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Theorem sadasslem 14131
Description: Lemma for sadass 14132. (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 3714 . . . . . . . . . . 11  |-  ( A  i^i  ( 0..^ N ) )  C_  A
2 sadasslem.1 . . . . . . . . . . 11  |-  ( ph  ->  A  C_  NN0 )
31, 2syl5ss 3510 . . . . . . . . . 10  |-  ( ph  ->  ( A  i^i  (
0..^ N ) ) 
C_  NN0 )
4 fzofi 12086 . . . . . . . . . . . 12  |-  ( 0..^ N )  e.  Fin
54a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( 0..^ N )  e.  Fin )
6 inss2 3715 . . . . . . . . . . 11  |-  ( A  i^i  ( 0..^ N ) )  C_  (
0..^ N )
7 ssfi 7759 . . . . . . . . . . 11  |-  ( ( ( 0..^ N )  e.  Fin  /\  ( A  i^i  ( 0..^ N ) )  C_  (
0..^ N ) )  ->  ( A  i^i  ( 0..^ N ) )  e.  Fin )
85, 6, 7sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  ( A  i^i  (
0..^ N ) )  e.  Fin )
9 elfpw 7840 . . . . . . . . . 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 664 . . . . . . . . 9  |-  ( ph  ->  ( A  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) )
11 bitsf1o 14106 . . . . . . . . . . 11  |-  (bits  |`  NN0 ) : NN0
-1-1-onto-> ( ~P NN0  i^i  Fin )
12 f1ocnv 5834 . . . . . . . . . . 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 5822 . . . . . . . . . . 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 6031 . . . . . . . . 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 10875 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  CC )
18 inss1 3714 . . . . . . . . . . 11  |-  ( B  i^i  ( 0..^ N ) )  C_  B
19 sadasslem.2 . . . . . . . . . . 11  |-  ( ph  ->  B  C_  NN0 )
2018, 19syl5ss 3510 . . . . . . . . . 10  |-  ( ph  ->  ( B  i^i  (
0..^ N ) ) 
C_  NN0 )
21 inss2 3715 . . . . . . . . . . 11  |-  ( B  i^i  ( 0..^ N ) )  C_  (
0..^ N )
22 ssfi 7759 . . . . . . . . . . 11  |-  ( ( ( 0..^ N )  e.  Fin  /\  ( B  i^i  ( 0..^ N ) )  C_  (
0..^ N ) )  ->  ( B  i^i  ( 0..^ N ) )  e.  Fin )
235, 21, 22sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  ( B  i^i  (
0..^ N ) )  e.  Fin )
24 elfpw 7840 . . . . . . . . . 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 664 . . . . . . . . 9  |-  ( ph  ->  ( B  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) )
2614ffvelrni 6031 . . . . . . . . 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 10875 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  CC )
29 inss1 3714 . . . . . . . . . . 11  |-  ( C  i^i  ( 0..^ N ) )  C_  C
30 sadasslem.3 . . . . . . . . . . 11  |-  ( ph  ->  C  C_  NN0 )
3129, 30syl5ss 3510 . . . . . . . . . 10  |-  ( ph  ->  ( C  i^i  (
0..^ N ) ) 
C_  NN0 )
32 inss2 3715 . . . . . . . . . . 11  |-  ( C  i^i  ( 0..^ N ) )  C_  (
0..^ N )
33 ssfi 7759 . . . . . . . . . . 11  |-  ( ( ( 0..^ N )  e.  Fin  /\  ( C  i^i  ( 0..^ N ) )  C_  (
0..^ N ) )  ->  ( C  i^i  ( 0..^ N ) )  e.  Fin )
345, 32, 33sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  ( C  i^i  (
0..^ N ) )  e.  Fin )
35 elfpw 7840 . . . . . . . . . 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 664 . . . . . . . . 9  |-  ( ph  ->  ( C  i^i  (
0..^ N ) )  e.  ( ~P NN0  i^i 
Fin ) )
3714ffvelrni 6031 . . . . . . . . 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 10875 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  CC )
4017, 28, 39addassd 9635 . . . . . 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 6311 . . . . 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 3714 . . . . . . . . . 10  |-  ( ( A sadd  B )  i^i  ( 0..^ N ) )  C_  ( A sadd  B )
43 sadcl 14123 . . . . . . . . . . 11  |-  ( ( A  C_  NN0  /\  B  C_ 
NN0 )  ->  ( A sadd  B )  C_  NN0 )
442, 19, 43syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( A sadd  B ) 
C_  NN0 )
4542, 44syl5ss 3510 . . . . . . . . 9  |-  ( ph  ->  ( ( A sadd  B
)  i^i  ( 0..^ N ) )  C_  NN0 )
46 inss2 3715 . . . . . . . . . 10  |-  ( ( A sadd  B )  i^i  ( 0..^ N ) )  C_  ( 0..^ N )
47 ssfi 7759 . . . . . . . . . 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 662 . . . . . . . . 9  |-  ( ph  ->  ( ( A sadd  B
)  i^i  ( 0..^ N ) )  e. 
Fin )
49 elfpw 7840 . . . . . . . . 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 664 . . . . . . . 8  |-  ( ph  ->  ( ( A sadd  B
)  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )
5114ffvelrni 6031 . . . . . . . 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 10874 . . . . . 6  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  B
)  i^i  ( 0..^ N ) ) )  e.  RR )
5416nn0red 10874 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  e.  RR )
5527nn0red 10874 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) )  e.  RR )
5654, 55readdcld 9640 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  ( A  i^i  ( 0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( B  i^i  (
0..^ N ) ) ) )  e.  RR )
5738nn0red 10874 . . . . . 6  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) )  e.  RR )
58 2rp 11250 . . . . . . . 8  |-  2  e.  RR+
5958a1i 11 . . . . . . 7  |-  ( ph  ->  2  e.  RR+ )
60 sadasslem.4 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
6160nn0zd 10988 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
6259, 61rpexpcld 12335 . . . . . 6  |-  ( ph  ->  ( 2 ^ N
)  e.  RR+ )
63 eqid 2457 . . . . . . 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 2457 . . . . . . 7  |-  `' (bits  |`  NN0 )  =  `' (bits  |`  NN0 )
652, 19, 63, 60, 64sadadd3 14122 . . . . . 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 2458 . . . . . 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 12045 . . . . 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 3714 . . . . . . . . . 10  |-  ( ( B sadd  C )  i^i  ( 0..^ N ) )  C_  ( B sadd  C )
69 sadcl 14123 . . . . . . . . . . 11  |-  ( ( B  C_  NN0  /\  C  C_ 
NN0 )  ->  ( B sadd  C )  C_  NN0 )
7019, 30, 69syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( B sadd  C ) 
C_  NN0 )
7168, 70syl5ss 3510 . . . . . . . . 9  |-  ( ph  ->  ( ( B sadd  C
)  i^i  ( 0..^ N ) )  C_  NN0 )
72 inss2 3715 . . . . . . . . . 10  |-  ( ( B sadd  C )  i^i  ( 0..^ N ) )  C_  ( 0..^ N )
73 ssfi 7759 . . . . . . . . . 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 662 . . . . . . . . 9  |-  ( ph  ->  ( ( B sadd  C
)  i^i  ( 0..^ N ) )  e. 
Fin )
75 elfpw 7840 . . . . . . . . 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 664 . . . . . . . 8  |-  ( ph  ->  ( ( B sadd  C
)  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )
7714ffvelrni 6031 . . . . . . . 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 10874 . . . . . 6  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( B sadd  C
)  i^i  ( 0..^ N ) ) )  e.  RR )
8055, 57readdcld 9640 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  ( B  i^i  ( 0..^ N ) ) )  +  ( `' (bits  |`  NN0 ) `  ( C  i^i  (
0..^ N ) ) ) )  e.  RR )
81 eqidd 2458 . . . . . 6  |-  ( ph  ->  ( ( `' (bits  |`  NN0 ) `  ( A  i^i  ( 0..^ N ) ) )  mod  ( 2 ^ N
) )  =  ( ( `' (bits  |`  NN0 ) `  ( A  i^i  (
0..^ N ) ) )  mod  ( 2 ^ N ) ) )
82 eqid 2457 . . . . . . 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 14122 . . . . . 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 12045 . . . . 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 2508 . . . 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 2457 . . . . 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 14122 . . . 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 2457 . . . . 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 14122 . . . 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 2508 . . 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 3714 . . . . . . . 8  |-  ( ( ( A sadd  B ) sadd 
C )  i^i  (
0..^ N ) ) 
C_  ( ( A sadd 
B ) sadd  C )
92 sadcl 14123 . . . . . . . . 9  |-  ( ( ( A sadd  B ) 
C_  NN0  /\  C  C_  NN0 )  ->  ( ( A sadd  B ) sadd  C ) 
C_  NN0 )
9344, 30, 92syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( ( A sadd  B
) sadd  C )  C_  NN0 )
9491, 93syl5ss 3510 . . . . . . 7  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  C_  NN0 )
95 inss2 3715 . . . . . . . 8  |-  ( ( ( A sadd  B ) sadd 
C )  i^i  (
0..^ N ) ) 
C_  ( 0..^ N )
96 ssfi 7759 . . . . . . . 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 662 . . . . . . 7  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  e.  Fin )
98 elfpw 7840 . . . . . . 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 664 . . . . . 6  |-  ( ph  ->  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) )  e.  ( ~P NN0  i^i  Fin ) )
10014ffvelrni 6031 . . . . . 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 10874 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  RR )
103101nn0ge0d 10876 . . . 4  |-  ( ph  ->  0  <_  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) )
104 fvres 5886 . . . . . . . . 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 6184 . . . . . . . . 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 663 . . . . . . . 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 2500 . . . . . . 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 3549 . . . . . 6  |-  ( ph  ->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) ) ) 
C_  ( 0..^ N ) )
110101nn0zd 10988 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  ZZ )
111 bitsfzo 14096 . . . . . . 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 661 . . . . . 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 232 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( ( A sadd 
B ) sadd  C )  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) ) )
114 elfzolt2 11834 . . . . 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 12022 . . . 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 1229 . . 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 3714 . . . . . . . 8  |-  ( ( A sadd  ( B sadd  C
) )  i^i  (
0..^ N ) ) 
C_  ( A sadd  ( B sadd  C ) )
119 sadcl 14123 . . . . . . . . 9  |-  ( ( A  C_  NN0  /\  ( B sadd  C )  C_  NN0 )  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
1202, 70, 119syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( A sadd  ( B sadd 
C ) )  C_  NN0 )
121118, 120syl5ss 3510 . . . . . . 7  |-  ( ph  ->  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  C_  NN0 )
122 inss2 3715 . . . . . . . 8  |-  ( ( A sadd  ( B sadd  C
) )  i^i  (
0..^ N ) ) 
C_  ( 0..^ N )
123 ssfi 7759 . . . . . . . 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 662 . . . . . . 7  |-  ( ph  ->  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  e.  Fin )
125 elfpw 7840 . . . . . . 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 664 . . . . . 6  |-  ( ph  ->  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) )  e.  ( ~P
NN0  i^i  Fin )
)
12714ffvelrni 6031 . . . . . 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 10874 . . . 4  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  RR )
130 2nn 10714 . . . . . . 7  |-  2  e.  NN
131130a1i 11 . . . . . 6  |-  ( ph  ->  2  e.  NN )
132131, 60nnexpcld 12333 . . . . 5  |-  ( ph  ->  ( 2 ^ N
)  e.  NN )
133132nnrpd 11280 . . . 4  |-  ( ph  ->  ( 2 ^ N
)  e.  RR+ )
134128nn0ge0d 10876 . . . 4  |-  ( ph  ->  0  <_  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )
135 fvres 5886 . . . . . . . . 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 6184 . . . . . . . . 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 663 . . . . . . . 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 2500 . . . . . . 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 3549 . . . . . 6  |-  ( ph  ->  (bits `  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) ) )  C_  ( 0..^ N ) )
141128nn0zd 10988 . . . . . . 7  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  ZZ )
142 bitsfzo 14096 . . . . . . 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 661 . . . . . 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 232 . . . . 5  |-  ( ph  ->  ( `' (bits  |`  NN0 ) `  ( ( A sadd  ( B sadd  C ) )  i^i  ( 0..^ N ) ) )  e.  ( 0..^ ( 2 ^ N ) ) )
145 elfzolt2 11834 . . . . 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 12022 . . . 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 1229 . . 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 2506 . 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 5821 . . . . 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 6171 . . . 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 670 . . 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 661 . 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 210 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 184    /\ wa 369    = wceq 1395  caddwcad 1446    e. wcel 1819    i^i cin 3470    C_ wss 3471   (/)c0 3793   ifcif 3944   ~Pcpw 4015   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007    |` cres 5010   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1oc1o 7141   2oc2o 7142   Fincfn 7535   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    < clt 9645    <_ cle 9646    - cmin 9824   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   RR+crp 11245  ..^cfzo 11820    mod cmo 11998    seqcseq 12109   ^cexp 12168  bitscbits 14080   sadd csad 14081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1364  df-tru 1398  df-fal 1401  df-had 1447  df-cad 1448  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-sum 13520  df-dvds 13998  df-bits 14083  df-sad 14112
This theorem is referenced by:  sadass  14132
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