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Theorem smumul 14467
Description: For sequences that correspond to valid integers, the sequence multiplication function produces the sequence for the product. This is effectively a proof of the correctness of the multiplication process, implemented in terms of logic gates for df-sad 14425, whose correctness is verified in sadadd 14441.

Outside this range, the sequences cannot be representing integers, but the smul function still "works". This extended function is best interpreted in terms of the ring structure of the 2-adic integers. (Contributed by Mario Carneiro, 22-Sep-2016.)

Assertion
Ref Expression
smumul  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( (bits `  A
) smul  (bits `  B )
)  =  (bits `  ( A  x.  B
) ) )

Proof of Theorem smumul
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 bitsss 14399 . . . . . 6  |-  (bits `  A )  C_  NN0
2 bitsss 14399 . . . . . 6  |-  (bits `  B )  C_  NN0
3 smucl 14458 . . . . . 6  |-  ( ( (bits `  A )  C_ 
NN0  /\  (bits `  B
)  C_  NN0 )  -> 
( (bits `  A
) smul  (bits `  B )
)  C_  NN0 )
41, 2, 3mp2an 678 . . . . 5  |-  ( (bits `  A ) smul  (bits `  B ) )  C_  NN0
54sseli 3428 . . . 4  |-  ( k  e.  ( (bits `  A ) smul  (bits `  B
) )  ->  k  e.  NN0 )
65a1i 11 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( k  e.  ( (bits `  A ) smul  (bits `  B ) )  ->  k  e.  NN0 ) )
7 bitsss 14399 . . . . 5  |-  (bits `  ( A  x.  B
) )  C_  NN0
87sseli 3428 . . . 4  |-  ( k  e.  (bits `  ( A  x.  B )
)  ->  k  e.  NN0 )
98a1i 11 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( k  e.  (bits `  ( A  x.  B
) )  ->  k  e.  NN0 ) )
10 simpll 760 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  A  e.  ZZ )
11 simplr 762 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  B  e.  ZZ )
12 simpr 463 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  k  e.  NN0 )
13 1nn0 10885 . . . . . . . . . . . . . 14  |-  1  e.  NN0
1413a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  1  e.  NN0 )
1512, 14nn0addcld 10929 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( k  +  1 )  e.  NN0 )
1610, 11, 15smumullem 14466 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( ( (bits `  A )  i^i  (
0..^ ( k  +  1 ) ) ) smul  (bits `  B )
)  =  (bits `  ( ( A  mod  ( 2 ^ (
k  +  1 ) ) )  x.  B
) ) )
1716ineq1d 3633 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( ( ( (bits `  A )  i^i  ( 0..^ ( k  +  1 ) ) ) smul  (bits `  B
) )  i^i  (
0..^ ( k  +  1 ) ) )  =  ( (bits `  ( ( A  mod  ( 2 ^ (
k  +  1 ) ) )  x.  B
) )  i^i  (
0..^ ( k  +  1 ) ) ) )
18 2nn 10767 . . . . . . . . . . . . . . . 16  |-  2  e.  NN
1918a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  2  e.  NN )
2019, 15nnexpcld 12437 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( 2 ^ ( k  +  1 ) )  e.  NN )
2110, 20zmodcld 12117 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( A  mod  ( 2 ^ (
k  +  1 ) ) )  e.  NN0 )
2221nn0zd 11038 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( A  mod  ( 2 ^ (
k  +  1 ) ) )  e.  ZZ )
2322, 11zmulcld 11046 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( ( A  mod  ( 2 ^ ( k  +  1 ) ) )  x.  B )  e.  ZZ )
24 bitsmod 14410 . . . . . . . . . . 11  |-  ( ( ( ( A  mod  ( 2 ^ (
k  +  1 ) ) )  x.  B
)  e.  ZZ  /\  ( k  +  1 )  e.  NN0 )  ->  (bits `  ( (
( A  mod  (
2 ^ ( k  +  1 ) ) )  x.  B )  mod  ( 2 ^ ( k  +  1 ) ) ) )  =  ( (bits `  ( ( A  mod  ( 2 ^ (
k  +  1 ) ) )  x.  B
) )  i^i  (
0..^ ( k  +  1 ) ) ) )
2523, 15, 24syl2anc 667 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  (bits `  (
( ( A  mod  ( 2 ^ (
k  +  1 ) ) )  x.  B
)  mod  ( 2 ^ ( k  +  1 ) ) ) )  =  ( (bits `  ( ( A  mod  ( 2 ^ (
k  +  1 ) ) )  x.  B
) )  i^i  (
0..^ ( k  +  1 ) ) ) )
2617, 25eqtr4d 2488 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( ( ( (bits `  A )  i^i  ( 0..^ ( k  +  1 ) ) ) smul  (bits `  B
) )  i^i  (
0..^ ( k  +  1 ) ) )  =  (bits `  (
( ( A  mod  ( 2 ^ (
k  +  1 ) ) )  x.  B
)  mod  ( 2 ^ ( k  +  1 ) ) ) ) )
27 inass 3642 . . . . . . . . . . . . 13  |-  ( ( (bits `  A )  i^i  ( 0..^ ( k  +  1 ) ) )  i^i  ( 0..^ ( k  +  1 ) ) )  =  ( (bits `  A
)  i^i  ( (
0..^ ( k  +  1 ) )  i^i  ( 0..^ ( k  +  1 ) ) ) )
28 inidm 3641 . . . . . . . . . . . . . 14  |-  ( ( 0..^ ( k  +  1 ) )  i^i  ( 0..^ ( k  +  1 ) ) )  =  ( 0..^ ( k  +  1 ) )
2928ineq2i 3631 . . . . . . . . . . . . 13  |-  ( (bits `  A )  i^i  (
( 0..^ ( k  +  1 ) )  i^i  ( 0..^ ( k  +  1 ) ) ) )  =  ( (bits `  A
)  i^i  ( 0..^ ( k  +  1 ) ) )
3027, 29eqtri 2473 . . . . . . . . . . . 12  |-  ( ( (bits `  A )  i^i  ( 0..^ ( k  +  1 ) ) )  i^i  ( 0..^ ( k  +  1 ) ) )  =  ( (bits `  A
)  i^i  ( 0..^ ( k  +  1 ) ) )
3130oveq1i 6300 . . . . . . . . . . 11  |-  ( ( ( (bits `  A
)  i^i  ( 0..^ ( k  +  1 ) ) )  i^i  ( 0..^ ( k  +  1 ) ) ) smul  ( (bits `  B )  i^i  (
0..^ ( k  +  1 ) ) ) )  =  ( ( (bits `  A )  i^i  ( 0..^ ( k  +  1 ) ) ) smul  ( (bits `  B )  i^i  (
0..^ ( k  +  1 ) ) ) )
3231ineq1i 3630 . . . . . . . . . 10  |-  ( ( ( ( (bits `  A )  i^i  (
0..^ ( k  +  1 ) ) )  i^i  ( 0..^ ( k  +  1 ) ) ) smul  ( (bits `  B )  i^i  (
0..^ ( k  +  1 ) ) ) )  i^i  ( 0..^ ( k  +  1 ) ) )  =  ( ( ( (bits `  A )  i^i  (
0..^ ( k  +  1 ) ) ) smul  ( (bits `  B
)  i^i  ( 0..^ ( k  +  1 ) ) ) )  i^i  ( 0..^ ( k  +  1 ) ) )
33 inss1 3652 . . . . . . . . . . . 12  |-  ( (bits `  A )  i^i  (
0..^ ( k  +  1 ) ) ) 
C_  (bits `  A
)
341a1i 11 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  (bits `  A
)  C_  NN0 )
3533, 34syl5ss 3443 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( (bits `  A )  i^i  (
0..^ ( k  +  1 ) ) ) 
C_  NN0 )
362a1i 11 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  (bits `  B
)  C_  NN0 )
3735, 36, 15smueq 14465 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( ( ( (bits `  A )  i^i  ( 0..^ ( k  +  1 ) ) ) smul  (bits `  B
) )  i^i  (
0..^ ( k  +  1 ) ) )  =  ( ( ( ( (bits `  A
)  i^i  ( 0..^ ( k  +  1 ) ) )  i^i  ( 0..^ ( k  +  1 ) ) ) smul  ( (bits `  B )  i^i  (
0..^ ( k  +  1 ) ) ) )  i^i  ( 0..^ ( k  +  1 ) ) ) )
3834, 36, 15smueq 14465 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( ( (bits `  A ) smul  (bits `  B ) )  i^i  ( 0..^ ( k  +  1 ) ) )  =  ( ( ( (bits `  A
)  i^i  ( 0..^ ( k  +  1 ) ) ) smul  (
(bits `  B )  i^i  ( 0..^ ( k  +  1 ) ) ) )  i^i  (
0..^ ( k  +  1 ) ) ) )
3932, 37, 383eqtr4a 2511 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( ( ( (bits `  A )  i^i  ( 0..^ ( k  +  1 ) ) ) smul  (bits `  B
) )  i^i  (
0..^ ( k  +  1 ) ) )  =  ( ( (bits `  A ) smul  (bits `  B ) )  i^i  ( 0..^ ( k  +  1 ) ) ) )
4020nnrpd 11339 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( 2 ^ ( k  +  1 ) )  e.  RR+ )
4110zred 11040 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  A  e.  RR )
42 modabs2 12131 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( 2 ^ (
k  +  1 ) )  e.  RR+ )  ->  ( ( A  mod  ( 2 ^ (
k  +  1 ) ) )  mod  (
2 ^ ( k  +  1 ) ) )  =  ( A  mod  ( 2 ^ ( k  +  1 ) ) ) )
4341, 40, 42syl2anc 667 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( ( A  mod  ( 2 ^ ( k  +  1 ) ) )  mod  ( 2 ^ (
k  +  1 ) ) )  =  ( A  mod  ( 2 ^ ( k  +  1 ) ) ) )
44 eqidd 2452 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( B  mod  ( 2 ^ (
k  +  1 ) ) )  =  ( B  mod  ( 2 ^ ( k  +  1 ) ) ) )
4522, 10, 11, 11, 40, 43, 44modmul12d 12144 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( ( ( A  mod  ( 2 ^ ( k  +  1 ) ) )  x.  B )  mod  ( 2 ^ (
k  +  1 ) ) )  =  ( ( A  x.  B
)  mod  ( 2 ^ ( k  +  1 ) ) ) )
4645fveq2d 5869 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  (bits `  (
( ( A  mod  ( 2 ^ (
k  +  1 ) ) )  x.  B
)  mod  ( 2 ^ ( k  +  1 ) ) ) )  =  (bits `  ( ( A  x.  B )  mod  (
2 ^ ( k  +  1 ) ) ) ) )
4726, 39, 463eqtr3d 2493 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( ( (bits `  A ) smul  (bits `  B ) )  i^i  ( 0..^ ( k  +  1 ) ) )  =  (bits `  ( ( A  x.  B )  mod  (
2 ^ ( k  +  1 ) ) ) ) )
4810, 11zmulcld 11046 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( A  x.  B )  e.  ZZ )
49 bitsmod 14410 . . . . . . . . 9  |-  ( ( ( A  x.  B
)  e.  ZZ  /\  ( k  +  1 )  e.  NN0 )  ->  (bits `  ( ( A  x.  B )  mod  ( 2 ^ (
k  +  1 ) ) ) )  =  ( (bits `  ( A  x.  B )
)  i^i  ( 0..^ ( k  +  1 ) ) ) )
5048, 15, 49syl2anc 667 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  (bits `  (
( A  x.  B
)  mod  ( 2 ^ ( k  +  1 ) ) ) )  =  ( (bits `  ( A  x.  B
) )  i^i  (
0..^ ( k  +  1 ) ) ) )
5147, 50eqtrd 2485 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( ( (bits `  A ) smul  (bits `  B ) )  i^i  ( 0..^ ( k  +  1 ) ) )  =  ( (bits `  ( A  x.  B
) )  i^i  (
0..^ ( k  +  1 ) ) ) )
5251eleq2d 2514 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( k  e.  ( ( (bits `  A ) smul  (bits `  B
) )  i^i  (
0..^ ( k  +  1 ) ) )  <-> 
k  e.  ( (bits `  ( A  x.  B
) )  i^i  (
0..^ ( k  +  1 ) ) ) ) )
53 elin 3617 . . . . . 6  |-  ( k  e.  ( ( (bits `  A ) smul  (bits `  B ) )  i^i  ( 0..^ ( k  +  1 ) ) )  <->  ( k  e.  ( (bits `  A
) smul  (bits `  B )
)  /\  k  e.  ( 0..^ ( k  +  1 ) ) ) )
54 elin 3617 . . . . . 6  |-  ( k  e.  ( (bits `  ( A  x.  B
) )  i^i  (
0..^ ( k  +  1 ) ) )  <-> 
( k  e.  (bits `  ( A  x.  B
) )  /\  k  e.  ( 0..^ ( k  +  1 ) ) ) )
5552, 53, 543bitr3g 291 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( ( k  e.  ( (bits `  A ) smul  (bits `  B
) )  /\  k  e.  ( 0..^ ( k  +  1 ) ) )  <->  ( k  e.  (bits `  ( A  x.  B ) )  /\  k  e.  ( 0..^ ( k  +  1 ) ) ) ) )
56 nn0uz 11193 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
5712, 56syl6eleq 2539 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  k  e.  (
ZZ>= `  0 ) )
58 eluzfz2b 11808 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  0
)  <->  k  e.  ( 0 ... k ) )
5957, 58sylib 200 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  k  e.  ( 0 ... k ) )
6012nn0zd 11038 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  k  e.  ZZ )
61 fzval3 11983 . . . . . . . 8  |-  ( k  e.  ZZ  ->  (
0 ... k )  =  ( 0..^ ( k  +  1 ) ) )
6260, 61syl 17 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( 0 ... k )  =  ( 0..^ ( k  +  1 ) ) )
6359, 62eleqtrd 2531 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  k  e.  ( 0..^ ( k  +  1 ) ) )
6463biantrud 510 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( k  e.  ( (bits `  A
) smul  (bits `  B )
)  <->  ( k  e.  ( (bits `  A
) smul  (bits `  B )
)  /\  k  e.  ( 0..^ ( k  +  1 ) ) ) ) )
6563biantrud 510 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( k  e.  (bits `  ( A  x.  B ) )  <->  ( k  e.  (bits `  ( A  x.  B ) )  /\  k  e.  ( 0..^ ( k  +  1 ) ) ) ) )
6655, 64, 653bitr4d 289 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( k  e.  ( (bits `  A
) smul  (bits `  B )
)  <->  k  e.  (bits `  ( A  x.  B
) ) ) )
6766ex 436 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( k  e.  NN0  ->  ( k  e.  ( (bits `  A ) smul  (bits `  B ) )  <-> 
k  e.  (bits `  ( A  x.  B
) ) ) ) )
686, 9, 67pm5.21ndd 356 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( k  e.  ( (bits `  A ) smul  (bits `  B ) )  <-> 
k  e.  (bits `  ( A  x.  B
) ) ) )
6968eqrdv 2449 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( (bits `  A
) smul  (bits `  B )
)  =  (bits `  ( A  x.  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    i^i cin 3403    C_ wss 3404   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   RR+crp 11302   ...cfz 11784  ..^cfzo 11915    mod cmo 12096   ^cexp 12272  bitscbits 14392   smul csmu 14394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-xor 1406  df-tru 1447  df-fal 1450  df-had 1497  df-cad 1510  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-dvds 14306  df-bits 14395  df-sad 14425  df-smu 14450
This theorem is referenced by: (None)
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