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Theorem seqfveq2 12221
Description: Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
seqfveq2.1  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
seqfveq2.2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
seqfveq2.3  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
seqfveq2.4  |-  ( (
ph  /\  k  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  k )  =  ( G `  k ) )
Assertion
Ref Expression
seqfveq2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  (  seq K ( 
.+  ,  G ) `
 N ) )
Distinct variable groups:    k, F    k, G    k, K    k, N    ph, k
Allowed substitution hints:    .+ ( k)    M( k)

Proof of Theorem seqfveq2
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seqfveq2.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
2 eluzfz2 11794 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 17 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2492 . . . . . 6  |-  ( x  =  K  ->  (
x  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5872 . . . . . . 7  |-  ( x  =  K  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  K
) )
6 fveq2 5872 . . . . . . 7  |-  ( x  =  K  ->  (  seq K (  .+  ,  G ) `  x
)  =  (  seq K (  .+  ,  G ) `  K
) )
75, 6eqeq12d 2442 . . . . . 6  |-  ( x  =  K  ->  (
(  seq M (  .+  ,  F ) `  x
)  =  (  seq K (  .+  ,  G ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq K ( 
.+  ,  G ) `
 K ) ) )
84, 7imbi12d 321 . . . . 5  |-  ( x  =  K  ->  (
( x  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  x )  =  (  seq K ( 
.+  ,  G ) `
 x ) )  <-> 
( K  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq K ( 
.+  ,  G ) `
 K ) ) ) )
98imbi2d 317 . . . 4  |-  ( x  =  K  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq K (  .+  ,  G ) `  x
) ) )  <->  ( ph  ->  ( K  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq K ( 
.+  ,  G ) `
 K ) ) ) ) )
10 eleq1 2492 . . . . . 6  |-  ( x  =  n  ->  (
x  e.  ( K ... N )  <->  n  e.  ( K ... N ) ) )
11 fveq2 5872 . . . . . . 7  |-  ( x  =  n  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  n
) )
12 fveq2 5872 . . . . . . 7  |-  ( x  =  n  ->  (  seq K (  .+  ,  G ) `  x
)  =  (  seq K (  .+  ,  G ) `  n
) )
1311, 12eqeq12d 2442 . . . . . 6  |-  ( x  =  n  ->  (
(  seq M (  .+  ,  F ) `  x
)  =  (  seq K (  .+  ,  G ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  n )  =  (  seq K ( 
.+  ,  G ) `
 n ) ) )
1410, 13imbi12d 321 . . . . 5  |-  ( x  =  n  ->  (
( x  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  x )  =  (  seq K ( 
.+  ,  G ) `
 x ) )  <-> 
( n  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  n )  =  (  seq K ( 
.+  ,  G ) `
 n ) ) ) )
1514imbi2d 317 . . . 4  |-  ( x  =  n  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq K (  .+  ,  G ) `  x
) ) )  <->  ( ph  ->  ( n  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  n )  =  (  seq K ( 
.+  ,  G ) `
 n ) ) ) ) )
16 eleq1 2492 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
x  e.  ( K ... N )  <->  ( n  +  1 )  e.  ( K ... N
) ) )
17 fveq2 5872 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  (
n  +  1 ) ) )
18 fveq2 5872 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (  seq K (  .+  ,  G ) `  x
)  =  (  seq K (  .+  ,  G ) `  (
n  +  1 ) ) )
1917, 18eqeq12d 2442 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
(  seq M (  .+  ,  F ) `  x
)  =  (  seq K (  .+  ,  G ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  ( n  +  1 ) )  =  (  seq K
(  .+  ,  G
) `  ( n  +  1 ) ) ) )
2016, 19imbi12d 321 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
( x  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  x )  =  (  seq K ( 
.+  ,  G ) `
 x ) )  <-> 
( ( n  + 
1 )  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  ( n  +  1 ) )  =  (  seq K
(  .+  ,  G
) `  ( n  +  1 ) ) ) ) )
2120imbi2d 317 . . . 4  |-  ( x  =  ( n  + 
1 )  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq K (  .+  ,  G ) `  x
) ) )  <->  ( ph  ->  ( ( n  + 
1 )  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  ( n  +  1 ) )  =  (  seq K
(  .+  ,  G
) `  ( n  +  1 ) ) ) ) ) )
22 eleq1 2492 . . . . . 6  |-  ( x  =  N  ->  (
x  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
23 fveq2 5872 . . . . . . 7  |-  ( x  =  N  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq M (  .+  ,  F ) `  N
) )
24 fveq2 5872 . . . . . . 7  |-  ( x  =  N  ->  (  seq K (  .+  ,  G ) `  x
)  =  (  seq K (  .+  ,  G ) `  N
) )
2523, 24eqeq12d 2442 . . . . . 6  |-  ( x  =  N  ->  (
(  seq M (  .+  ,  F ) `  x
)  =  (  seq K (  .+  ,  G ) `  x
)  <->  (  seq M
(  .+  ,  F
) `  N )  =  (  seq K ( 
.+  ,  G ) `
 N ) ) )
2622, 25imbi12d 321 . . . . 5  |-  ( x  =  N  ->  (
( x  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  x )  =  (  seq K ( 
.+  ,  G ) `
 x ) )  <-> 
( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  N )  =  (  seq K ( 
.+  ,  G ) `
 N ) ) ) )
2726imbi2d 317 . . . 4  |-  ( x  =  N  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  x
)  =  (  seq K (  .+  ,  G ) `  x
) ) )  <->  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  N )  =  (  seq K ( 
.+  ,  G ) `
 N ) ) ) ) )
28 seqfveq2.2 . . . . . . 7  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
29 seqfveq2.1 . . . . . . . . 9  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
30 eluzelz 11157 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
3129, 30syl 17 . . . . . . . 8  |-  ( ph  ->  K  e.  ZZ )
32 seq1 12212 . . . . . . . 8  |-  ( K  e.  ZZ  ->  (  seq K (  .+  ,  G ) `  K
)  =  ( G `
 K ) )
3331, 32syl 17 . . . . . . 7  |-  ( ph  ->  (  seq K ( 
.+  ,  G ) `
 K )  =  ( G `  K
) )
3428, 33eqtr4d 2464 . . . . . 6  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  (  seq K ( 
.+  ,  G ) `
 K ) )
3534a1d 26 . . . . 5  |-  ( ph  ->  ( K  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq K ( 
.+  ,  G ) `
 K ) ) )
3635a1i 11 . . . 4  |-  ( K  e.  ZZ  ->  ( ph  ->  ( K  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq K (  .+  ,  G ) `  K
) ) ) )
37 peano2fzr 11799 . . . . . . . . . 10  |-  ( ( n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) )  ->  n  e.  ( K ... N ) )
3837adantl 467 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( K ... N ) )
3938expr 618 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  +  1 )  e.  ( K ... N )  ->  n  e.  ( K ... N
) ) )
4039imim1d 78 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  n
)  =  (  seq K (  .+  ,  G ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  n
)  =  (  seq K (  .+  ,  G ) `  n
) ) ) )
41 oveq1 6303 . . . . . . . . . 10  |-  ( (  seq M (  .+  ,  F ) `  n
)  =  (  seq K (  .+  ,  G ) `  n
)  ->  ( (  seq M (  .+  ,  F ) `  n
)  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq K
(  .+  ,  G
) `  n )  .+  ( F `  (
n  +  1 ) ) ) )
42 simpl 458 . . . . . . . . . . . . 13  |-  ( ( n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) )  ->  n  e.  ( ZZ>= `  K ) )
43 uztrn 11164 . . . . . . . . . . . . 13  |-  ( ( n  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
4442, 29, 43syl2anr 480 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( ZZ>= `  M )
)
45 seqp1 12214 . . . . . . . . . . . 12  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq M (  .+  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  .+  ,  F ) `  n
)  .+  ( F `  ( n  +  1 ) ) ) )
4644, 45syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq M (  .+  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  .+  ,  F ) `  n
)  .+  ( F `  ( n  +  1 ) ) ) )
47 seqp1 12214 . . . . . . . . . . . . 13  |-  ( n  e.  ( ZZ>= `  K
)  ->  (  seq K (  .+  ,  G ) `  (
n  +  1 ) )  =  ( (  seq K (  .+  ,  G ) `  n
)  .+  ( G `  ( n  +  1 ) ) ) )
4847ad2antrl 732 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq K (  .+  ,  G ) `  (
n  +  1 ) )  =  ( (  seq K (  .+  ,  G ) `  n
)  .+  ( G `  ( n  +  1 ) ) ) )
49 eluzp1p1 11173 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
5049ad2antrl 732 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
51 elfzuz3 11784 . . . . . . . . . . . . . . . 16  |-  ( ( n  +  1 )  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
5251ad2antll 733 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
53 elfzuzb 11781 . . . . . . . . . . . . . . 15  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  <->  ( (
n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) )  /\  N  e.  ( ZZ>= `  ( n  +  1 ) ) ) )
5450, 52, 53sylanbrc 668 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
55 seqfveq2.4 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  k )  =  ( G `  k ) )
5655ralrimiva 2837 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. k  e.  ( ( K  +  1 ) ... N ) ( F `  k
)  =  ( G `
 k ) )
5756adantr 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  A. k  e.  ( ( K  + 
1 ) ... N
) ( F `  k )  =  ( G `  k ) )
58 fveq2 5872 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
59 fveq2 5872 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
6058, 59eqeq12d 2442 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  =  ( G `
 k )  <->  ( F `  ( n  +  1 ) )  =  ( G `  ( n  +  1 ) ) ) )
6160rspcv 3175 . . . . . . . . . . . . . 14  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  ->  ( A. k  e.  (
( K  +  1 ) ... N ) ( F `  k
)  =  ( G `
 k )  -> 
( F `  (
n  +  1 ) )  =  ( G `
 ( n  + 
1 ) ) ) )
6254, 57, 61sylc 62 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( F `  ( n  +  1 ) )  =  ( G `  ( n  +  1 ) ) )
6362oveq2d 6312 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq K (  .+  ,  G ) `  n
)  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq K
(  .+  ,  G
) `  n )  .+  ( G `  (
n  +  1 ) ) ) )
6448, 63eqtr4d 2464 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq K (  .+  ,  G ) `  (
n  +  1 ) )  =  ( (  seq K (  .+  ,  G ) `  n
)  .+  ( F `  ( n  +  1 ) ) ) )
6546, 64eqeq12d 2442 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ) `  (
n  +  1 ) )  =  (  seq K (  .+  ,  G ) `  (
n  +  1 ) )  <->  ( (  seq M (  .+  ,  F ) `  n
)  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq K
(  .+  ,  G
) `  n )  .+  ( F `  (
n  +  1 ) ) ) ) )
6641, 65syl5ibr 224 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ) `  n
)  =  (  seq K (  .+  ,  G ) `  n
)  ->  (  seq M (  .+  ,  F ) `  (
n  +  1 ) )  =  (  seq K (  .+  ,  G ) `  (
n  +  1 ) ) ) )
6766expr 618 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  +  1 )  e.  ( K ... N )  ->  (
(  seq M (  .+  ,  F ) `  n
)  =  (  seq K (  .+  ,  G ) `  n
)  ->  (  seq M (  .+  ,  F ) `  (
n  +  1 ) )  =  (  seq K (  .+  ,  G ) `  (
n  +  1 ) ) ) ) )
6867a2d 29 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  n
)  =  (  seq K (  .+  ,  G ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  (
n  +  1 ) )  =  (  seq K (  .+  ,  G ) `  (
n  +  1 ) ) ) ) )
6940, 68syld 45 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  n
)  =  (  seq K (  .+  ,  G ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  (
n  +  1 ) )  =  (  seq K (  .+  ,  G ) `  (
n  +  1 ) ) ) ) )
7069expcom 436 . . . . 5  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( ( n  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ) `  n
)  =  (  seq K (  .+  ,  G ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  (
n  +  1 ) )  =  (  seq K (  .+  ,  G ) `  (
n  +  1 ) ) ) ) ) )
7170a2d 29 . . . 4  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( ( ph  ->  ( n  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ) `  n
)  =  (  seq K (  .+  ,  G ) `  n
) ) )  -> 
( ph  ->  ( ( n  +  1 )  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  (
n  +  1 ) )  =  (  seq K (  .+  ,  G ) `  (
n  +  1 ) ) ) ) ) )
729, 15, 21, 27, 36, 71uzind4 11206 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  N )  =  (  seq K ( 
.+  ,  G ) `
 N ) ) ) )
731, 72mpcom 37 . 2  |-  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  N )  =  (  seq K ( 
.+  ,  G ) `
 N ) ) )
743, 73mpd 15 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  (  seq K ( 
.+  ,  G ) `
 N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   ` cfv 5592  (class class class)co 6296   1c1 9529    + caddc 9531   ZZcz 10926   ZZ>=cuz 11148   ...cfz 11771    seqcseq 12199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-seq 12200
This theorem is referenced by:  seqfeq2  12222  seqfveq  12223  seqz  12247
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