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

Theorem sadasslem 12661
Description: Lemma for sadass 12662. (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 3389 . . . . . . . . . . 11  |-  ( A  i^i  ( 0..^ N ) )  C_  A
2 sadasslem.1 . . . . . . . . . . 11  |-  ( ph  ->  A  C_  NN0 )
31, 2syl5ss 3190 . . . . . . . . . 10  |-  ( ph  ->  ( A  i^i  (
0..^ N ) ) 
C_  NN0 )
4 fzofi 11036 . . . . . . . . . . . 12  |-  ( 0..^ N )  e.  Fin
54a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  ( 0..^ N )  e.  Fin )
6 inss2 3390 . . . . . . . . . . 11  |-  ( A  i^i  ( 0..^ N ) )  C_  (
0..^ N )
7 ssfi 7083 . . . . . . . . . . 11  |-  ( ( ( 0..^ N )  e.  Fin  /\  ( A  i^i  ( 0..^ N ) )  C_  (
0..^ N ) )  ->  ( A  i^i  ( 0..^ N ) )  e.  Fin )
85, 6, 7sylancl 643 . . . . . . . . . 10  |-  ( ph  ->  ( A  i^i  (
0..^ N ) )  e.  Fin )
9 elfpw 7157 . . . . . . . . . 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 645 . . . . . . . . 9  |-  ( ph  ->  ( A  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) )
11 bitsf1o 12636 . . . . . . . . . . 11  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
12 f1ocnv 5485 . . . . . . . . . . 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 5472 . . . . . . . . . . 11  |-  ( `' (bits  |`  NN0 ) : ( ~P NN0  i^i  Fin ) -1-1-onto-> NN0  ->  `' (bits  |` 
NN0 ) : ( ~P NN0  i^i  Fin )
--> NN0 )
1411, 12, 13mp2b 9 . . . . . . . . . 10  |-  `' (bits  |`  NN0 ) : ( ~P NN0  i^i  Fin )
--> NN0
1514ffvelrni 5664 . . . . . . . . 9  |-  ( ( A  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  NN0 )
1610, 15syl 15 . . . . . . . 8  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  NN0 )
1716nn0cnd 10020 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  CC )
18 inss1 3389 . . . . . . . . . . 11  |-  ( B  i^i  ( 0..^ N ) )  C_  B
19 sadasslem.2 . . . . . . . . . . 11  |-  ( ph  ->  B  C_  NN0 )
2018, 19syl5ss 3190 . . . . . . . . . 10  |-  ( ph  ->  ( B  i^i  (
0..^ N ) ) 
C_  NN0 )
21 inss2 3390 . . . . . . . . . . 11  |-  ( B  i^i  ( 0..^ N ) )  C_  (
0..^ N )
22 ssfi 7083 . . . . . . . . . . 11  |-  ( ( ( 0..^ N )  e.  Fin  /\  ( B  i^i  ( 0..^ N ) )  C_  (
0..^ N ) )  ->  ( B  i^i  ( 0..^ N ) )  e.  Fin )
235, 21, 22sylancl 643 . . . . . . . . . 10  |-  ( ph  ->  ( B  i^i  (
0..^ N ) )  e.  Fin )
24 elfpw 7157 . . . . . . . . . 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 645 . . . . . . . . 9  |-  ( ph  ->  ( B  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) )
2614ffvelrni 5664 . . . . . . . . 9  |-  ( ( B  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  NN0 )
2725, 26syl 15 . . . . . . . 8  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  NN0 )
2827nn0cnd 10020 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  CC )
29 inss1 3389 . . . . . . . . . . 11  |-  ( C  i^i  ( 0..^ N ) )  C_  C
30 sadasslem.3 . . . . . . . . . . 11  |-  ( ph  ->  C  C_  NN0 )
3129, 30syl5ss 3190 . . . . . . . . . 10  |-  ( ph  ->  ( C  i^i  (
0..^ N ) ) 
C_  NN0 )
32 inss2 3390 . . . . . . . . . . 11  |-  ( C  i^i  ( 0..^ N ) )  C_  (
0..^ N )
33 ssfi 7083 . . . . . . . . . . 11  |-  ( ( ( 0..^ N )  e.  Fin  /\  ( C  i^i  ( 0..^ N ) )  C_  (
0..^ N ) )  ->  ( C  i^i  ( 0..^ N ) )  e.  Fin )
345, 32, 33sylancl 643 . . . . . . . . . 10  |-  ( ph  ->  ( C  i^i  (
0..^ N ) )  e.  Fin )
35 elfpw 7157 . . . . . . . . . 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 645 . . . . . . . . 9  |-  ( ph  ->  ( C  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) )
3714ffvelrni 5664 . . . . . . . . 9  |-  ( ( C  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  NN0 )
3836, 37syl 15 . . . . . . . 8  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  NN0 )
3938nn0cnd 10020 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  CC )
4017, 28, 39addassd 8857 . . . . . 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 5873 . . . . 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 3389 . . . . . . . . . 10  |-  ( ( A sadd  B )  i^i  ( 0..^ N ) )  C_  ( A sadd  B )
43 sadcl 12653 . . . . . . . . . . 11  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0 )  ->  ( A sadd  B )  C_  NN0 )
442, 19, 43syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( A sadd  B ) 
C_  NN0 )
4542, 44syl5ss 3190 . . . . . . . . 9  |-  ( ph  ->  ( ( A sadd  B
)  i^i  ( 0..^ N ) )  C_  NN0 )
46 inss2 3390 . . . . . . . . . 10  |-  ( ( A sadd  B )  i^i  ( 0..^ N ) )  C_  ( 0..^ N )
47 ssfi 7083 . . . . . . . . . 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 643 . . . . . . . . 9  |-  ( ph  ->  ( ( A sadd  B
)  i^i  ( 0..^ N ) )  e. 
Fin )
49 elfpw 7157 . . . . . . . . 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 645 . . . . . . . 8  |-  ( ph  ->  ( ( A sadd  B
)  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )
5114ffvelrni 5664 . . . . . . . 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 15 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  B
)  i^i  ( 0..^ N ) ) )  e.  NN0 )
5352nn0red 10019 . . . . . 6  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  B
)  i^i  ( 0..^ N ) ) )  e.  RR )
5416nn0red 10019 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  RR )
5527nn0red 10019 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  RR )
5654, 55readdcld 8862 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  ( A  i^i  ( 0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) ) )  e.  RR )
5738nn0red 10019 . . . . . 6  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  RR )
58 2rp 10359 . . . . . . . 8  |-  2  e.  RR+
5958a1i 10 . . . . . . 7  |-  ( ph  ->  2  e.  RR+ )
60 sadasslem.4 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
6160nn0zd 10115 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
6259, 61rpexpcld 11268 . . . . . 6  |-  ( ph  ->  ( 2 ^ N
)  e.  RR+ )
63 eqid 2283 . . . . . . 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 2283 . . . . . . 7  |-  `' (bits  |`  NN0 )  =  `' (bits  |`  NN0 )
652, 19, 63, 60, 64sadadd3 12652 . . . . . 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 2284 . . . . . 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 11005 . . . . 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 3389 . . . . . . . . . 10  |-  ( ( B sadd  C )  i^i  ( 0..^ N ) )  C_  ( B sadd  C )
69 sadcl 12653 . . . . . . . . . . 11  |-  ( ( B  C_  NN0  /\  C  C_ 
NN0 )  ->  ( B sadd  C )  C_  NN0 )
7019, 30, 69syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( B sadd  C ) 
C_  NN0 )
7168, 70syl5ss 3190 . . . . . . . . 9  |-  ( ph  ->  ( ( B sadd  C
)  i^i  ( 0..^ N ) )  C_  NN0 )
72 inss2 3390 . . . . . . . . . 10  |-  ( ( B sadd  C )  i^i  ( 0..^ N ) )  C_  ( 0..^ N )
73 ssfi 7083 . . . . . . . . . 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 643 . . . . . . . . 9  |-  ( ph  ->  ( ( B sadd  C
)  i^i  ( 0..^ N ) )  e. 
Fin )
75 elfpw 7157 . . . . . . . . 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 645 . . . . . . . 8  |-  ( ph  ->  ( ( B sadd  C
)  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )
7714ffvelrni 5664 . . . . . . . 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 15 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( B sadd  C
)  i^i  ( 0..^ N ) ) )  e.  NN0 )
7978nn0red 10019 . . . . . 6  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( B sadd  C
)  i^i  ( 0..^ N ) ) )  e.  RR )
8055, 57readdcld 8862 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  ( B  i^i  ( 0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) ) )  e.  RR )
81 eqidd 2284 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  ( A  i^i  ( 0..^ N ) ) )  mod  ( 2 ^ N
) )  =  ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) ) )
82 eqid 2283 . . . . . . 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 12652 . . . . . 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 11005 . . . . 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 2325 . . . 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 2283 . . . . 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 12652 . . . 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 2283 . . . . 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 12652 . . . 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 2325 . . 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 3389 . . . . . . . 8  |-  ( ( ( A sadd  B ) sadd 
C )  i^i  (
0..^ N ) ) 
C_  ( ( A sadd 
B ) sadd  C )
92 sadcl 12653 . . . . . . . . 9  |-  ( ( ( A sadd  B ) 
C_  NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B ) sadd  C ) 
C_  NN0 )
9344, 30, 92syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( ( A sadd  B
) sadd  C )  C_  NN0 )
9491, 93syl5ss 3190 . . . . . . 7  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  C_  NN0 )
95 inss2 3390 . . . . . . . 8  |-  ( ( ( A sadd  B ) sadd 
C )  i^i  (
0..^ N ) ) 
C_  ( 0..^ N )
96 ssfi 7083 . . . . . . . 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 643 . . . . . . 7  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  e.  Fin )
98 elfpw 7157 . . . . . . 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 645 . . . . . 6  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )
10014ffvelrni 5664 . . . . . 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 15 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e. 
NN0 )
102101nn0red 10019 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  RR )
103101nn0ge0d 10021 . . . 4  |-  ( ph  ->  0  <_  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) )
104 fvres 5542 . . . . . . . . 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 15 . . . . . . . 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 5793 . . . . . . . . 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 644 . . . . . . . 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 2317 . . . . . . 7  |-  ( ph  ->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) )  =  ( ( ( A sadd  B ) sadd  C
)  i^i  ( 0..^ N ) ) )
10995a1i 10 . . . . . . 7  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  C_  (
0..^ N ) )
110108, 109eqsstrd 3212 . . . . . 6  |-  ( ph  ->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) ) 
C_  ( 0..^ N ) )
111101nn0zd 10115 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  ZZ )
112 bitsfzo 12626 . . . . . . 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 ) ) )
113111, 60, 112syl2anc 642 . . . . . 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 ) ) )
114110, 113mpbird 223 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) ) )
115 elfzolt2 10883 . . . . 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 ) )
116114, 115syl 15 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  < 
( 2 ^ N
) )
117 modid 10993 . . . 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 ) ) ) )
118102, 62, 103, 116, 117syl22anc 1183 . . 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 ) ) ) )
119 inss1 3389 . . . . . . . 8  |-  ( ( A sadd  ( B sadd  C
) )  i^i  (
0..^ N ) ) 
C_  ( A sadd  ( B sadd  C ) )
120 sadcl 12653 . . . . . . . . 9  |-  ( ( A  C_  NN0  /\  ( B sadd  C )  C_  NN0 )  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
1212, 70, 120syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
122119, 121syl5ss 3190 . . . . . . 7  |-  ( ph  ->  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  C_  NN0 )
123 inss2 3390 . . . . . . . 8  |-  ( ( A sadd  ( B sadd  C
) )  i^i  (
0..^ N ) ) 
C_  ( 0..^ N )
124 ssfi 7083 . . . . . . . 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 )
1255, 123, 124sylancl 643 . . . . . . 7  |-  ( ph  ->  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  e.  Fin )
126 elfpw 7157 . . . . . . 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 ) )
127122, 125, 126sylanbrc 645 . . . . . 6  |-  ( ph  ->  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  e.  ( ~P
NN0  i^i  Fin )
)
12814ffvelrni 5664 . . . . . 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 )
129127, 128syl 15 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  NN0 )
130129nn0red 10019 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  RR )
131 2nn 9877 . . . . . . 7  |-  2  e.  NN
132131a1i 10 . . . . . 6  |-  ( ph  ->  2  e.  NN )
133132, 60nnexpcld 11266 . . . . 5  |-  ( ph  ->  ( 2 ^ N
)  e.  NN )
134133nnrpd 10389 . . . 4  |-  ( ph  ->  ( 2 ^ N
)  e.  RR+ )
135129nn0ge0d 10021 . . . 4  |-  ( ph  ->  0  <_  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )
136 fvres 5542 . . . . . . . . 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 ) ) ) ) )
137129, 136syl 15 . . . . . . . 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 ) ) ) ) )
138 f1ocnvfv2 5793 . . . . . . . . 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 ) ) )
13911, 127, 138sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( (bits  |`  NN0 ) `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )  =  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )
140137, 139eqtr3d 2317 . . . . . . 7  |-  ( ph  ->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )  =  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )
141123a1i 10 . . . . . . 7  |-  ( ph  ->  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  C_  ( 0..^ N ) )
142140, 141eqsstrd 3212 . . . . . 6  |-  ( ph  ->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )  C_  ( 0..^ N ) )
143129nn0zd 10115 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  ZZ )
144 bitsfzo 12626 . . . . . . 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 ) ) )
145143, 60, 144syl2anc 642 . . . . . 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 ) ) )
146142, 145mpbird 223 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) ) )
147 elfzolt2 10883 . . . . 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 ) )
148146, 147syl 15 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  <  (
2 ^ N ) )
149 modid 10993 . . . 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 ) ) ) )
150130, 134, 135, 148, 149syl22anc 1183 . . 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 ) ) ) )
15190, 118, 1503eqtr3d 2323 . 2  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  =  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )
152 f1of1 5471 . . . . 5  |-  ( `' (bits  |`  NN0 ) : ( ~P NN0  i^i  Fin ) -1-1-onto-> NN0  ->  `' (bits  |` 
NN0 ) : ( ~P NN0  i^i  Fin ) -1-1-> NN0 )
15311, 12, 152mp2b 9 . . . 4  |-  `' (bits  |`  NN0 ) : ( ~P NN0  i^i  Fin ) -1-1-> NN0
154 f1fveq 5786 . . . 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 ) ) ) )
155153, 154mpan 651 . . 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 ) ) ) )
15699, 127, 155syl2anc 642 . 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 ) ) ) )
157151, 156mpbid 201 1  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  =  ( ( A sadd  ( B sadd 
C ) )  i^i  ( 0..^ N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358  caddwcad 1369    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   (/)c0 3455   ifcif 3565   ~Pcpw 3625   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688    |` cres 4691   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   Fincfn 6863   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   RR+crp 10354  ..^cfzo 10870    mod cmo 10973    seq cseq 11046   ^cexp 11104  bitscbits 12610   sadd csad 12611
This theorem is referenced by:  sadass  12662
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  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-xor 1296  df-tru 1310  df-had 1370  df-cad 1371  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-int 3863  df-iun 3907  df-disj 3994  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-se 4353  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  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-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-bits 12613  df-sad 12642
  Copyright terms: Public domain W3C validator