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Theorem sadasslem 14444
Description: Lemma for sadass 14445. (Contributed by Mario Carneiro, 9-Sep-2016.)
Hypotheses
Ref Expression
sadasslem.1  |-  ( ph  ->  A  C_  NN0 )
sadasslem.2  |-  ( ph  ->  B  C_  NN0 )
sadasslem.3  |-  ( ph  ->  C  C_  NN0 )
sadasslem.4  |-  ( ph  ->  N  e.  NN0 )
Assertion
Ref Expression
sadasslem  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  =  ( ( A sadd  ( B sadd 
C ) )  i^i  ( 0..^ N ) ) )

Proof of Theorem sadasslem
Dummy variables  c  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3652 . . . . . . . . . . 11  |-  ( A  i^i  ( 0..^ N ) )  C_  A
2 sadasslem.1 . . . . . . . . . . 11  |-  ( ph  ->  A  C_  NN0 )
31, 2syl5ss 3443 . . . . . . . . . 10  |-  ( ph  ->  ( A  i^i  (
0..^ N ) ) 
C_  NN0 )
4 fzofi 12187 . . . . . . . . . . . 12  |-  ( 0..^ N )  e.  Fin
54a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( 0..^ N )  e.  Fin )
6 inss2 3653 . . . . . . . . . . 11  |-  ( A  i^i  ( 0..^ N ) )  C_  (
0..^ N )
7 ssfi 7792 . . . . . . . . . . 11  |-  ( ( ( 0..^ N )  e.  Fin  /\  ( A  i^i  ( 0..^ N ) )  C_  (
0..^ N ) )  ->  ( A  i^i  ( 0..^ N ) )  e.  Fin )
85, 6, 7sylancl 668 . . . . . . . . . 10  |-  ( ph  ->  ( A  i^i  (
0..^ N ) )  e.  Fin )
9 elfpw 7876 . . . . . . . . . 10  |-  ( ( A  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  <->  ( ( A  i^i  ( 0..^ N ) )  C_  NN0  /\  ( A  i^i  (
0..^ N ) )  e.  Fin ) )
103, 8, 9sylanbrc 670 . . . . . . . . 9  |-  ( ph  ->  ( A  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) )
11 bitsf1o 14419 . . . . . . . . . . 11  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
12 f1ocnv 5826 . . . . . . . . . . 11  |-  ( (bits  |`  NN0 ) : NN0 -1-1-onto-> ( ~P NN0  i^i  Fin )  ->  `' (bits  |`  NN0 ) : ( ~P NN0  i^i 
Fin ) -1-1-onto-> NN0 )
13 f1of 5814 . . . . . . . . . . 11  |-  ( `' (bits  |`  NN0 ) : ( ~P NN0  i^i  Fin ) -1-1-onto-> NN0  ->  `' (bits  |` 
NN0 ) : ( ~P NN0  i^i  Fin )
--> NN0 )
1411, 12, 13mp2b 10 . . . . . . . . . 10  |-  `' (bits  |`  NN0 ) : ( ~P NN0  i^i  Fin )
--> NN0
1514ffvelrni 6021 . . . . . . . . 9  |-  ( ( A  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  NN0 )
1610, 15syl 17 . . . . . . . 8  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  NN0 )
1716nn0cnd 10927 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  CC )
18 inss1 3652 . . . . . . . . . . 11  |-  ( B  i^i  ( 0..^ N ) )  C_  B
19 sadasslem.2 . . . . . . . . . . 11  |-  ( ph  ->  B  C_  NN0 )
2018, 19syl5ss 3443 . . . . . . . . . 10  |-  ( ph  ->  ( B  i^i  (
0..^ N ) ) 
C_  NN0 )
21 inss2 3653 . . . . . . . . . . 11  |-  ( B  i^i  ( 0..^ N ) )  C_  (
0..^ N )
22 ssfi 7792 . . . . . . . . . . 11  |-  ( ( ( 0..^ N )  e.  Fin  /\  ( B  i^i  ( 0..^ N ) )  C_  (
0..^ N ) )  ->  ( B  i^i  ( 0..^ N ) )  e.  Fin )
235, 21, 22sylancl 668 . . . . . . . . . 10  |-  ( ph  ->  ( B  i^i  (
0..^ N ) )  e.  Fin )
24 elfpw 7876 . . . . . . . . . 10  |-  ( ( B  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  <->  ( ( B  i^i  ( 0..^ N ) )  C_  NN0  /\  ( B  i^i  (
0..^ N ) )  e.  Fin ) )
2520, 23, 24sylanbrc 670 . . . . . . . . 9  |-  ( ph  ->  ( B  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) )
2614ffvelrni 6021 . . . . . . . . 9  |-  ( ( B  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  NN0 )
2725, 26syl 17 . . . . . . . 8  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  NN0 )
2827nn0cnd 10927 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  CC )
29 inss1 3652 . . . . . . . . . . 11  |-  ( C  i^i  ( 0..^ N ) )  C_  C
30 sadasslem.3 . . . . . . . . . . 11  |-  ( ph  ->  C  C_  NN0 )
3129, 30syl5ss 3443 . . . . . . . . . 10  |-  ( ph  ->  ( C  i^i  (
0..^ N ) ) 
C_  NN0 )
32 inss2 3653 . . . . . . . . . . 11  |-  ( C  i^i  ( 0..^ N ) )  C_  (
0..^ N )
33 ssfi 7792 . . . . . . . . . . 11  |-  ( ( ( 0..^ N )  e.  Fin  /\  ( C  i^i  ( 0..^ N ) )  C_  (
0..^ N ) )  ->  ( C  i^i  ( 0..^ N ) )  e.  Fin )
345, 32, 33sylancl 668 . . . . . . . . . 10  |-  ( ph  ->  ( C  i^i  (
0..^ N ) )  e.  Fin )
35 elfpw 7876 . . . . . . . . . 10  |-  ( ( C  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  <->  ( ( C  i^i  ( 0..^ N ) )  C_  NN0  /\  ( C  i^i  (
0..^ N ) )  e.  Fin ) )
3631, 34, 35sylanbrc 670 . . . . . . . . 9  |-  ( ph  ->  ( C  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) )
3714ffvelrni 6021 . . . . . . . . 9  |-  ( ( C  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  NN0 )
3836, 37syl 17 . . . . . . . 8  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  NN0 )
3938nn0cnd 10927 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  CC )
4017, 28, 39addassd 9665 . . . . . 6  |-  ( ph  ->  ( ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) ) )  =  ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  +  ( ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) ) ) ) )
4140oveq1d 6305 . . . . 5  |-  ( ph  ->  ( ( ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) ) )  mod  (
2 ^ N ) )  =  ( ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  +  ( ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) ) ) )  mod  ( 2 ^ N
) ) )
42 inss1 3652 . . . . . . . . . 10  |-  ( ( A sadd  B )  i^i  ( 0..^ N ) )  C_  ( A sadd  B )
43 sadcl 14436 . . . . . . . . . . 11  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0 )  ->  ( A sadd  B )  C_  NN0 )
442, 19, 43syl2anc 667 . . . . . . . . . 10  |-  ( ph  ->  ( A sadd  B ) 
C_  NN0 )
4542, 44syl5ss 3443 . . . . . . . . 9  |-  ( ph  ->  ( ( A sadd  B
)  i^i  ( 0..^ N ) )  C_  NN0 )
46 inss2 3653 . . . . . . . . . 10  |-  ( ( A sadd  B )  i^i  ( 0..^ N ) )  C_  ( 0..^ N )
47 ssfi 7792 . . . . . . . . . 10  |-  ( ( ( 0..^ N )  e.  Fin  /\  (
( A sadd  B )  i^i  ( 0..^ N ) )  C_  ( 0..^ N ) )  -> 
( ( A sadd  B
)  i^i  ( 0..^ N ) )  e. 
Fin )
485, 46, 47sylancl 668 . . . . . . . . 9  |-  ( ph  ->  ( ( A sadd  B
)  i^i  ( 0..^ N ) )  e. 
Fin )
49 elfpw 7876 . . . . . . . . 9  |-  ( ( ( A sadd  B )  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) 
<->  ( ( ( A sadd 
B )  i^i  (
0..^ N ) ) 
C_  NN0  /\  (
( A sadd  B )  i^i  ( 0..^ N ) )  e.  Fin )
)
5045, 48, 49sylanbrc 670 . . . . . . . 8  |-  ( ph  ->  ( ( A sadd  B
)  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )
5114ffvelrni 6021 . . . . . . . 8  |-  ( ( ( A sadd  B )  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  ->  ( `' (bits  |`  NN0 ) `  (
( A sadd  B )  i^i  ( 0..^ N ) ) )  e.  NN0 )
5250, 51syl 17 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  B
)  i^i  ( 0..^ N ) ) )  e.  NN0 )
5352nn0red 10926 . . . . . 6  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  B
)  i^i  ( 0..^ N ) ) )  e.  RR )
5416nn0red 10926 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  RR )
5527nn0red 10926 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  RR )
5654, 55readdcld 9670 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  ( A  i^i  ( 0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) ) )  e.  RR )
5738nn0red 10926 . . . . . 6  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  RR )
58 2rp 11307 . . . . . . . 8  |-  2  e.  RR+
5958a1i 11 . . . . . . 7  |-  ( ph  ->  2  e.  RR+ )
60 sadasslem.4 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
6160nn0zd 11038 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
6259, 61rpexpcld 12439 . . . . . 6  |-  ( ph  ->  ( 2 ^ N
)  e.  RR+ )
63 eqid 2451 . . . . . . 7  |-  seq 0
( ( c  e.  2o ,  m  e. 
NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/) 
e.  c ) ,  1o ,  (/) ) ) ,  ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  A ,  m  e.  B ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) )
64 eqid 2451 . . . . . . 7  |-  `' (bits  |`  NN0 )  =  `' (bits  |`  NN0 )
652, 19, 63, 60, 64sadadd3 14435 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  (
( A sadd  B )  i^i  ( 0..^ N ) ) )  mod  (
2 ^ N ) )  =  ( ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) ) )  mod  (
2 ^ N ) ) )
66 eqidd 2452 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  ( C  i^i  ( 0..^ N ) ) )  mod  ( 2 ^ N
) )  =  ( ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) ) )
6753, 56, 57, 57, 62, 65, 66modadd12d 12146 . . . . 5  |-  ( ph  ->  ( ( ( `' (bits  |`  NN0 ) `  ( ( A sadd  B
)  i^i  ( 0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  ( 0..^ N ) ) ) )  mod  ( 2 ^ N ) )  =  ( ( ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) ) )  mod  (
2 ^ N ) ) )
68 inss1 3652 . . . . . . . . . 10  |-  ( ( B sadd  C )  i^i  ( 0..^ N ) )  C_  ( B sadd  C )
69 sadcl 14436 . . . . . . . . . . 11  |-  ( ( B  C_  NN0  /\  C  C_ 
NN0 )  ->  ( B sadd  C )  C_  NN0 )
7019, 30, 69syl2anc 667 . . . . . . . . . 10  |-  ( ph  ->  ( B sadd  C ) 
C_  NN0 )
7168, 70syl5ss 3443 . . . . . . . . 9  |-  ( ph  ->  ( ( B sadd  C
)  i^i  ( 0..^ N ) )  C_  NN0 )
72 inss2 3653 . . . . . . . . . 10  |-  ( ( B sadd  C )  i^i  ( 0..^ N ) )  C_  ( 0..^ N )
73 ssfi 7792 . . . . . . . . . 10  |-  ( ( ( 0..^ N )  e.  Fin  /\  (
( B sadd  C )  i^i  ( 0..^ N ) )  C_  ( 0..^ N ) )  -> 
( ( B sadd  C
)  i^i  ( 0..^ N ) )  e. 
Fin )
745, 72, 73sylancl 668 . . . . . . . . 9  |-  ( ph  ->  ( ( B sadd  C
)  i^i  ( 0..^ N ) )  e. 
Fin )
75 elfpw 7876 . . . . . . . . 9  |-  ( ( ( B sadd  C )  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) 
<->  ( ( ( B sadd 
C )  i^i  (
0..^ N ) ) 
C_  NN0  /\  (
( B sadd  C )  i^i  ( 0..^ N ) )  e.  Fin )
)
7671, 74, 75sylanbrc 670 . . . . . . . 8  |-  ( ph  ->  ( ( B sadd  C
)  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )
7714ffvelrni 6021 . . . . . . . 8  |-  ( ( ( B sadd  C )  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  ->  ( `' (bits  |`  NN0 ) `  (
( B sadd  C )  i^i  ( 0..^ N ) ) )  e.  NN0 )
7876, 77syl 17 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( B sadd  C
)  i^i  ( 0..^ N ) ) )  e.  NN0 )
7978nn0red 10926 . . . . . 6  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( B sadd  C
)  i^i  ( 0..^ N ) ) )  e.  RR )
8055, 57readdcld 9670 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  ( B  i^i  ( 0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) ) )  e.  RR )
81 eqidd 2452 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  ( A  i^i  ( 0..^ N ) ) )  mod  ( 2 ^ N
) )  =  ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) ) )
82 eqid 2451 . . . . . . 7  |-  seq 0
( ( c  e.  2o ,  m  e. 
NN0  |->  if (cadd ( m  e.  B ,  m  e.  C ,  (/) 
e.  c ) ,  1o ,  (/) ) ) ,  ( n  e. 
NN0  |->  if ( n  =  0 ,  (/) ,  ( n  -  1 ) ) ) )  =  seq 0 ( ( c  e.  2o ,  m  e.  NN0  |->  if (cadd ( m  e.  B ,  m  e.  C ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) )
8319, 30, 82, 60, 64sadadd3 14435 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  (
( B sadd  C )  i^i  ( 0..^ N ) ) )  mod  (
2 ^ N ) )  =  ( ( ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) ) )  mod  (
2 ^ N ) ) )
8454, 54, 79, 80, 62, 81, 83modadd12d 12146 . . . . 5  |-  ( ph  ->  ( ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( ( B sadd  C
)  i^i  ( 0..^ N ) ) ) )  mod  ( 2 ^ N ) )  =  ( ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  +  ( ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) ) ) )  mod  ( 2 ^ N
) ) )
8541, 67, 843eqtr4d 2495 . . . 4  |-  ( ph  ->  ( ( ( `' (bits  |`  NN0 ) `  ( ( A sadd  B
)  i^i  ( 0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  ( 0..^ N ) ) ) )  mod  ( 2 ^ N ) )  =  ( ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( ( B sadd  C
)  i^i  ( 0..^ N ) ) ) )  mod  ( 2 ^ N ) ) )
86 eqid 2451 . . . . 5  |-  seq 0
( ( c  e.  2o ,  m  e. 
NN0  |->  if (cadd ( m  e.  ( A sadd 
B ) ,  m  e.  C ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) )  =  seq 0
( ( c  e.  2o ,  m  e. 
NN0  |->  if (cadd ( m  e.  ( A sadd 
B ) ,  m  e.  C ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) )
8744, 30, 86, 60, 64sadadd3 14435 . . . 4  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) )  =  ( ( ( `' (bits  |`  NN0 ) `  ( ( A sadd  B
)  i^i  ( 0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  ( 0..^ N ) ) ) )  mod  ( 2 ^ N ) ) )
88 eqid 2451 . . . . 5  |-  seq 0
( ( c  e.  2o ,  m  e. 
NN0  |->  if (cadd ( m  e.  A ,  m  e.  ( B sadd  C ) ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) )  =  seq 0
( ( c  e.  2o ,  m  e. 
NN0  |->  if (cadd ( m  e.  A ,  m  e.  ( B sadd  C ) ,  (/)  e.  c ) ,  1o ,  (/) ) ) ,  ( n  e.  NN0  |->  if ( n  =  0 ,  (/) ,  ( n  - 
1 ) ) ) )
892, 70, 88, 60, 64sadadd3 14435 . . . 4  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  (
( A sadd  ( B sadd  C ) )  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) )  =  ( ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( ( B sadd  C
)  i^i  ( 0..^ N ) ) ) )  mod  ( 2 ^ N ) ) )
9085, 87, 893eqtr4d 2495 . . 3  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) )  =  ( ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  mod  (
2 ^ N ) ) )
91 inss1 3652 . . . . . . . 8  |-  ( ( ( A sadd  B ) sadd 
C )  i^i  (
0..^ N ) ) 
C_  ( ( A sadd 
B ) sadd  C )
92 sadcl 14436 . . . . . . . . 9  |-  ( ( ( A sadd  B ) 
C_  NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B ) sadd  C ) 
C_  NN0 )
9344, 30, 92syl2anc 667 . . . . . . . 8  |-  ( ph  ->  ( ( A sadd  B
) sadd  C )  C_  NN0 )
9491, 93syl5ss 3443 . . . . . . 7  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  C_  NN0 )
95 inss2 3653 . . . . . . . 8  |-  ( ( ( A sadd  B ) sadd 
C )  i^i  (
0..^ N ) ) 
C_  ( 0..^ N )
96 ssfi 7792 . . . . . . . 8  |-  ( ( ( 0..^ N )  e.  Fin  /\  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) ) 
C_  ( 0..^ N ) )  ->  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) )  e.  Fin )
975, 95, 96sylancl 668 . . . . . . 7  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  e.  Fin )
98 elfpw 7876 . . . . . . 7  |-  ( ( ( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin )  <->  ( (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) ) 
C_  NN0  /\  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) )  e.  Fin ) )
9994, 97, 98sylanbrc 670 . . . . . 6  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )
10014ffvelrni 6021 . . . . . 6  |-  ( ( ( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin )  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e. 
NN0 )
10199, 100syl 17 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e. 
NN0 )
102101nn0red 10926 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  RR )
103101nn0ge0d 10928 . . . 4  |-  ( ph  ->  0  <_  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) )
104 fvres 5879 . . . . . . . . 9  |-  ( ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e. 
NN0  ->  ( (bits  |`  NN0 ) `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) )  =  (bits `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) ) )
105101, 104syl 17 . . . . . . . 8  |-  ( ph  ->  ( (bits  |`  NN0 ) `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) )  =  (bits `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) ) )
106 f1ocnvfv2 6176 . . . . . . . . 9  |-  ( ( (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )  /\  ( ( ( A sadd  B ) sadd  C
)  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )  ->  (
(bits  |`  NN0 ) `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) )  =  ( ( ( A sadd  B ) sadd  C
)  i^i  ( 0..^ N ) ) )
10711, 99, 106sylancr 669 . . . . . . . 8  |-  ( ph  ->  ( (bits  |`  NN0 ) `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) )  =  ( ( ( A sadd  B ) sadd  C
)  i^i  ( 0..^ N ) ) )
108105, 107eqtr3d 2487 . . . . . . 7  |-  ( ph  ->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) )  =  ( ( ( A sadd  B ) sadd  C
)  i^i  ( 0..^ N ) ) )
109108, 95syl6eqss 3482 . . . . . 6  |-  ( ph  ->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) ) 
C_  ( 0..^ N ) )
110101nn0zd 11038 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  ZZ )
111 bitsfzo 14409 . . . . . . 7  |-  ( ( ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  ZZ  /\  N  e. 
NN0 )  ->  (
( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) )  <-> 
(bits `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) ) 
C_  ( 0..^ N ) ) )
112110, 60, 111syl2anc 667 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N
) )  <->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) ) 
C_  ( 0..^ N ) ) )
113109, 112mpbird 236 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) ) )
114 elfzolt2 11929 . . . . 5  |-  ( ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) )  ->  ( `' (bits  |`  NN0 ) `  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) ) )  <  ( 2 ^ N ) )
115113, 114syl 17 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  < 
( 2 ^ N
) )
116 modid 12121 . . . 4  |-  ( ( ( ( `' (bits  |`  NN0 ) `  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) ) )  e.  RR  /\  ( 2 ^ N
)  e.  RR+ )  /\  ( 0  <_  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  /\  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  < 
( 2 ^ N
) ) )  -> 
( ( `' (bits  |`  NN0 ) `  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) )  =  ( `' (bits  |`  NN0 ) `  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) ) ) )
117102, 62, 103, 115, 116syl22anc 1269 . . 3  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) )  =  ( `' (bits  |`  NN0 ) `  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) ) ) )
118 inss1 3652 . . . . . . . 8  |-  ( ( A sadd  ( B sadd  C
) )  i^i  (
0..^ N ) ) 
C_  ( A sadd  ( B sadd  C ) )
119 sadcl 14436 . . . . . . . . 9  |-  ( ( A  C_  NN0  /\  ( B sadd  C )  C_  NN0 )  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
1202, 70, 119syl2anc 667 . . . . . . . 8  |-  ( ph  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
121118, 120syl5ss 3443 . . . . . . 7  |-  ( ph  ->  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  C_  NN0 )
122 inss2 3653 . . . . . . . 8  |-  ( ( A sadd  ( B sadd  C
) )  i^i  (
0..^ N ) ) 
C_  ( 0..^ N )
123 ssfi 7792 . . . . . . . 8  |-  ( ( ( 0..^ N )  e.  Fin  /\  (
( A sadd  ( B sadd  C ) )  i^i  (
0..^ N ) ) 
C_  ( 0..^ N ) )  ->  (
( A sadd  ( B sadd  C ) )  i^i  (
0..^ N ) )  e.  Fin )
1245, 122, 123sylancl 668 . . . . . . 7  |-  ( ph  ->  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  e.  Fin )
125 elfpw 7876 . . . . . . 7  |-  ( ( ( A sadd  ( B sadd 
C ) )  i^i  ( 0..^ N ) )  e.  ( ~P
NN0  i^i  Fin )  <->  ( ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  C_  NN0  /\  (
( A sadd  ( B sadd  C ) )  i^i  (
0..^ N ) )  e.  Fin ) )
126121, 124, 125sylanbrc 670 . . . . . 6  |-  ( ph  ->  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  e.  ( ~P
NN0  i^i  Fin )
)
12714ffvelrni 6021 . . . . . 6  |-  ( ( ( A sadd  ( B sadd 
C ) )  i^i  ( 0..^ N ) )  e.  ( ~P
NN0  i^i  Fin )  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  NN0 )
128126, 127syl 17 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  NN0 )
129128nn0red 10926 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  RR )
130 2nn 10767 . . . . . . 7  |-  2  e.  NN
131130a1i 11 . . . . . 6  |-  ( ph  ->  2  e.  NN )
132131, 60nnexpcld 12437 . . . . 5  |-  ( ph  ->  ( 2 ^ N
)  e.  NN )
133132nnrpd 11339 . . . 4  |-  ( ph  ->  ( 2 ^ N
)  e.  RR+ )
134128nn0ge0d 10928 . . . 4  |-  ( ph  ->  0  <_  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )
135 fvres 5879 . . . . . . . . 9  |-  ( ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  NN0  ->  ( (bits  |`  NN0 ) `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )  =  (bits `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) ) )
136128, 135syl 17 . . . . . . . 8  |-  ( ph  ->  ( (bits  |`  NN0 ) `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )  =  (bits `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) ) )
137 f1ocnvfv2 6176 . . . . . . . . 9  |-  ( ( (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )  /\  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )  ->  (
(bits  |`  NN0 ) `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )  =  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )
13811, 126, 137sylancr 669 . . . . . . . 8  |-  ( ph  ->  ( (bits  |`  NN0 ) `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )  =  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )
139136, 138eqtr3d 2487 . . . . . . 7  |-  ( ph  ->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )  =  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )
140139, 122syl6eqss 3482 . . . . . 6  |-  ( ph  ->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )  C_  ( 0..^ N ) )
141128nn0zd 11038 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  ZZ )
142 bitsfzo 14409 . . . . . . 7  |-  ( ( ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  ZZ  /\  N  e.  NN0 )  ->  ( ( `' (bits  |`  NN0 ) `  (
( A sadd  ( B sadd  C ) )  i^i  (
0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N
) )  <->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )  C_  ( 0..^ N ) ) )
143141, 60, 142syl2anc 667 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  (
( A sadd  ( B sadd  C ) )  i^i  (
0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N
) )  <->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )  C_  ( 0..^ N ) ) )
144140, 143mpbird 236 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) ) )
145 elfzolt2 11929 . . . . 5  |-  ( ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) )  -> 
( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  <  (
2 ^ N ) )
146144, 145syl 17 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  <  (
2 ^ N ) )
147 modid 12121 . . . 4  |-  ( ( ( ( `' (bits  |`  NN0 ) `  (
( A sadd  ( B sadd  C ) )  i^i  (
0..^ N ) ) )  e.  RR  /\  ( 2 ^ N
)  e.  RR+ )  /\  ( 0  <_  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  /\  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  <  (
2 ^ N ) ) )  ->  (
( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  mod  (
2 ^ N ) )  =  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )
148129, 133, 134, 146, 147syl22anc 1269 . . 3  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  (
( A sadd  ( B sadd  C ) )  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) )  =  ( `' (bits  |`  NN0 ) `  (
( A sadd  ( B sadd  C ) )  i^i  (
0..^ N ) ) ) )
14990, 117, 1483eqtr3d 2493 . 2  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  =  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )
150 f1of1 5813 . . . . 5  |-  ( `' (bits  |`  NN0 ) : ( ~P NN0  i^i  Fin ) -1-1-onto-> NN0  ->  `' (bits  |` 
NN0 ) : ( ~P NN0  i^i  Fin ) -1-1-> NN0 )
15111, 12, 150mp2b 10 . . . 4  |-  `' (bits  |`  NN0 ) : ( ~P NN0  i^i  Fin ) -1-1-> NN0
152 f1fveq 6163 . . . 4  |-  ( ( `' (bits  |`  NN0 ) : ( ~P NN0  i^i 
Fin ) -1-1-> NN0  /\  ( ( ( ( A sadd  B ) sadd  C
)  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  /\  ( ( A sadd  ( B sadd  C
) )  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) ) )  ->  ( ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  =  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  <->  ( (
( A sadd  B ) sadd  C )  i^i  ( 0..^ N ) )  =  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )
153151, 152mpan 676 . . 3  |-  ( ( ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  /\  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )  ->  (
( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  =  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  <->  ( (
( A sadd  B ) sadd  C )  i^i  ( 0..^ N ) )  =  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )
15499, 126, 153syl2anc 667 . 2  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  (
( ( A sadd  B
) sadd  C )  i^i  (
0..^ N ) ) )  =  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  <->  ( (
( A sadd  B ) sadd  C )  i^i  ( 0..^ N ) )  =  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )
155149, 154mpbid 214 1  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  =  ( ( A sadd  ( B sadd 
C ) )  i^i  ( 0..^ N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444  caddwcad 1509    e. wcel 1887    i^i cin 3403    C_ wss 3404   (/)c0 3731   ifcif 3881   ~Pcpw 3951   class class class wbr 4402    |-> cmpt 4461   `'ccnv 4833    |` cres 4836   -->wf 5578   -1-1->wf1 5579   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   1oc1o 7175   2oc2o 7176   Fincfn 7569   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676    - cmin 9860   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   RR+crp 11302  ..^cfzo 11915    mod cmo 12096    seqcseq 12213   ^cexp 12272  bitscbits 14392   sadd csad 14393
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
This theorem is referenced by:  sadass  14445
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