 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
58	21, 57	ax-mp 5	. . . . . . . . . . . . . 14 |- ( 0 ... 0 ) = { 0 }
59	14, 14	oveq12i 6323	. . . . . . . . . . . . . . . . 17 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = ( 0 + 0 )
60		00id 9821	. . . . . . . . . . . . . . . . 17 |- ( 0 + 0 ) = 0
61	59, 60	eqtri 2452	. . . . . . . . . . . . . . . 16 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) = 0
62	24	4sqlem4a 14900	. . . . . . . . . . . . . . . . 17 |- ( ( 0 e. ZZ[_i] /\ 0 e. ZZ[_i] ) -> ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S )
63	23, 23, 62	mp2an 677	. . . . . . . . . . . . . . . 16 |- ( ( ( abs ` 0 ) ^ 2 ) + ( ( abs ` 0 ) ^ 2 ) ) e. S
64	61, 63	eqeltrri 2509	. . . . . . . . . . . . . . 15 |- 0 e. S
65		snssi 4150	. . . . . . . . . . . . . . 15 |- ( 0 e. S -> { 0 } C_ S )
66	64, 65	ax-mp 5	. . . . . . . . . . . . . 14 |- { 0 } C_ S
67	58, 66	eqsstri 3500	. . . . . . . . . . . . 13 |- ( 0 ... 0 ) C_ S
68	67	a1i 11	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... 0 ) C_ S )
69		0p1e1 10734	. . . . . . . . . . . . . 14 |- ( 0 + 1 ) = 1
70	69	oveq1i 6321	. . . . . . . . . . . . 13 |- ( ( 0 + 1 ) ... ( j - 1 ) ) = ( 1 ... ( j - 1 ) )
71		simpr 463	. . . . . . . . . . . . . 14 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> A. m e. ( 1 ... ( j - 1 ) ) m e. S )
72		dfss3 3460	. . . . . . . . . . . . . 14 |- ( ( 1 ... ( j - 1 ) ) C_ S <-> A. m e. ( 1 ... ( j - 1 ) ) m e. S )
73	71, 72	sylibr 216	. . . . . . . . . . . . 13 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 1 ... ( j - 1 ) ) C_ S )
74	70, 73	syl5eqss 3514	. . . . . . . . . . . 12 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( 0 + 1 ) ... ( j - 1 ) ) C_ S )
75	68, 74	unssd 3648	. . . . . . . . . . 11 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... ( j - 1 ) ) ) C_ S )
76	56, 75	eqsstrd 3504	. . . . . . . . . 10 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> ( 0 ... ( 2 x. ( ( j - 1 ) / 2 ) ) ) C_ S )
77		oveq1 6318	. . . . . . . . . . . 12 |- ( k = i -> ( k x. j ) = ( i x. j ) )
78	77	eleq1d 2492	. . . . . . . . . . 11 |- ( k = i -> ( ( k x. j ) e. S <-> ( i x. j ) e. S ) )
79	78	cbvrabv 3084	. . . . . . . . . 10 |- { k e. NN | ( k x. j ) e. S } = { i e. NN | ( i x. j ) e. S }
80		eqid 2423	. . . . . . . . . 10 |- inf ( { k e. NN | ( k x. j ) e. S } , RR , < ) = inf ( { k e. NN | ( k x. j ) e. S } , RR , < )
81	24, 31, 47, 33, 76, 79, 80	4sqlem18 14917	. . . . . . . . 9 |- ( ( j e. ( Prime \ { 2 } ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. S )
82	29, 81	sylanbr 476	. . . . . . . 8 |- ( ( ( j e. Prime /\ j =/= 2 ) /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. S )
83	82	an32s 812	. . . . . . 7 |- ( ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) /\ j =/= 2 ) -> j e. S )
84	10, 10	oveq12i 6323	. . . . . . . . . 10 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) = ( 1 + 1 )
85		df-2 10681	. . . . . . . . . 10 |- 2 = ( 1 + 1 )
86	84, 85	eqtr4i 2455	. . . . . . . . 9 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) = 2
87	24	4sqlem4a 14900	. . . . . . . . . 10 |- ( ( 1 e. ZZ[_i] /\ 1 e. ZZ[_i] ) -> ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) e. S )
88	20, 20, 87	mp2an 677	. . . . . . . . 9 |- ( ( ( abs ` 1 ) ^ 2 ) + ( ( abs ` 1 ) ^ 2 ) ) e. S
89	86, 88	eqeltrri 2509	. . . . . . . 8 |- 2 e. S
90	89	a1i 11	. . . . . . 7 |- ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> 2 e. S )
91	28, 83, 90	pm2.61ne 2740	. . . . . 6 |- ( ( j e. Prime /\ A. m e. ( 1 ... ( j - 1 ) ) m e. S ) -> j e. S )
92	24	mul4sq 14903	. . . . . . 7 |- ( ( m e. S /\ i e. S ) -> ( m x. i ) e. S )
93	92	a1i 11	. . . . . 6 |- ( ( m e. ( ZZ>= ` 2 ) /\ i e. ( ZZ>= ` 2 ) ) -> ( ( m e. S /\ i e. S ) -> ( m x. i ) e. S ) )
94	2, 3, 4, 5, 6, 27, 91, 93	prmind2 14640	. . . . 5 |- ( k e. NN -> k e. S )
95		id 23	. . . . . 6 |- ( k = 0 -> k = 0 )
96	95, 64	syl6eqel 2520	. . . . 5 |- ( k = 0 -> k e. S )
97	94, 96	jaoi 381	. . . 4 |- ( ( k e. NN \/ k = 0 ) -> k e. S )
98	1, 97	sylbi 199	. . 3 |- ( k e. NN0 -> k e. S )
99	98	ssriv 3474	. 2 |- NN0 C_ S
100	24	4sqlem1 14897	. 2 |- S C_ NN0
101	99, 100	eqssi 3486	1 |- NN0 = S

Syntax hints:   -> wi 4   \/ wo 370   /\ wa 371   = wceq 1438   e. wcel 1873   {cab 2408   =/= wne 2619   A.wral 2776   E.wrex 2777   {crab 2780   \ cdif 3439   u. cun 3440   C_ wss 3442   {csn 4004   ` cfv 5607  (class class class)co 6311  infcinf 7970   CCcc 9550   RRcr 9551   0cc0 9552   1c1 9553   + caddc 9555   x. cmul 9557   < clt 9688   - cmin 9873   / cdiv 10282   NNcn 10622   2c2 10672   NN0cn0 10882   ZZcz 10950   ZZ>=cuz 11172   ...cfz 11797   ^cexp 12284   abscabs 13303   Primecprime 14627   ZZ[_i]cgz 14878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-8 1875  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-rep 4542  ax-sep 4552  ax-nul 4561  ax-pow 4608  ax-pr 4666  ax-un 6603  ax-cnex 9608  ax-resscn 9609  ax-1cn 9610  ax-icn 9611  ax-addcl 9612  ax-addrcl 9613  ax-mulcl 9614  ax-mulrcl 9615  ax-mulcom 9616  ax-addass 9617  ax-mulass 9618  ax-distr 9619  ax-i2m1 9620  ax-1ne0 9621  ax-1rid 9622  ax-rnegex 9623  ax-rrecex 9624  ax-cnre 9625  ax-pre-lttri 9626  ax-pre-lttrn 9627  ax-pre-ltadd 9628  ax-pre-mulgt0 9629  ax-pre-sup 9630

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3087  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3918  df-pw 3989  df-sn 4005  df-pr 4007  df-tp 4009  df-op 4011  df-uni 4226  df-int 4262  df-iun 4307  df-br 4430  df-opab 4489  df-mpt 4490  df-tr 4525  df-eprel 4770  df-id 4774  df-po 4780  df-so 4781  df-fr 4818  df-we 4820  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-pred 5405  df-ord 5451  df-on 5452  df-lim 5453  df-suc 5454  df-iota 5571  df-fun 5609  df-fn 5610  df-f 5611  df-f1 5612  df-fo 5613  df-f1o 5614  df-fv 5615  df-riota 6273  df-ov 6314  df-oprab 6315  df-mpt2 6316  df-om 6713  df-1st 6813  df-2nd 6814  df-wrecs 7045  df-recs 7107  df-rdg 7145  df-1o 7199  df-2o 7200  df-oadd 7203  df-er 7380  df-en 7587  df-dom 7588  df-sdom 7589  df-fin 7590  df-sup 7971  df-inf 7972  df-card 8387  df-cda 8611  df-pnf 9690  df-mnf 9691  df-xr 9692  df-ltxr 9693  df-le 9694  df-sub 9875  df-neg 9876  df-div 10283  df-nn 10623  df-2 10681  df-3 10682  df-4 10683  df-n0 10883  df-z 10951  df-uz 11173  df-rp 11316  df-fz 11798  df-fl 12040  df-mod 12109  df-seq 12226  df-exp 12285  df-hash 12528  df-cj 13168  df-re 13169  df-im 13170  df-sqrt 13304  df-abs 13305  df-dvds 14311  df-gcd 14474  df-prm 14628  df-gz 14879

This theorem is referenced by:  4sq  14919