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Theorem sadaddlem 14414
Description: Lemma for sadadd 14415. (Contributed by Mario Carneiro, 9-Sep-2016.)
Hypotheses
Ref Expression
sadaddlem.c  |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  (bits `  A ) ,  m  e.  (bits `  B ) ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) )
sadaddlem.k  |-  K  =  `' (bits  |`  NN0 )
sadaddlem.1  |-  ( ph  ->  A  e.  ZZ )
sadaddlem.2  |-  ( ph  ->  B  e.  ZZ )
sadaddlem.3  |-  ( ph  ->  N  e.  NN0 )
Assertion
Ref Expression
sadaddlem  |-  ( ph  ->  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) )  =  (bits `  (
( A  +  B
)  mod  ( 2 ^ N ) ) ) )
Distinct variable groups:    m, c, n    A, c, m    B, c, m    n, N
Allowed substitution hints:    ph( m, n, c)    A( n)    B( n)    C( m, n, c)    K( m, n, c)    N( m, c)

Proof of Theorem sadaddlem
StepHypRef Expression
1 sadaddlem.k . . . . . . . . . . . . 13  |-  K  =  `' (bits  |`  NN0 )
21fveq1i 5882 . . . . . . . . . . . 12  |-  ( K `
 ( (bits `  A )  i^i  (
0..^ N ) ) )  =  ( `' (bits  |`  NN0 ) `  ( (bits `  A )  i^i  ( 0..^ N ) ) )
3 sadaddlem.1 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A  e.  ZZ )
4 2nn 10767 . . . . . . . . . . . . . . . . . 18  |-  2  e.  NN
54a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  2  e.  NN )
6 sadaddlem.3 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  e.  NN0 )
75, 6nnexpcld 12434 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 2 ^ N
)  e.  NN )
83, 7zmodcld 12114 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A  mod  (
2 ^ N ) )  e.  NN0 )
9 fvres 5895 . . . . . . . . . . . . . . 15  |-  ( ( A  mod  ( 2 ^ N ) )  e.  NN0  ->  ( (bits  |`  NN0 ) `  ( A  mod  ( 2 ^ N ) ) )  =  (bits `  ( A  mod  ( 2 ^ N ) ) ) )
108, 9syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (bits  |`  NN0 ) `  ( A  mod  (
2 ^ N ) ) )  =  (bits `  ( A  mod  (
2 ^ N ) ) ) )
11 bitsmod 14384 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( A  mod  ( 2 ^ N
) ) )  =  ( (bits `  A
)  i^i  ( 0..^ N ) ) )
123, 6, 11syl2anc 665 . . . . . . . . . . . . . 14  |-  ( ph  ->  (bits `  ( A  mod  ( 2 ^ N
) ) )  =  ( (bits `  A
)  i^i  ( 0..^ N ) ) )
1310, 12eqtrd 2470 . . . . . . . . . . . . 13  |-  ( ph  ->  ( (bits  |`  NN0 ) `  ( A  mod  (
2 ^ N ) ) )  =  ( (bits `  A )  i^i  ( 0..^ N ) ) )
14 bitsf1o 14393 . . . . . . . . . . . . . 14  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
15 f1ocnvfv 6192 . . . . . . . . . . . . . 14  |-  ( ( (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )  /\  ( A  mod  ( 2 ^ N
) )  e.  NN0 )  ->  ( ( (bits  |`  NN0 ) `  ( A  mod  ( 2 ^ N ) ) )  =  ( (bits `  A )  i^i  (
0..^ N ) )  ->  ( `' (bits  |`  NN0 ) `  (
(bits `  A )  i^i  ( 0..^ N ) ) )  =  ( A  mod  ( 2 ^ N ) ) ) )
1614, 8, 15sylancr 667 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( (bits  |`  NN0 ) `  ( A  mod  (
2 ^ N ) ) )  =  ( (bits `  A )  i^i  ( 0..^ N ) )  ->  ( `' (bits  |`  NN0 ) `  ( (bits `  A )  i^i  ( 0..^ N ) ) )  =  ( A  mod  ( 2 ^ N ) ) ) )
1713, 16mpd 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( (bits `  A
)  i^i  ( 0..^ N ) ) )  =  ( A  mod  ( 2 ^ N
) ) )
182, 17syl5eq 2482 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  (
(bits `  A )  i^i  ( 0..^ N ) ) )  =  ( A  mod  ( 2 ^ N ) ) )
1918oveq2d 6321 . . . . . . . . . 10  |-  ( ph  ->  ( A  -  ( K `  ( (bits `  A )  i^i  (
0..^ N ) ) ) )  =  ( A  -  ( A  mod  ( 2 ^ N ) ) ) )
2019oveq1d 6320 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  ( K `  ( (bits `  A )  i^i  (
0..^ N ) ) ) )  /  (
2 ^ N ) )  =  ( ( A  -  ( A  mod  ( 2 ^ N ) ) )  /  ( 2 ^ N ) ) )
213zred 11040 . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR )
227nnrpd 11339 . . . . . . . . . 10  |-  ( ph  ->  ( 2 ^ N
)  e.  RR+ )
23 moddifz 12106 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( 2 ^ N
)  e.  RR+ )  ->  ( ( A  -  ( A  mod  ( 2 ^ N ) ) )  /  ( 2 ^ N ) )  e.  ZZ )
2421, 22, 23syl2anc 665 . . . . . . . . 9  |-  ( ph  ->  ( ( A  -  ( A  mod  ( 2 ^ N ) ) )  /  ( 2 ^ N ) )  e.  ZZ )
2520, 24eqeltrd 2517 . . . . . . . 8  |-  ( ph  ->  ( ( A  -  ( K `  ( (bits `  A )  i^i  (
0..^ N ) ) ) )  /  (
2 ^ N ) )  e.  ZZ )
267nnzd 11039 . . . . . . . . 9  |-  ( ph  ->  ( 2 ^ N
)  e.  ZZ )
277nnne0d 10654 . . . . . . . . 9  |-  ( ph  ->  ( 2 ^ N
)  =/=  0 )
28 inss1 3688 . . . . . . . . . . . . . 14  |-  ( (bits `  A )  i^i  (
0..^ N ) ) 
C_  (bits `  A
)
29 bitsss 14374 . . . . . . . . . . . . . 14  |-  (bits `  A )  C_  NN0
3028, 29sstri 3479 . . . . . . . . . . . . 13  |-  ( (bits `  A )  i^i  (
0..^ N ) ) 
C_  NN0
31 fzofi 12184 . . . . . . . . . . . . . 14  |-  ( 0..^ N )  e.  Fin
32 inss2 3689 . . . . . . . . . . . . . 14  |-  ( (bits `  A )  i^i  (
0..^ N ) ) 
C_  ( 0..^ N )
33 ssfi 7798 . . . . . . . . . . . . . 14  |-  ( ( ( 0..^ N )  e.  Fin  /\  (
(bits `  A )  i^i  ( 0..^ N ) )  C_  ( 0..^ N ) )  -> 
( (bits `  A
)  i^i  ( 0..^ N ) )  e. 
Fin )
3431, 32, 33mp2an 676 . . . . . . . . . . . . 13  |-  ( (bits `  A )  i^i  (
0..^ N ) )  e.  Fin
35 elfpw 7882 . . . . . . . . . . . . 13  |-  ( ( (bits `  A )  i^i  ( 0..^ N ) )  e.  ( ~P
NN0  i^i  Fin )  <->  ( ( (bits `  A
)  i^i  ( 0..^ N ) )  C_  NN0 
/\  ( (bits `  A )  i^i  (
0..^ N ) )  e.  Fin ) )
3630, 34, 35mpbir2an 928 . . . . . . . . . . . 12  |-  ( (bits `  A )  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin )
37 f1ocnv 5843 . . . . . . . . . . . . . . 15  |-  ( (bits  |`  NN0 ) : NN0 -1-1-onto-> ( ~P NN0  i^i  Fin )  ->  `' (bits  |`  NN0 ) : ( ~P NN0  i^i 
Fin ) -1-1-onto-> NN0 )
38 f1of 5831 . . . . . . . . . . . . . . 15  |-  ( `' (bits  |`  NN0 ) : ( ~P NN0  i^i  Fin ) -1-1-onto-> NN0  ->  `' (bits  |` 
NN0 ) : ( ~P NN0  i^i  Fin )
--> NN0 )
3914, 37, 38mp2b 10 . . . . . . . . . . . . . 14  |-  `' (bits  |`  NN0 ) : ( ~P NN0  i^i  Fin )
--> NN0
401feq1i 5738 . . . . . . . . . . . . . 14  |-  ( K : ( ~P NN0  i^i 
Fin ) --> NN0  <->  `' (bits  |` 
NN0 ) : ( ~P NN0  i^i  Fin )
--> NN0 )
4139, 40mpbir 212 . . . . . . . . . . . . 13  |-  K :
( ~P NN0  i^i  Fin ) --> NN0
4241ffvelrni 6036 . . . . . . . . . . . 12  |-  ( ( (bits `  A )  i^i  ( 0..^ N ) )  e.  ( ~P
NN0  i^i  Fin )  ->  ( K `  (
(bits `  A )  i^i  ( 0..^ N ) ) )  e.  NN0 )
4336, 42mp1i 13 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  (
(bits `  A )  i^i  ( 0..^ N ) ) )  e.  NN0 )
4443nn0zd 11038 . . . . . . . . . 10  |-  ( ph  ->  ( K `  (
(bits `  A )  i^i  ( 0..^ N ) ) )  e.  ZZ )
453, 44zsubcld 11045 . . . . . . . . 9  |-  ( ph  ->  ( A  -  ( K `  ( (bits `  A )  i^i  (
0..^ N ) ) ) )  e.  ZZ )
46 dvdsval2 14286 . . . . . . . . 9  |-  ( ( ( 2 ^ N
)  e.  ZZ  /\  ( 2 ^ N
)  =/=  0  /\  ( A  -  ( K `  ( (bits `  A )  i^i  (
0..^ N ) ) ) )  e.  ZZ )  ->  ( ( 2 ^ N )  ||  ( A  -  ( K `  ( (bits `  A )  i^i  (
0..^ N ) ) ) )  <->  ( ( A  -  ( K `  ( (bits `  A
)  i^i  ( 0..^ N ) ) ) )  /  ( 2 ^ N ) )  e.  ZZ ) )
4726, 27, 45, 46syl3anc 1264 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ N )  ||  ( A  -  ( K `  ( (bits `  A
)  i^i  ( 0..^ N ) ) ) )  <->  ( ( A  -  ( K `  ( (bits `  A )  i^i  ( 0..^ N ) ) ) )  / 
( 2 ^ N
) )  e.  ZZ ) )
4825, 47mpbird 235 . . . . . . 7  |-  ( ph  ->  ( 2 ^ N
)  ||  ( A  -  ( K `  ( (bits `  A )  i^i  ( 0..^ N ) ) ) ) )
491fveq1i 5882 . . . . . . . . . . . 12  |-  ( K `
 ( (bits `  B )  i^i  (
0..^ N ) ) )  =  ( `' (bits  |`  NN0 ) `  ( (bits `  B )  i^i  ( 0..^ N ) ) )
50 sadaddlem.2 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  B  e.  ZZ )
5150, 7zmodcld 12114 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( B  mod  (
2 ^ N ) )  e.  NN0 )
52 fvres 5895 . . . . . . . . . . . . . . 15  |-  ( ( B  mod  ( 2 ^ N ) )  e.  NN0  ->  ( (bits  |`  NN0 ) `  ( B  mod  ( 2 ^ N ) ) )  =  (bits `  ( B  mod  ( 2 ^ N ) ) ) )
5351, 52syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( (bits  |`  NN0 ) `  ( B  mod  (
2 ^ N ) ) )  =  (bits `  ( B  mod  (
2 ^ N ) ) ) )
54 bitsmod 14384 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( B  mod  ( 2 ^ N
) ) )  =  ( (bits `  B
)  i^i  ( 0..^ N ) ) )
5550, 6, 54syl2anc 665 . . . . . . . . . . . . . 14  |-  ( ph  ->  (bits `  ( B  mod  ( 2 ^ N
) ) )  =  ( (bits `  B
)  i^i  ( 0..^ N ) ) )
5653, 55eqtrd 2470 . . . . . . . . . . . . 13  |-  ( ph  ->  ( (bits  |`  NN0 ) `  ( B  mod  (
2 ^ N ) ) )  =  ( (bits `  B )  i^i  ( 0..^ N ) ) )
57 f1ocnvfv 6192 . . . . . . . . . . . . . 14  |-  ( ( (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )  /\  ( B  mod  ( 2 ^ N
) )  e.  NN0 )  ->  ( ( (bits  |`  NN0 ) `  ( B  mod  ( 2 ^ N ) ) )  =  ( (bits `  B )  i^i  (
0..^ N ) )  ->  ( `' (bits  |`  NN0 ) `  (
(bits `  B )  i^i  ( 0..^ N ) ) )  =  ( B  mod  ( 2 ^ N ) ) ) )
5814, 51, 57sylancr 667 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( (bits  |`  NN0 ) `  ( B  mod  (
2 ^ N ) ) )  =  ( (bits `  B )  i^i  ( 0..^ N ) )  ->  ( `' (bits  |`  NN0 ) `  ( (bits `  B )  i^i  ( 0..^ N ) ) )  =  ( B  mod  ( 2 ^ N ) ) ) )
5956, 58mpd 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( (bits `  B
)  i^i  ( 0..^ N ) ) )  =  ( B  mod  ( 2 ^ N
) ) )
6049, 59syl5eq 2482 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  (
(bits `  B )  i^i  ( 0..^ N ) ) )  =  ( B  mod  ( 2 ^ N ) ) )
6160oveq2d 6321 . . . . . . . . . 10  |-  ( ph  ->  ( B  -  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) )  =  ( B  -  ( B  mod  ( 2 ^ N ) ) ) )
6261oveq1d 6320 . . . . . . . . 9  |-  ( ph  ->  ( ( B  -  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) )  /  (
2 ^ N ) )  =  ( ( B  -  ( B  mod  ( 2 ^ N ) ) )  /  ( 2 ^ N ) ) )
6350zred 11040 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR )
64 moddifz 12106 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  ( 2 ^ N
)  e.  RR+ )  ->  ( ( B  -  ( B  mod  ( 2 ^ N ) ) )  /  ( 2 ^ N ) )  e.  ZZ )
6563, 22, 64syl2anc 665 . . . . . . . . 9  |-  ( ph  ->  ( ( B  -  ( B  mod  ( 2 ^ N ) ) )  /  ( 2 ^ N ) )  e.  ZZ )
6662, 65eqeltrd 2517 . . . . . . . 8  |-  ( ph  ->  ( ( B  -  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) )  /  (
2 ^ N ) )  e.  ZZ )
67 inss1 3688 . . . . . . . . . . . . . 14  |-  ( (bits `  B )  i^i  (
0..^ N ) ) 
C_  (bits `  B
)
68 bitsss 14374 . . . . . . . . . . . . . 14  |-  (bits `  B )  C_  NN0
6967, 68sstri 3479 . . . . . . . . . . . . 13  |-  ( (bits `  B )  i^i  (
0..^ N ) ) 
C_  NN0
70 inss2 3689 . . . . . . . . . . . . . 14  |-  ( (bits `  B )  i^i  (
0..^ N ) ) 
C_  ( 0..^ N )
71 ssfi 7798 . . . . . . . . . . . . . 14  |-  ( ( ( 0..^ N )  e.  Fin  /\  (
(bits `  B )  i^i  ( 0..^ N ) )  C_  ( 0..^ N ) )  -> 
( (bits `  B
)  i^i  ( 0..^ N ) )  e. 
Fin )
7231, 70, 71mp2an 676 . . . . . . . . . . . . 13  |-  ( (bits `  B )  i^i  (
0..^ N ) )  e.  Fin
73 elfpw 7882 . . . . . . . . . . . . 13  |-  ( ( (bits `  B )  i^i  ( 0..^ N ) )  e.  ( ~P
NN0  i^i  Fin )  <->  ( ( (bits `  B
)  i^i  ( 0..^ N ) )  C_  NN0 
/\  ( (bits `  B )  i^i  (
0..^ N ) )  e.  Fin ) )
7469, 72, 73mpbir2an 928 . . . . . . . . . . . 12  |-  ( (bits `  B )  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin )
7541ffvelrni 6036 . . . . . . . . . . . 12  |-  ( ( (bits `  B )  i^i  ( 0..^ N ) )  e.  ( ~P
NN0  i^i  Fin )  ->  ( K `  (
(bits `  B )  i^i  ( 0..^ N ) ) )  e.  NN0 )
7674, 75mp1i 13 . . . . . . . . . . 11  |-  ( ph  ->  ( K `  (
(bits `  B )  i^i  ( 0..^ N ) ) )  e.  NN0 )
7776nn0zd 11038 . . . . . . . . . 10  |-  ( ph  ->  ( K `  (
(bits `  B )  i^i  ( 0..^ N ) ) )  e.  ZZ )
7850, 77zsubcld 11045 . . . . . . . . 9  |-  ( ph  ->  ( B  -  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) )  e.  ZZ )
79 dvdsval2 14286 . . . . . . . . 9  |-  ( ( ( 2 ^ N
)  e.  ZZ  /\  ( 2 ^ N
)  =/=  0  /\  ( B  -  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) )  e.  ZZ )  ->  ( ( 2 ^ N )  ||  ( B  -  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) )  <->  ( ( B  -  ( K `  ( (bits `  B
)  i^i  ( 0..^ N ) ) ) )  /  ( 2 ^ N ) )  e.  ZZ ) )
8026, 27, 78, 79syl3anc 1264 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ N )  ||  ( B  -  ( K `  ( (bits `  B
)  i^i  ( 0..^ N ) ) ) )  <->  ( ( B  -  ( K `  ( (bits `  B )  i^i  ( 0..^ N ) ) ) )  / 
( 2 ^ N
) )  e.  ZZ ) )
8166, 80mpbird 235 . . . . . . 7  |-  ( ph  ->  ( 2 ^ N
)  ||  ( B  -  ( K `  ( (bits `  B )  i^i  ( 0..^ N ) ) ) ) )
82 dvds2add 14312 . . . . . . . 8  |-  ( ( ( 2 ^ N
)  e.  ZZ  /\  ( A  -  ( K `  ( (bits `  A )  i^i  (
0..^ N ) ) ) )  e.  ZZ  /\  ( B  -  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) )  e.  ZZ )  ->  ( ( ( 2 ^ N ) 
||  ( A  -  ( K `  ( (bits `  A )  i^i  (
0..^ N ) ) ) )  /\  (
2 ^ N ) 
||  ( B  -  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) ) )  -> 
( 2 ^ N
)  ||  ( ( A  -  ( K `  ( (bits `  A
)  i^i  ( 0..^ N ) ) ) )  +  ( B  -  ( K `  ( (bits `  B )  i^i  ( 0..^ N ) ) ) ) ) ) )
8326, 45, 78, 82syl3anc 1264 . . . . . . 7  |-  ( ph  ->  ( ( ( 2 ^ N )  ||  ( A  -  ( K `  ( (bits `  A )  i^i  (
0..^ N ) ) ) )  /\  (
2 ^ N ) 
||  ( B  -  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) ) )  -> 
( 2 ^ N
)  ||  ( ( A  -  ( K `  ( (bits `  A
)  i^i  ( 0..^ N ) ) ) )  +  ( B  -  ( K `  ( (bits `  B )  i^i  ( 0..^ N ) ) ) ) ) ) )
8448, 81, 83mp2and 683 . . . . . 6  |-  ( ph  ->  ( 2 ^ N
)  ||  ( ( A  -  ( K `  ( (bits `  A
)  i^i  ( 0..^ N ) ) ) )  +  ( B  -  ( K `  ( (bits `  B )  i^i  ( 0..^ N ) ) ) ) ) )
853zcnd 11041 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
8650zcnd 11041 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
8743nn0cnd 10927 . . . . . . 7  |-  ( ph  ->  ( K `  (
(bits `  A )  i^i  ( 0..^ N ) ) )  e.  CC )
8876nn0cnd 10927 . . . . . . 7  |-  ( ph  ->  ( K `  (
(bits `  B )  i^i  ( 0..^ N ) ) )  e.  CC )
8985, 86, 87, 88addsub4d 10032 . . . . . 6  |-  ( ph  ->  ( ( A  +  B )  -  (
( K `  (
(bits `  A )  i^i  ( 0..^ N ) ) )  +  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) ) )  =  ( ( A  -  ( K `  ( (bits `  A )  i^i  (
0..^ N ) ) ) )  +  ( B  -  ( K `
 ( (bits `  B )  i^i  (
0..^ N ) ) ) ) ) )
9084, 89breqtrrd 4452 . . . . 5  |-  ( ph  ->  ( 2 ^ N
)  ||  ( ( A  +  B )  -  ( ( K `
 ( (bits `  A )  i^i  (
0..^ N ) ) )  +  ( K `
 ( (bits `  B )  i^i  (
0..^ N ) ) ) ) ) )
913, 50zaddcld 11044 . . . . . 6  |-  ( ph  ->  ( A  +  B
)  e.  ZZ )
9244, 77zaddcld 11044 . . . . . 6  |-  ( ph  ->  ( ( K `  ( (bits `  A )  i^i  ( 0..^ N ) ) )  +  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) )  e.  ZZ )
93 moddvds 14290 . . . . . 6  |-  ( ( ( 2 ^ N
)  e.  NN  /\  ( A  +  B
)  e.  ZZ  /\  ( ( K `  ( (bits `  A )  i^i  ( 0..^ N ) ) )  +  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) )  e.  ZZ )  ->  ( ( ( A  +  B )  mod  ( 2 ^ N ) )  =  ( ( ( K `
 ( (bits `  A )  i^i  (
0..^ N ) ) )  +  ( K `
 ( (bits `  B )  i^i  (
0..^ N ) ) ) )  mod  (
2 ^ N ) )  <->  ( 2 ^ N )  ||  (
( A  +  B
)  -  ( ( K `  ( (bits `  A )  i^i  (
0..^ N ) ) )  +  ( K `
 ( (bits `  B )  i^i  (
0..^ N ) ) ) ) ) ) )
947, 91, 92, 93syl3anc 1264 . . . . 5  |-  ( ph  ->  ( ( ( A  +  B )  mod  ( 2 ^ N
) )  =  ( ( ( K `  ( (bits `  A )  i^i  ( 0..^ N ) ) )  +  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) )  mod  (
2 ^ N ) )  <->  ( 2 ^ N )  ||  (
( A  +  B
)  -  ( ( K `  ( (bits `  A )  i^i  (
0..^ N ) ) )  +  ( K `
 ( (bits `  B )  i^i  (
0..^ N ) ) ) ) ) ) )
9590, 94mpbird 235 . . . 4  |-  ( ph  ->  ( ( A  +  B )  mod  (
2 ^ N ) )  =  ( ( ( K `  (
(bits `  A )  i^i  ( 0..^ N ) ) )  +  ( K `  ( (bits `  B )  i^i  (
0..^ N ) ) ) )  mod  (
2 ^ N ) ) )
9629a1i 11 . . . . 5  |-  ( ph  ->  (bits `  A )  C_ 
NN0 )
9768a1i 11 . . . . 5  |-  ( ph  ->  (bits `  B )  C_ 
NN0 )
98 sadaddlem.c . . . . 5  |-  C  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  (bits `  A ) ,  m  e.  (bits `  B ) ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) )
9996, 97, 98, 6, 1sadadd3 14409 . . . 4  |-  ( ph  ->  ( ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) )  =  ( ( ( K `  ( (bits `  A )  i^i  (
0..^ N ) ) )  +  ( K `
 ( (bits `  B )  i^i  (
0..^ N ) ) ) )  mod  (
2 ^ N ) ) )
100 inss1 3688 . . . . . . . . 9  |-  ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) )  C_  (
(bits `  A ) sadd  (bits `  B ) )
101 sadcl 14410 . . . . . . . . . 10  |-  ( ( (bits `  A )  C_ 
NN0  /\  (bits `  B
)  C_  NN0 )  -> 
( (bits `  A
) sadd  (bits `  B )
)  C_  NN0 )
10229, 68, 101mp2an 676 . . . . . . . . 9  |-  ( (bits `  A ) sadd  (bits `  B ) )  C_  NN0
103100, 102sstri 3479 . . . . . . . 8  |-  ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) )  C_  NN0
104 inss2 3689 . . . . . . . . 9  |-  ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) )  C_  (
0..^ N )
105 ssfi 7798 . . . . . . . . 9  |-  ( ( ( 0..^ N )  e.  Fin  /\  (
( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) )  C_  ( 0..^ N ) )  ->  ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) )  e.  Fin )
10631, 104, 105mp2an 676 . . . . . . . 8  |-  ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) )  e.  Fin
107 elfpw 7882 . . . . . . . 8  |-  ( ( ( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  <->  ( ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) )  C_  NN0  /\  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) )  e.  Fin ) )
108103, 106, 107mpbir2an 928 . . . . . . 7  |-  ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )
10941ffvelrni 6036 . . . . . . 7  |-  ( ( ( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  ->  ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) )  e.  NN0 )
110108, 109mp1i 13 . . . . . 6  |-  ( ph  ->  ( K `  (
( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) ) )  e.  NN0 )
111110nn0red 10926 . . . . 5  |-  ( ph  ->  ( K `  (
( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) ) )  e.  RR )
112110nn0ge0d 10928 . . . . 5  |-  ( ph  ->  0  <_  ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) ) )
1131fveq1i 5882 . . . . . . . . . 10  |-  ( K `
 ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) ) )  =  ( `' (bits  |`  NN0 ) `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) )
114113fveq2i 5884 . . . . . . . . 9  |-  ( (bits  |`  NN0 ) `  ( K `  ( (
(bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) ) ) )  =  ( (bits  |`  NN0 ) `  ( `' (bits  |`  NN0 ) `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) ) )
115 fvres 5895 . . . . . . . . . 10  |-  ( ( K `  ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) ) )  e. 
NN0  ->  ( (bits  |`  NN0 ) `  ( K `  (
( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) ) ) )  =  (bits `  ( K `  ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) ) ) ) )
116110, 115syl 17 . . . . . . . . 9  |-  ( ph  ->  ( (bits  |`  NN0 ) `  ( K `  (
( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) ) ) )  =  (bits `  ( K `  ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) ) ) ) )
117108a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) )
118 f1ocnvfv2 6191 . . . . . . . . . 10  |-  ( ( (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )  /\  ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) )  e.  ( ~P
NN0  i^i  Fin )
)  ->  ( (bits  |` 
NN0 ) `  ( `' (bits  |`  NN0 ) `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) ) )  =  ( ( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) ) )
11914, 117, 118sylancr 667 . . . . . . . . 9  |-  ( ph  ->  ( (bits  |`  NN0 ) `  ( `' (bits  |`  NN0 ) `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) ) )  =  ( ( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) ) )
120114, 116, 1193eqtr3a 2494 . . . . . . . 8  |-  ( ph  ->  (bits `  ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) ) )  =  ( ( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) ) )
121120, 104syl6eqss 3520 . . . . . . 7  |-  ( ph  ->  (bits `  ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) ) )  C_  (
0..^ N ) )
122110nn0zd 11038 . . . . . . . 8  |-  ( ph  ->  ( K `  (
( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) ) )  e.  ZZ )
123 bitsfzo 14383 . . . . . . . 8  |-  ( ( ( K `  (
( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) ) )  e.  ZZ  /\  N  e.  NN0 )  ->  (
( K `  (
( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) )  <->  (bits `  ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) ) )  C_  (
0..^ N ) ) )
124122, 6, 123syl2anc 665 . . . . . . 7  |-  ( ph  ->  ( ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N
) )  <->  (bits `  ( K `  ( (
(bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) ) ) ) 
C_  ( 0..^ N ) ) )
125121, 124mpbird 235 . . . . . 6  |-  ( ph  ->  ( K `  (
( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) ) )
126 elfzolt2 11927 . . . . . 6  |-  ( ( K `  ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) )  ->  ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) )  <  ( 2 ^ N ) )
127125, 126syl 17 . . . . 5  |-  ( ph  ->  ( K `  (
( (bits `  A
) sadd  (bits `  B )
)  i^i  ( 0..^ N ) ) )  <  ( 2 ^ N ) )
128 modid 12118 . . . . 5  |-  ( ( ( ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) )  e.  RR  /\  ( 2 ^ N
)  e.  RR+ )  /\  ( 0  <_  ( K `  ( (
(bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) ) )  /\  ( K `  ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) ) )  < 
( 2 ^ N
) ) )  -> 
( ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) )  =  ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) ) )
129111, 22, 112, 127, 128syl22anc 1265 . . . 4  |-  ( ph  ->  ( ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) )  =  ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) ) )
13095, 99, 1293eqtr2d 2476 . . 3  |-  ( ph  ->  ( ( A  +  B )  mod  (
2 ^ N ) )  =  ( K `
 ( ( (bits `  A ) sadd  (bits `  B ) )  i^i  ( 0..^ N ) ) ) )
131130fveq2d 5885 . 2  |-  ( ph  ->  (bits `  ( ( A  +  B )  mod  ( 2 ^ N
) ) )  =  (bits `  ( K `  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) ) ) ) )
132131, 120eqtr2d 2471 1  |-  ( ph  ->  ( ( (bits `  A ) sadd  (bits `  B
) )  i^i  (
0..^ N ) )  =  (bits `  (
( A  +  B
)  mod  ( 2 ^ N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437  caddwcad 1504    e. wcel 1870    =/= wne 2625    i^i cin 3441    C_ wss 3442   (/)c0 3767   ifcif 3915   ~Pcpw 3985   class class class wbr 4426    |-> cmpt 4484   `'ccnv 4853    |` cres 4856   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   1oc1o 7183   2oc2o 7184   Fincfn 7577   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541    < clt 9674    <_ cle 9675    - cmin 9859    / cdiv 10268   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   RR+crp 11302  ..^cfzo 11913    mod cmo 12093    seqcseq 12210   ^cexp 12269    || cdvds 14283  bitscbits 14367   sadd csad 14368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-xor 1401  df-tru 1440  df-fal 1443  df-had 1492  df-cad 1505  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  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 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-dvds 14284  df-bits 14370  df-sad 14399
This theorem is referenced by:  sadadd  14415
  Copyright terms: Public domain W3C validator