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Theorem sadasslem 12909
Description: Lemma for sadass 12910. (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 3504 . . . . . . . . . . 11  |-  ( A  i^i  ( 0..^ N ) )  C_  A
2 sadasslem.1 . . . . . . . . . . 11  |-  ( ph  ->  A  C_  NN0 )
31, 2syl5ss 3302 . . . . . . . . . 10  |-  ( ph  ->  ( A  i^i  (
0..^ N ) ) 
C_  NN0 )
4 fzofi 11240 . . . . . . . . . . . 12  |-  ( 0..^ N )  e.  Fin
54a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( 0..^ N )  e.  Fin )
6 inss2 3505 . . . . . . . . . . 11  |-  ( A  i^i  ( 0..^ N ) )  C_  (
0..^ N )
7 ssfi 7265 . . . . . . . . . . 11  |-  ( ( ( 0..^ N )  e.  Fin  /\  ( A  i^i  ( 0..^ N ) )  C_  (
0..^ N ) )  ->  ( A  i^i  ( 0..^ N ) )  e.  Fin )
85, 6, 7sylancl 644 . . . . . . . . . 10  |-  ( ph  ->  ( A  i^i  (
0..^ N ) )  e.  Fin )
9 elfpw 7343 . . . . . . . . . 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 646 . . . . . . . . 9  |-  ( ph  ->  ( A  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) )
11 bitsf1o 12884 . . . . . . . . . . 11  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
12 f1ocnv 5627 . . . . . . . . . . 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 5614 . . . . . . . . . . 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 5808 . . . . . . . . 9  |-  ( ( A  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  NN0 )
1610, 15syl 16 . . . . . . . 8  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  NN0 )
1716nn0cnd 10208 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  CC )
18 inss1 3504 . . . . . . . . . . 11  |-  ( B  i^i  ( 0..^ N ) )  C_  B
19 sadasslem.2 . . . . . . . . . . 11  |-  ( ph  ->  B  C_  NN0 )
2018, 19syl5ss 3302 . . . . . . . . . 10  |-  ( ph  ->  ( B  i^i  (
0..^ N ) ) 
C_  NN0 )
21 inss2 3505 . . . . . . . . . . 11  |-  ( B  i^i  ( 0..^ N ) )  C_  (
0..^ N )
22 ssfi 7265 . . . . . . . . . . 11  |-  ( ( ( 0..^ N )  e.  Fin  /\  ( B  i^i  ( 0..^ N ) )  C_  (
0..^ N ) )  ->  ( B  i^i  ( 0..^ N ) )  e.  Fin )
235, 21, 22sylancl 644 . . . . . . . . . 10  |-  ( ph  ->  ( B  i^i  (
0..^ N ) )  e.  Fin )
24 elfpw 7343 . . . . . . . . . 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 646 . . . . . . . . 9  |-  ( ph  ->  ( B  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) )
2614ffvelrni 5808 . . . . . . . . 9  |-  ( ( B  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  NN0 )
2725, 26syl 16 . . . . . . . 8  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  NN0 )
2827nn0cnd 10208 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  CC )
29 inss1 3504 . . . . . . . . . . 11  |-  ( C  i^i  ( 0..^ N ) )  C_  C
30 sadasslem.3 . . . . . . . . . . 11  |-  ( ph  ->  C  C_  NN0 )
3129, 30syl5ss 3302 . . . . . . . . . 10  |-  ( ph  ->  ( C  i^i  (
0..^ N ) ) 
C_  NN0 )
32 inss2 3505 . . . . . . . . . . 11  |-  ( C  i^i  ( 0..^ N ) )  C_  (
0..^ N )
33 ssfi 7265 . . . . . . . . . . 11  |-  ( ( ( 0..^ N )  e.  Fin  /\  ( C  i^i  ( 0..^ N ) )  C_  (
0..^ N ) )  ->  ( C  i^i  ( 0..^ N ) )  e.  Fin )
345, 32, 33sylancl 644 . . . . . . . . . 10  |-  ( ph  ->  ( C  i^i  (
0..^ N ) )  e.  Fin )
35 elfpw 7343 . . . . . . . . . 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 646 . . . . . . . . 9  |-  ( ph  ->  ( C  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) )
3714ffvelrni 5808 . . . . . . . . 9  |-  ( ( C  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin )  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  NN0 )
3836, 37syl 16 . . . . . . . 8  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  NN0 )
3938nn0cnd 10208 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  CC )
4017, 28, 39addassd 9043 . . . . . 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 6035 . . . . 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 3504 . . . . . . . . . 10  |-  ( ( A sadd  B )  i^i  ( 0..^ N ) )  C_  ( A sadd  B )
43 sadcl 12901 . . . . . . . . . . 11  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0 )  ->  ( A sadd  B )  C_  NN0 )
442, 19, 43syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( A sadd  B ) 
C_  NN0 )
4542, 44syl5ss 3302 . . . . . . . . 9  |-  ( ph  ->  ( ( A sadd  B
)  i^i  ( 0..^ N ) )  C_  NN0 )
46 inss2 3505 . . . . . . . . . 10  |-  ( ( A sadd  B )  i^i  ( 0..^ N ) )  C_  ( 0..^ N )
47 ssfi 7265 . . . . . . . . . 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 644 . . . . . . . . 9  |-  ( ph  ->  ( ( A sadd  B
)  i^i  ( 0..^ N ) )  e. 
Fin )
49 elfpw 7343 . . . . . . . . 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 646 . . . . . . . 8  |-  ( ph  ->  ( ( A sadd  B
)  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )
5114ffvelrni 5808 . . . . . . . 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 16 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  B
)  i^i  ( 0..^ N ) ) )  e.  NN0 )
5352nn0red 10207 . . . . . 6  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  B
)  i^i  ( 0..^ N ) ) )  e.  RR )
5416nn0red 10207 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  RR )
5527nn0red 10207 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  RR )
5654, 55readdcld 9048 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  ( A  i^i  ( 0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) ) )  e.  RR )
5738nn0red 10207 . . . . . 6  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  RR )
58 2rp 10549 . . . . . . . 8  |-  2  e.  RR+
5958a1i 11 . . . . . . 7  |-  ( ph  ->  2  e.  RR+ )
60 sadasslem.4 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
6160nn0zd 10305 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
6259, 61rpexpcld 11473 . . . . . 6  |-  ( ph  ->  ( 2 ^ N
)  e.  RR+ )
63 eqid 2387 . . . . . . 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 2387 . . . . . . 7  |-  `' (bits  |`  NN0 )  =  `' (bits  |`  NN0 )
652, 19, 63, 60, 64sadadd3 12900 . . . . . 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 2388 . . . . . 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 11209 . . . . 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 3504 . . . . . . . . . 10  |-  ( ( B sadd  C )  i^i  ( 0..^ N ) )  C_  ( B sadd  C )
69 sadcl 12901 . . . . . . . . . . 11  |-  ( ( B  C_  NN0  /\  C  C_ 
NN0 )  ->  ( B sadd  C )  C_  NN0 )
7019, 30, 69syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( B sadd  C ) 
C_  NN0 )
7168, 70syl5ss 3302 . . . . . . . . 9  |-  ( ph  ->  ( ( B sadd  C
)  i^i  ( 0..^ N ) )  C_  NN0 )
72 inss2 3505 . . . . . . . . . 10  |-  ( ( B sadd  C )  i^i  ( 0..^ N ) )  C_  ( 0..^ N )
73 ssfi 7265 . . . . . . . . . 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 644 . . . . . . . . 9  |-  ( ph  ->  ( ( B sadd  C
)  i^i  ( 0..^ N ) )  e. 
Fin )
75 elfpw 7343 . . . . . . . . 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 646 . . . . . . . 8  |-  ( ph  ->  ( ( B sadd  C
)  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )
7714ffvelrni 5808 . . . . . . . 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 16 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( B sadd  C
)  i^i  ( 0..^ N ) ) )  e.  NN0 )
7978nn0red 10207 . . . . . 6  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( B sadd  C
)  i^i  ( 0..^ N ) ) )  e.  RR )
8055, 57readdcld 9048 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  ( B  i^i  ( 0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) ) )  e.  RR )
81 eqidd 2388 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  ( A  i^i  ( 0..^ N ) ) )  mod  ( 2 ^ N
) )  =  ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) ) )
82 eqid 2387 . . . . . . 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 12900 . . . . . 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 11209 . . . . 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 2429 . . . 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 2387 . . . . 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 12900 . . . 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 2387 . . . . 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 12900 . . . 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 2429 . . 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 3504 . . . . . . . 8  |-  ( ( ( A sadd  B ) sadd 
C )  i^i  (
0..^ N ) ) 
C_  ( ( A sadd 
B ) sadd  C )
92 sadcl 12901 . . . . . . . . 9  |-  ( ( ( A sadd  B ) 
C_  NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B ) sadd  C ) 
C_  NN0 )
9344, 30, 92syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( ( A sadd  B
) sadd  C )  C_  NN0 )
9491, 93syl5ss 3302 . . . . . . 7  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  C_  NN0 )
95 inss2 3505 . . . . . . . 8  |-  ( ( ( A sadd  B ) sadd 
C )  i^i  (
0..^ N ) ) 
C_  ( 0..^ N )
96 ssfi 7265 . . . . . . . 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 644 . . . . . . 7  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  e.  Fin )
98 elfpw 7343 . . . . . . 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 646 . . . . . 6  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )
10014ffvelrni 5808 . . . . . 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 16 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e. 
NN0 )
102101nn0red 10207 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  RR )
103101nn0ge0d 10209 . . . 4  |-  ( ph  ->  0  <_  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) )
104 fvres 5685 . . . . . . . . 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 16 . . . . . . . 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 5954 . . . . . . . . 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 645 . . . . . . . 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 2421 . . . . . . 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 3341 . . . . . 6  |-  ( ph  ->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) ) 
C_  ( 0..^ N ) )
110101nn0zd 10305 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  ZZ )
111 bitsfzo 12874 . . . . . . 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 643 . . . . . 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 224 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) ) )
114 elfzolt2 11078 . . . . 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 16 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  < 
( 2 ^ N
) )
116 modid 11197 . . . 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 1185 . . 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 3504 . . . . . . . 8  |-  ( ( A sadd  ( B sadd  C
) )  i^i  (
0..^ N ) ) 
C_  ( A sadd  ( B sadd  C ) )
119 sadcl 12901 . . . . . . . . 9  |-  ( ( A  C_  NN0  /\  ( B sadd  C )  C_  NN0 )  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
1202, 70, 119syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
121118, 120syl5ss 3302 . . . . . . 7  |-  ( ph  ->  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  C_  NN0 )
122 inss2 3505 . . . . . . . 8  |-  ( ( A sadd  ( B sadd  C
) )  i^i  (
0..^ N ) ) 
C_  ( 0..^ N )
123 ssfi 7265 . . . . . . . 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 644 . . . . . . 7  |-  ( ph  ->  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  e.  Fin )
125 elfpw 7343 . . . . . . 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 646 . . . . . 6  |-  ( ph  ->  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  e.  ( ~P
NN0  i^i  Fin )
)
12714ffvelrni 5808 . . . . . 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 16 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  NN0 )
129128nn0red 10207 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  RR )
130 2nn 10065 . . . . . . 7  |-  2  e.  NN
131130a1i 11 . . . . . 6  |-  ( ph  ->  2  e.  NN )
132131, 60nnexpcld 11471 . . . . 5  |-  ( ph  ->  ( 2 ^ N
)  e.  NN )
133132nnrpd 10579 . . . 4  |-  ( ph  ->  ( 2 ^ N
)  e.  RR+ )
134128nn0ge0d 10209 . . . 4  |-  ( ph  ->  0  <_  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )
135 fvres 5685 . . . . . . . . 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 16 . . . . . . . 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 5954 . . . . . . . . 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 645 . . . . . . . 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 2421 . . . . . . 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 3341 . . . . . 6  |-  ( ph  ->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )  C_  ( 0..^ N ) )
141128nn0zd 10305 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  ZZ )
142 bitsfzo 12874 . . . . . . 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 643 . . . . . 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 224 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) ) )
145 elfzolt2 11078 . . . . 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 16 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  <  (
2 ^ N ) )
147 modid 11197 . . . 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 1185 . . 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 2427 . 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 5613 . . . . 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 5947 . . . 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 652 . . 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 643 . 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 202 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 177    /\ wa 359  caddwcad 1385    = wceq 1649    e. wcel 1717    i^i cin 3262    C_ wss 3263   (/)c0 3571   ifcif 3682   ~Pcpw 3742   class class class wbr 4153    e. cmpt 4207   `'ccnv 4817    |` cres 4820   -->wf 5390   -1-1->wf1 5391   -1-1-onto->wf1o 5393   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   1oc1o 6653   2oc2o 6654   Fincfn 7045   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    < clt 9053    <_ cle 9054    - cmin 9223   NNcn 9932   2c2 9981   NN0cn0 10153   ZZcz 10214   RR+crp 10544  ..^cfzo 11065    mod cmo 11177    seq cseq 11250   ^cexp 11309  bitscbits 12858   sadd csad 12859
This theorem is referenced by:  sadass  12910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1311  df-tru 1325  df-had 1386  df-cad 1387  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-disj 4124  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407  df-dvds 12780  df-bits 12861  df-sad 12890
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