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Theorem mul4sq 13001
Description: Euler's four-square identity: The product of two sums of four squares is also a sum of four squares. This is usually quoted as an explicit formula involving eight real variables; we save some time by working with complex numbers (gaussian integers) instead, so that we only have to work with four variables, and also hiding the actual formula for the product in the proof of mul4sqlem 13000. (For the curious, the explicit formula that is used is  (  |  a  |  ^ 2  +  |  b  |  ^
2 ) (  |  c  |  ^ 2  +  |  d  |  ^ 2 )  =  |  a *  x.  c  +  b  x.  d *  |  ^ 2  +  | 
a *  x.  d  -  b  x.  c
*  |  ^ 2.) (Contributed by Mario Carneiro, 14-Jul-2014.)
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
mul4sq  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  x.  B
)  e.  S )
Distinct variable groups:    w, n, x, y, z    B, n    A, n    S, n
Allowed substitution hints:    A( x, y, z, w)    B( x, y, z, w)    S( x, y, z, w)

Proof of Theorem mul4sq
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 4sq.1 . . 3  |-  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 ) ) ) }
214sqlem4 12999 . 2  |-  ( A  e.  S  <->  E. a  e.  ZZ [ _i ]  E. b  e.  ZZ [ _i ]  A  =  ( ( ( abs `  a ) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) ) )
314sqlem4 12999 . 2  |-  ( B  e.  S  <->  E. c  e.  ZZ [ _i ]  E. d  e.  ZZ [ _i ]  B  =  ( ( ( abs `  c ) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) ) )
4 reeanv 2707 . . 3  |-  ( E. a  e.  ZZ [
_i ]  E. c  e.  ZZ [ _i ] 
( E. b  e.  ZZ [ _i ]  A  =  ( (
( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  /\  E. d  e.  ZZ [ _i ]  B  =  (
( ( abs `  c
) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) ) )  <->  ( E. a  e.  ZZ [ _i ]  E. b  e.  ZZ [ _i ]  A  =  ( ( ( abs `  a ) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  /\  E. c  e.  ZZ [ _i ]  E. d  e.  ZZ [ _i ]  B  =  ( ( ( abs `  c ) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) ) ) )
5 reeanv 2707 . . . . 5  |-  ( E. b  e.  ZZ [
_i ]  E. d  e.  ZZ [ _i ] 
( A  =  ( ( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  /\  B  =  ( ( ( abs `  c ) ^ 2 )  +  ( ( abs `  d
) ^ 2 ) ) )  <->  ( E. b  e.  ZZ [ _i ]  A  =  (
( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  /\  E. d  e.  ZZ [ _i ]  B  =  (
( ( abs `  c
) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) ) ) )
6 simpll 730 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  a  e.  ZZ [ _i ] )
7 gzabssqcl 12988 . . . . . . . . . . . . 13  |-  ( a  e.  ZZ [ _i ]  ->  ( ( abs `  a ) ^ 2 )  e.  NN0 )
86, 7syl 15 . . . . . . . . . . . 12  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( abs `  a ) ^ 2 )  e.  NN0 )
9 simprl 732 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  b  e.  ZZ [ _i ] )
10 gzabssqcl 12988 . . . . . . . . . . . . 13  |-  ( b  e.  ZZ [ _i ]  ->  ( ( abs `  b ) ^ 2 )  e.  NN0 )
119, 10syl 15 . . . . . . . . . . . 12  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( abs `  b ) ^ 2 )  e.  NN0 )
128, 11nn0addcld 10022 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( ( abs `  a ) ^ 2 )  +  ( ( abs `  b
) ^ 2 ) )  e.  NN0 )
1312nn0cnd 10020 . . . . . . . . . 10  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( ( abs `  a ) ^ 2 )  +  ( ( abs `  b
) ^ 2 ) )  e.  CC )
1413div1d 9528 . . . . . . . . 9  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( ( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  /  1
)  =  ( ( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) ) )
15 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  c  e.  ZZ [ _i ] )
16 gzabssqcl 12988 . . . . . . . . . . . . 13  |-  ( c  e.  ZZ [ _i ]  ->  ( ( abs `  c ) ^ 2 )  e.  NN0 )
1715, 16syl 15 . . . . . . . . . . . 12  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( abs `  c ) ^ 2 )  e.  NN0 )
18 simprr 733 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  d  e.  ZZ [ _i ] )
19 gzabssqcl 12988 . . . . . . . . . . . . 13  |-  ( d  e.  ZZ [ _i ]  ->  ( ( abs `  d ) ^ 2 )  e.  NN0 )
2018, 19syl 15 . . . . . . . . . . . 12  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( abs `  d ) ^ 2 )  e.  NN0 )
2117, 20nn0addcld 10022 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( ( abs `  c ) ^ 2 )  +  ( ( abs `  d
) ^ 2 ) )  e.  NN0 )
2221nn0cnd 10020 . . . . . . . . . 10  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( ( abs `  c ) ^ 2 )  +  ( ( abs `  d
) ^ 2 ) )  e.  CC )
2322div1d 9528 . . . . . . . . 9  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( ( ( abs `  c
) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) )  /  1
)  =  ( ( ( abs `  c
) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) ) )
2414, 23oveq12d 5876 . . . . . . . 8  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( ( ( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  /  1
)  x.  ( ( ( ( abs `  c
) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) )  /  1
) )  =  ( ( ( ( abs `  a ) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  x.  ( ( ( abs `  c ) ^ 2 )  +  ( ( abs `  d
) ^ 2 ) ) ) )
25 eqid 2283 . . . . . . . . 9  |-  ( ( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  =  ( ( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )
26 eqid 2283 . . . . . . . . 9  |-  ( ( ( abs `  c
) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) )  =  ( ( ( abs `  c
) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) )
27 1nn 9757 . . . . . . . . . 10  |-  1  e.  NN
2827a1i 10 . . . . . . . . 9  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  1  e.  NN )
29 gzsubcl 12987 . . . . . . . . . . . . 13  |-  ( ( a  e.  ZZ [
_i ]  /\  c  e.  ZZ [ _i ]
)  ->  ( a  -  c )  e.  ZZ [ _i ]
)
3029adantr 451 . . . . . . . . . . . 12  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( a  -  c )  e.  ZZ [ _i ] )
31 gzcn 12979 . . . . . . . . . . . 12  |-  ( ( a  -  c )  e.  ZZ [ _i ]  ->  ( a  -  c )  e.  CC )
3230, 31syl 15 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( a  -  c )  e.  CC )
3332div1d 9528 . . . . . . . . . 10  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( a  -  c )  / 
1 )  =  ( a  -  c ) )
3433, 30eqeltrd 2357 . . . . . . . . 9  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( a  -  c )  / 
1 )  e.  ZZ [ _i ] )
35 gzsubcl 12987 . . . . . . . . . . . . 13  |-  ( ( b  e.  ZZ [
_i ]  /\  d  e.  ZZ [ _i ]
)  ->  ( b  -  d )  e.  ZZ [ _i ]
)
3635adantl 452 . . . . . . . . . . . 12  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( b  -  d )  e.  ZZ [ _i ] )
37 gzcn 12979 . . . . . . . . . . . 12  |-  ( ( b  -  d )  e.  ZZ [ _i ]  ->  ( b  -  d )  e.  CC )
3836, 37syl 15 . . . . . . . . . . 11  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( b  -  d )  e.  CC )
3938div1d 9528 . . . . . . . . . 10  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( b  -  d )  / 
1 )  =  ( b  -  d ) )
4039, 36eqeltrd 2357 . . . . . . . . 9  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( b  -  d )  / 
1 )  e.  ZZ [ _i ] )
4114, 12eqeltrd 2357 . . . . . . . . 9  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( ( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  /  1
)  e.  NN0 )
421, 6, 9, 15, 18, 25, 26, 28, 34, 40, 41mul4sqlem 13000 . . . . . . . 8  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( ( ( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  /  1
)  x.  ( ( ( ( abs `  c
) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) )  /  1
) )  e.  S
)
4324, 42eqeltrrd 2358 . . . . . . 7  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( ( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  x.  (
( ( abs `  c
) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) ) )  e.  S )
44 oveq12 5867 . . . . . . . 8  |-  ( ( A  =  ( ( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  /\  B  =  ( ( ( abs `  c ) ^ 2 )  +  ( ( abs `  d
) ^ 2 ) ) )  ->  ( A  x.  B )  =  ( ( ( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  x.  (
( ( abs `  c
) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) ) ) )
4544eleq1d 2349 . . . . . . 7  |-  ( ( A  =  ( ( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  /\  B  =  ( ( ( abs `  c ) ^ 2 )  +  ( ( abs `  d
) ^ 2 ) ) )  ->  (
( A  x.  B
)  e.  S  <->  ( (
( ( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  x.  (
( ( abs `  c
) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) ) )  e.  S ) )
4643, 45syl5ibrcom 213 . . . . . 6  |-  ( ( ( a  e.  ZZ [ _i ]  /\  c  e.  ZZ [ _i ]
)  /\  ( b  e.  ZZ [ _i ]  /\  d  e.  ZZ [ _i ] ) )  ->  ( ( A  =  ( ( ( abs `  a ) ^ 2 )  +  ( ( abs `  b
) ^ 2 ) )  /\  B  =  ( ( ( abs `  c ) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) ) )  ->  ( A  x.  B )  e.  S
) )
4746rexlimdvva 2674 . . . . 5  |-  ( ( a  e.  ZZ [
_i ]  /\  c  e.  ZZ [ _i ]
)  ->  ( E. b  e.  ZZ [ _i ]  E. d  e.  ZZ [ _i ]  ( A  =  ( ( ( abs `  a ) ^ 2 )  +  ( ( abs `  b
) ^ 2 ) )  /\  B  =  ( ( ( abs `  c ) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) ) )  ->  ( A  x.  B )  e.  S
) )
485, 47syl5bir 209 . . . 4  |-  ( ( a  e.  ZZ [
_i ]  /\  c  e.  ZZ [ _i ]
)  ->  ( ( E. b  e.  ZZ [ _i ]  A  =  ( ( ( abs `  a ) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  /\  E. d  e.  ZZ [ _i ]  B  =  ( (
( abs `  c
) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) ) )  -> 
( A  x.  B
)  e.  S ) )
4948rexlimivv 2672 . . 3  |-  ( E. a  e.  ZZ [
_i ]  E. c  e.  ZZ [ _i ] 
( E. b  e.  ZZ [ _i ]  A  =  ( (
( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  /\  E. d  e.  ZZ [ _i ]  B  =  (
( ( abs `  c
) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) ) )  -> 
( A  x.  B
)  e.  S )
504, 49sylbir 204 . 2  |-  ( ( E. a  e.  ZZ [ _i ]  E. b  e.  ZZ [ _i ]  A  =  ( (
( abs `  a
) ^ 2 )  +  ( ( abs `  b ) ^ 2 ) )  /\  E. c  e.  ZZ [ _i ]  E. d  e.  ZZ [ _i ]  B  =  ( ( ( abs `  c ) ^ 2 )  +  ( ( abs `  d ) ^ 2 ) ) )  ->  ( A  x.  B )  e.  S
)
512, 3, 50syl2anb 465 1  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  x.  B
)  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ^cexp 11104   abscabs 11719   ZZ [ _i ]cgz 12976
This theorem is referenced by:  4sqlem19  13010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-gz 12977
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