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Theorem 4sqlem19 14992
Description: Lemma for 4sq 14993. 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 14991. 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 14977  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.
StepHypRef Expression
1 elnn0 10895 . . . 4  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
2 eleq1 2537 . . . . . 6  |-  ( j  =  1  ->  (
j  e.  S  <->  1  e.  S ) )
3 eleq1 2537 . . . . . 6  |-  ( j  =  m  ->  (
j  e.  S  <->  m  e.  S ) )
4 eleq1 2537 . . . . . 6  |-  ( j  =  i  ->  (
j  e.  S  <->  i  e.  S ) )
5 eleq1 2537 . . . . . 6  |-  ( j  =  ( m  x.  i )  ->  (
j  e.  S  <->  ( m  x.  i )  e.  S
) )
6 eleq1 2537 . . . . . 6  |-  ( j  =  k  ->  (
j  e.  S  <->  k  e.  S ) )
7 abs1 13437 . . . . . . . . . . 11  |-  ( abs `  1 )  =  1
87oveq1i 6318 . . . . . . . . . 10  |-  ( ( abs `  1 ) ^ 2 )  =  ( 1 ^ 2 )
9 sq1 12407 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
108, 9eqtri 2493 . . . . . . . . 9  |-  ( ( abs `  1 ) ^ 2 )  =  1
11 abs0 13425 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
1211oveq1i 6318 . . . . . . . . . 10  |-  ( ( abs `  0 ) ^ 2 )  =  ( 0 ^ 2 )
13 sq0 12404 . . . . . . . . . 10  |-  ( 0 ^ 2 )  =  0
1412, 13eqtri 2493 . . . . . . . . 9  |-  ( ( abs `  0 ) ^ 2 )  =  0
1510, 14oveq12i 6320 . . . . . . . 8  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  ( 1  +  0 )
16 1p0e1 10744 . . . . . . . 8  |-  ( 1  +  0 )  =  1
1715, 16eqtri 2493 . . . . . . 7  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  1
18 1z 10991 . . . . . . . . 9  |-  1  e.  ZZ
19 zgz 14956 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  1  e.  ZZ[_i]
)
2018, 19ax-mp 5 . . . . . . . 8  |-  1  e.  ZZ[_i]
21 0z 10972 . . . . . . . . 9  |-  0  e.  ZZ
22 zgz 14956 . . . . . . . . 9  |-  ( 0  e.  ZZ  ->  0  e.  ZZ[_i]
)
2321, 22ax-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 ) ) ) }
25244sqlem4a 14974 . . . . . . . 8  |-  ( ( 1  e.  ZZ[_i]  /\  0  e.  ZZ[_i]
)  ->  ( (
( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S )
2620, 23, 25mp2an 686 . . . . . . 7  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S
2717, 26eqeltrri 2546 . . . . . 6  |-  1  e.  S
28 eleq1 2537 . . . . . . 7  |-  ( j  =  2  ->  (
j  e.  S  <->  2  e.  S ) )
29 eldifsn 4088 . . . . . . . . 9  |-  ( j  e.  ( Prime  \  {
2 } )  <->  ( j  e.  Prime  /\  j  =/=  2 ) )
30 oddprm 14844 . . . . . . . . . . 11  |-  ( j  e.  ( Prime  \  {
2 } )  -> 
( ( j  - 
1 )  /  2
)  e.  NN )
3130adantr 472 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
j  -  1 )  /  2 )  e.  NN )
32 eldifi 3544 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( Prime  \  {
2 } )  -> 
j  e.  Prime )
3332adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  Prime )
34 prmnn 14704 . . . . . . . . . . . . . . 15  |-  ( j  e.  Prime  ->  j  e.  NN )
35 nncn 10639 . . . . . . . . . . . . . . 15  |-  ( j  e.  NN  ->  j  e.  CC )
3633, 34, 353syl 18 . . . . . . . . . . . . . 14  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  CC )
37 ax-1cn 9615 . . . . . . . . . . . . . 14  |-  1  e.  CC
38 subcl 9894 . . . . . . . . . . . . . 14  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( j  -  1 )  e.  CC )
3936, 37, 38sylancl 675 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( j  -  1 )  e.  CC )
40 2cnd 10704 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  2  e.  CC )
41 2ne0 10724 . . . . . . . . . . . . . 14  |-  2  =/=  0
4241a1i 11 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  2  =/=  0 )
4339, 40, 42divcan2d 10407 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 2  x.  ( ( j  -  1 )  / 
2 ) )  =  ( j  -  1 ) )
4443oveq1d 6323 . . . . . . . . . . 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 9904 . . . . . . . . . . . 12  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  - 
1 )  +  1 )  =  j )
4636, 37, 45sylancl 675 . . . . . . . . . . 11  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
j  -  1 )  +  1 )  =  j )
4744, 46eqtr2d 2506 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  =  ( ( 2  x.  ( ( j  - 
1 )  /  2
) )  +  1 ) )
4843oveq2d 6324 . . . . . . . . . . . 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 10935 . . . . . . . . . . . . . . 15  |-  ( j  e.  NN  ->  (
j  -  1 )  e.  NN0 )
5033, 34, 493syl 18 . . . . . . . . . . . . . 14  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( j  -  1 )  e. 
NN0 )
51 elnn0uz 11220 . . . . . . . . . . . . . 14  |-  ( ( j  -  1 )  e.  NN0  <->  ( j  - 
1 )  e.  (
ZZ>= `  0 ) )
5250, 51sylib 201 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( j  -  1 )  e.  ( ZZ>= `  0 )
)
53 eluzfz1 11832 . . . . . . . . . . . . 13  |-  ( ( j  -  1 )  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... (
j  -  1 ) ) )
54 fzsplit 11851 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0 ... ( j  -  1 ) )  ->  (
0 ... ( j  - 
1 ) )  =  ( ( 0 ... 0 )  u.  (
( 0  +  1 ) ... ( j  -  1 ) ) ) )
5552, 53, 543syl 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 ) ) ) )
5648, 55eqtrd 2505 . . . . . . . . . . 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 11866 . . . . . . . . . . . . . . 15  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
5821, 57ax-mp 5 . . . . . . . . . . . . . 14  |-  ( 0 ... 0 )  =  { 0 }
5914, 14oveq12i 6320 . . . . . . . . . . . . . . . . 17  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  ( 0  +  0 )
60 00id 9826 . . . . . . . . . . . . . . . . 17  |-  ( 0  +  0 )  =  0
6159, 60eqtri 2493 . . . . . . . . . . . . . . . 16  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  0
62244sqlem4a 14974 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  ZZ[_i]  /\  0  e.  ZZ[_i]
)  ->  ( (
( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S )
6323, 23, 62mp2an 686 . . . . . . . . . . . . . . . 16  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S
6461, 63eqeltrri 2546 . . . . . . . . . . . . . . 15  |-  0  e.  S
65 snssi 4107 . . . . . . . . . . . . . . 15  |-  ( 0  e.  S  ->  { 0 }  C_  S )
6664, 65ax-mp 5 . . . . . . . . . . . . . 14  |-  { 0 }  C_  S
6758, 66eqsstri 3448 . . . . . . . . . . . . 13  |-  ( 0 ... 0 )  C_  S
6867a1i 11 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... 0 )  C_  S )
69 0p1e1 10743 . . . . . . . . . . . . . 14  |-  ( 0  +  1 )  =  1
7069oveq1i 6318 . . . . . . . . . . . . 13  |-  ( ( 0  +  1 ) ... ( j  - 
1 ) )  =  ( 1 ... (
j  -  1 ) )
71 simpr 468 . . . . . . . . . . . . . 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 3408 . . . . . . . . . . . . . 14  |-  ( ( 1 ... ( j  -  1 ) ) 
C_  S  <->  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)
7371, 72sylibr 217 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 1 ... ( j  - 
1 ) )  C_  S )
7470, 73syl5eqss 3462 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
0  +  1 ) ... ( j  - 
1 ) )  C_  S )
7568, 74unssd 3601 . . . . . . . . . . 11  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
0 ... 0 )  u.  ( ( 0  +  1 ) ... (
j  -  1 ) ) )  C_  S
)
7656, 75eqsstrd 3452 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... ( 2  x.  ( ( j  - 
1 )  /  2
) ) )  C_  S )
77 oveq1 6315 . . . . . . . . . . . 12  |-  ( k  =  i  ->  (
k  x.  j )  =  ( i  x.  j ) )
7877eleq1d 2533 . . . . . . . . . . 11  |-  ( k  =  i  ->  (
( k  x.  j
)  e.  S  <->  ( i  x.  j )  e.  S
) )
7978cbvrabv 3030 . . . . . . . . . 10  |-  { k  e.  NN  |  ( k  x.  j )  e.  S }  =  { i  e.  NN  |  ( i  x.  j )  e.  S }
80 eqid 2471 . . . . . . . . . 10  |- inf ( { k  e.  NN  | 
( k  x.  j
)  e.  S } ,  RR ,  <  )  = inf ( { k  e.  NN  |  ( k  x.  j )  e.  S } ,  RR ,  <  )
8124, 31, 47, 33, 76, 79, 804sqlem18 14991 . . . . . . . . 9  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  S )
8229, 81sylanbr 481 . . . . . . . 8  |-  ( ( ( j  e.  Prime  /\  j  =/=  2 )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  S )
8382an32s 821 . . . . . . 7  |-  ( ( ( j  e.  Prime  /\ 
A. m  e.  ( 1 ... ( j  -  1 ) ) m  e.  S )  /\  j  =/=  2
)  ->  j  e.  S )
8410, 10oveq12i 6320 . . . . . . . . . 10  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  ( 1  +  1 )
85 df-2 10690 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
8684, 85eqtr4i 2496 . . . . . . . . 9  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  2
87244sqlem4a 14974 . . . . . . . . . 10  |-  ( ( 1  e.  ZZ[_i]  /\  1  e.  ZZ[_i]
)  ->  ( (
( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  e.  S )
8820, 20, 87mp2an 686 . . . . . . . . 9  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  e.  S
8986, 88eqeltrri 2546 . . . . . . . 8  |-  2  e.  S
9089a1i 11 . . . . . . 7  |-  ( ( j  e.  Prime  /\  A. m  e.  ( 1 ... ( j  - 
1 ) ) m  e.  S )  -> 
2  e.  S )
9128, 83, 90pm2.61ne 2728 . . . . . 6  |-  ( ( j  e.  Prime  /\  A. m  e.  ( 1 ... ( j  - 
1 ) ) m  e.  S )  -> 
j  e.  S )
9224mul4sq 14977 . . . . . . 7  |-  ( ( m  e.  S  /\  i  e.  S )  ->  ( m  x.  i
)  e.  S )
9392a1i 11 . . . . . 6  |-  ( ( m  e.  ( ZZ>= ` 
2 )  /\  i  e.  ( ZZ>= `  2 )
)  ->  ( (
m  e.  S  /\  i  e.  S )  ->  ( m  x.  i
)  e.  S ) )
942, 3, 4, 5, 6, 27, 91, 93prmind2 14714 . . . . 5  |-  ( k  e.  NN  ->  k  e.  S )
95 id 22 . . . . . 6  |-  ( k  =  0  ->  k  =  0 )
9695, 64syl6eqel 2557 . . . . 5  |-  ( k  =  0  ->  k  e.  S )
9794, 96jaoi 386 . . . 4  |-  ( ( k  e.  NN  \/  k  =  0 )  ->  k  e.  S
)
981, 97sylbi 200 . . 3  |-  ( k  e.  NN0  ->  k  e.  S )
9998ssriv 3422 . 2  |-  NN0  C_  S
100244sqlem1 14971 . 2  |-  S  C_  NN0
10199, 100eqssi 3434 1  |-  NN0  =  S
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760    \ cdif 3387    u. cun 3388    C_ wss 3390   {csn 3959   ` cfv 5589  (class class class)co 6308  infcinf 7973   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    - cmin 9880    / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   ^cexp 12310   abscabs 13374   Primecprime 14701   ZZ[_i]cgz 14952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-prm 14702  df-gz 14953
This theorem is referenced by:  4sq  14993
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