MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  smumul Structured version   Unicode version

Theorem smumul 13710
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 13668, whose correctness is verified in sadadd 13684.

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 13643 . . . . . 6  |-  (bits `  A )  C_  NN0
2 bitsss 13643 . . . . . 6  |-  (bits `  B )  C_  NN0
3 smucl 13701 . . . . . 6  |-  ( ( (bits `  A )  C_ 
NN0  /\  (bits `  B
)  C_  NN0 )  -> 
( (bits `  A
) smul  (bits `  B )
)  C_  NN0 )
41, 2, 3mp2an 672 . . . . 5  |-  ( (bits `  A ) smul  (bits `  B ) )  C_  NN0
54sseli 3373 . . . 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 13643 . . . . 5  |-  (bits `  ( A  x.  B
) )  C_  NN0
87sseli 3373 . . . 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 753 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  A  e.  ZZ )
11 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  B  e.  ZZ )
12 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  k  e.  NN0 )
13 1nn0 10616 . . . . . . . . . . . . . 14  |-  1  e.  NN0
1413a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  1  e.  NN0 )
1512, 14nn0addcld 10661 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( k  +  1 )  e.  NN0 )
1610, 11, 15smumullem 13709 . . . . . . . . . . 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 3572 . . . . . . . . . 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 10500 . . . . . . . . . . . . . . . 16  |-  2  e.  NN
1918a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  2  e.  NN )
2019, 15nnexpcld 12050 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( 2 ^ ( k  +  1 ) )  e.  NN )
2110, 20zmodcld 11749 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( A  mod  ( 2 ^ (
k  +  1 ) ) )  e.  NN0 )
2221nn0zd 10766 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( A  mod  ( 2 ^ (
k  +  1 ) ) )  e.  ZZ )
2322, 11zmulcld 10774 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( ( A  mod  ( 2 ^ ( k  +  1 ) ) )  x.  B )  e.  ZZ )
24 bitsmod 13653 . . . . . . . . . . 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 661 . . . . . . . . . 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 2478 . . . . . . . . 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 3581 . . . . . . . . . . . . 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 3580 . . . . . . . . . . . . . 14  |-  ( ( 0..^ ( k  +  1 ) )  i^i  ( 0..^ ( k  +  1 ) ) )  =  ( 0..^ ( k  +  1 ) )
2928ineq2i 3570 . . . . . . . . . . . . 13  |-  ( (bits `  A )  i^i  (
( 0..^ ( k  +  1 ) )  i^i  ( 0..^ ( k  +  1 ) ) ) )  =  ( (bits `  A
)  i^i  ( 0..^ ( k  +  1 ) ) )
3027, 29eqtri 2463 . . . . . . . . . . . 12  |-  ( ( (bits `  A )  i^i  ( 0..^ ( k  +  1 ) ) )  i^i  ( 0..^ ( k  +  1 ) ) )  =  ( (bits `  A
)  i^i  ( 0..^ ( k  +  1 ) ) )
3130oveq1i 6122 . . . . . . . . . . 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 3569 . . . . . . . . . 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 3591 . . . . . . . . . . . 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 3388 . . . . . . . . . . 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 13708 . . . . . . . . . 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 13708 . . . . . . . . . 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 2501 . . . . . . . . 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 11047 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( 2 ^ ( k  +  1 ) )  e.  RR+ )
4110zred 10768 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  A  e.  RR )
42 modabs2 11763 . . . . . . . . . . . 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 661 . . . . . . . . . . 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 2444 . . . . . . . . . . 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 11774 . . . . . . . . . 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 5716 . . . . . . . . 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 2483 . . . . . . . 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 10774 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( A  x.  B )  e.  ZZ )
49 bitsmod 13653 . . . . . . . . 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 661 . . . . . . . 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 2475 . . . . . . 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 2510 . . . . . 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 3560 . . . . . 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 3560 . . . . . 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 287 . . . . 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 10916 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
5712, 56syl6eleq 2533 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  k  e.  (
ZZ>= `  0 ) )
58 eluzfz2b 11481 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  0
)  <->  k  e.  ( 0 ... k ) )
5957, 58sylib 196 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  k  e.  ( 0 ... k ) )
6012nn0zd 10766 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  k  e.  ZZ )
61 fzval3 11626 . . . . . . . 8  |-  ( k  e.  ZZ  ->  (
0 ... k )  =  ( 0..^ ( k  +  1 ) ) )
6260, 61syl 16 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( 0 ... k )  =  ( 0..^ ( k  +  1 ) ) )
6359, 62eleqtrd 2519 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  k  e.  ( 0..^ ( k  +  1 ) ) )
6463biantrud 507 . . . . 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 507 . . . . 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 285 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  k  e.  NN0 )  ->  ( k  e.  ( (bits `  A
) smul  (bits `  B )
)  <->  k  e.  (bits `  ( A  x.  B
) ) ) )
6766ex 434 . . 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 354 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( k  e.  ( (bits `  A ) smul  (bits `  B ) )  <-> 
k  e.  (bits `  ( A  x.  B
) ) ) )
6968eqrdv 2441 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 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3348    C_ wss 3349   ` cfv 5439  (class class class)co 6112   RRcr 9302   0cc0 9303   1c1 9304    + caddc 9306    x. cmul 9308   NNcn 10343   2c2 10392   NN0cn0 10600   ZZcz 10667   ZZ>=cuz 10882   RR+crp 11012   ...cfz 11458  ..^cfzo 11569    mod cmo 11729   ^cexp 11886  bitscbits 13636   smul csmu 13638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  df-tru 1372  df-fal 1375  df-had 1421  df-cad 1422  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-disj 4284  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-oi 7745  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-fz 11459  df-fzo 11570  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-hash 12125  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-clim 12987  df-sum 13185  df-dvds 13557  df-bits 13639  df-sad 13668  df-smu 13693
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator