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Theorem iseqoveq 9592
Description: Equality theorem for the sequence builder operation. This is similar to iseqeq2 9577 but  .+ and  Q only have to agree on elements of  S. (Contributed by Jim Kingdon, 21-Apr-2022.)
Hypotheses
Ref Expression
iseqoveq.m  |-  ( ph  ->  M  e.  ZZ )
iseqoveq.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
iseqoveq.eq  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( x Q y ) )
iseqoveq.plcl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
iseqoveq.qcl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x Q y )  e.  S )
Assertion
Ref Expression
iseqoveq  |-  ( ph  ->  seq M (  .+  ,  F ,  S )  =  seq M ( Q ,  F ,  S ) )
Distinct variable groups:    x,  .+ , y    x, F, y    x, M, y    x, Q, y   
x, S, y    ph, x, y

Proof of Theorem iseqoveq
Dummy variables  k  w  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2083 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 iseqoveq.m . . . 4  |-  ( ph  ->  M  e.  ZZ )
3 iseqoveq.f . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
4 iseqoveq.plcl . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
51, 2, 3, 4iseqfcl 9587 . . 3  |-  ( ph  ->  seq M (  .+  ,  F ,  S ) : ( ZZ>= `  M
) --> S )
6 ffn 5097 . . 3  |-  (  seq M (  .+  ,  F ,  S ) : ( ZZ>= `  M
) --> S  ->  seq M (  .+  ,  F ,  S )  Fn  ( ZZ>= `  M )
)
75, 6syl 14 . 2  |-  ( ph  ->  seq M (  .+  ,  F ,  S )  Fn  ( ZZ>= `  M
) )
8 iseqoveq.qcl . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x Q y )  e.  S )
91, 2, 3, 8iseqfcl 9587 . . 3  |-  ( ph  ->  seq M ( Q ,  F ,  S
) : ( ZZ>= `  M ) --> S )
10 ffn 5097 . . 3  |-  (  seq M ( Q ,  F ,  S ) : ( ZZ>= `  M
) --> S  ->  seq M ( Q ,  F ,  S )  Fn  ( ZZ>= `  M )
)
119, 10syl 14 . 2  |-  ( ph  ->  seq M ( Q ,  F ,  S
)  Fn  ( ZZ>= `  M ) )
12 fveq2 5229 . . . . . 6  |-  ( w  =  M  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 M ) )
13 fveq2 5229 . . . . . 6  |-  ( w  =  M  ->  (  seq M ( Q ,  F ,  S ) `  w )  =  (  seq M ( Q ,  F ,  S
) `  M )
)
1412, 13eqeq12d 2097 . . . . 5  |-  ( w  =  M  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( Q ,  F ,  S ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  M
)  =  (  seq M ( Q ,  F ,  S ) `  M ) ) )
1514imbi2d 228 . . . 4  |-  ( w  =  M  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M ( Q ,  F ,  S
) `  w )
)  <->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 M )  =  (  seq M ( Q ,  F ,  S ) `  M
) ) ) )
16 fveq2 5229 . . . . . 6  |-  ( w  =  k  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 k ) )
17 fveq2 5229 . . . . . 6  |-  ( w  =  k  ->  (  seq M ( Q ,  F ,  S ) `  w )  =  (  seq M ( Q ,  F ,  S
) `  k )
)
1816, 17eqeq12d 2097 . . . . 5  |-  ( w  =  k  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( Q ,  F ,  S ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  k
)  =  (  seq M ( Q ,  F ,  S ) `  k ) ) )
1918imbi2d 228 . . . 4  |-  ( w  =  k  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M ( Q ,  F ,  S
) `  w )
)  <->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 k )  =  (  seq M ( Q ,  F ,  S ) `  k
) ) ) )
20 fveq2 5229 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) ) )
21 fveq2 5229 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  (  seq M ( Q ,  F ,  S ) `  w )  =  (  seq M ( Q ,  F ,  S
) `  ( k  +  1 ) ) )
2220, 21eqeq12d 2097 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( Q ,  F ,  S ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  (
k  +  1 ) )  =  (  seq M ( Q ,  F ,  S ) `  ( k  +  1 ) ) ) )
2322imbi2d 228 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M ( Q ,  F ,  S
) `  w )
)  <->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 ( k  +  1 ) )  =  (  seq M ( Q ,  F ,  S ) `  (
k  +  1 ) ) ) ) )
24 fveq2 5229 . . . . . 6  |-  ( w  =  n  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M (  .+  ,  F ,  S ) `
 n ) )
25 fveq2 5229 . . . . . 6  |-  ( w  =  n  ->  (  seq M ( Q ,  F ,  S ) `  w )  =  (  seq M ( Q ,  F ,  S
) `  n )
)
2624, 25eqeq12d 2097 . . . . 5  |-  ( w  =  n  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq M ( Q ,  F ,  S ) `  w
)  <->  (  seq M
(  .+  ,  F ,  S ) `  n
)  =  (  seq M ( Q ,  F ,  S ) `  n ) ) )
2726imbi2d 228 . . . 4  |-  ( w  =  n  ->  (
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq M ( Q ,  F ,  S
) `  w )
)  <->  ( ph  ->  (  seq M (  .+  ,  F ,  S ) `
 n )  =  (  seq M ( Q ,  F ,  S ) `  n
) ) ) )
28 simpr 108 . . . . . . 7  |-  ( (
ph  /\  M  e.  ZZ )  ->  M  e.  ZZ )
293adantlr 461 . . . . . . 7  |-  ( ( ( ph  /\  M  e.  ZZ )  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
304adantlr 461 . . . . . . 7  |-  ( ( ( ph  /\  M  e.  ZZ )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  .+  y )  e.  S
)
3128, 29, 30iseq1 9585 . . . . . 6  |-  ( (
ph  /\  M  e.  ZZ )  ->  (  seq M (  .+  ,  F ,  S ) `  M )  =  ( F `  M ) )
328adantlr 461 . . . . . . 7  |-  ( ( ( ph  /\  M  e.  ZZ )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x Q y )  e.  S )
3328, 29, 32iseq1 9585 . . . . . 6  |-  ( (
ph  /\  M  e.  ZZ )  ->  (  seq M ( Q ,  F ,  S ) `  M )  =  ( F `  M ) )
3431, 33eqtr4d 2118 . . . . 5  |-  ( (
ph  /\  M  e.  ZZ )  ->  (  seq M (  .+  ,  F ,  S ) `  M )  =  (  seq M ( Q ,  F ,  S
) `  M )
)
3534expcom 114 . . . 4  |-  ( M  e.  ZZ  ->  ( ph  ->  (  seq M
(  .+  ,  F ,  S ) `  M
)  =  (  seq M ( Q ,  F ,  S ) `  M ) ) )
36 simpr 108 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  ( ZZ>= `  M )
)
3736adantr 270 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )
)  ->  k  e.  ( ZZ>= `  M )
)
383adantlr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
3938adantlr 461 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  /\  (  seq M (  .+  ,  F ,  S ) `
 k )  =  (  seq M ( Q ,  F ,  S ) `  k
) )  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
404adantlr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
4140adantlr 461 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  k  e.  ( ZZ>= `  M ) )  /\  (  seq M (  .+  ,  F ,  S ) `
 k )  =  (  seq M ( Q ,  F ,  S ) `  k
) )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( x  .+  y )  e.  S
)
4237, 39, 41iseqcl 9589 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )
)  ->  (  seq M (  .+  ,  F ,  S ) `  k )  e.  S
)
43 fveq2 5229 . . . . . . . . . . . 12  |-  ( x  =  ( k  +  1 )  ->  ( F `  x )  =  ( F `  ( k  +  1 ) ) )
4443eleq1d 2151 . . . . . . . . . . 11  |-  ( x  =  ( k  +  1 )  ->  (
( F `  x
)  e.  S  <->  ( F `  ( k  +  1 ) )  e.  S
) )
453ralrimiva 2439 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  S )
4645ad2antrr 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )
)  ->  A. x  e.  ( ZZ>= `  M )
( F `  x
)  e.  S )
47 peano2uz 8804 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( k  +  1 )  e.  ( ZZ>= `  M )
)
4847ad2antlr 473 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )
)  ->  ( k  +  1 )  e.  ( ZZ>= `  M )
)
4944, 46, 48rspcdva 2715 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )
)  ->  ( F `  ( k  +  1 ) )  e.  S
)
50 iseqoveq.eq . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( x Q y ) )
5150ralrimivva 2448 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x  .+  y )  =  ( x Q y ) )
5251ad2antrr 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )
)  ->  A. x  e.  S  A. y  e.  S  ( x  .+  y )  =  ( x Q y ) )
53 oveq1 5570 . . . . . . . . . . . 12  |-  ( x  =  (  seq M
(  .+  ,  F ,  S ) `  k
)  ->  ( x  .+  y )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  k
)  .+  y )
)
54 oveq1 5570 . . . . . . . . . . . 12  |-  ( x  =  (  seq M
(  .+  ,  F ,  S ) `  k
)  ->  ( x Q y )  =  ( (  seq M
(  .+  ,  F ,  S ) `  k
) Q y ) )
5553, 54eqeq12d 2097 . . . . . . . . . . 11  |-  ( x  =  (  seq M
(  .+  ,  F ,  S ) `  k
)  ->  ( (
x  .+  y )  =  ( x Q y )  <->  ( (  seq M (  .+  ,  F ,  S ) `  k )  .+  y
)  =  ( (  seq M (  .+  ,  F ,  S ) `
 k ) Q y ) ) )
56 oveq2 5571 . . . . . . . . . . . 12  |-  ( y  =  ( F `  ( k  +  1 ) )  ->  (
(  seq M (  .+  ,  F ,  S ) `
 k )  .+  y )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
57 oveq2 5571 . . . . . . . . . . . 12  |-  ( y  =  ( F `  ( k  +  1 ) )  ->  (
(  seq M (  .+  ,  F ,  S ) `
 k ) Q y )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  k
) Q ( F `
 ( k  +  1 ) ) ) )
5856, 57eqeq12d 2097 . . . . . . . . . . 11  |-  ( y  =  ( F `  ( k  +  1 ) )  ->  (
( (  seq M
(  .+  ,  F ,  S ) `  k
)  .+  y )  =  ( (  seq M (  .+  ,  F ,  S ) `  k ) Q y )  <->  ( (  seq M (  .+  ,  F ,  S ) `  k )  .+  ( F `  ( k  +  1 ) ) )  =  ( (  seq M (  .+  ,  F ,  S ) `
 k ) Q ( F `  (
k  +  1 ) ) ) ) )
5955, 58rspc2va 2722 . . . . . . . . . 10  |-  ( ( ( (  seq M
(  .+  ,  F ,  S ) `  k
)  e.  S  /\  ( F `  ( k  +  1 ) )  e.  S )  /\  A. x  e.  S  A. y  e.  S  (
x  .+  y )  =  ( x Q y ) )  -> 
( (  seq M
(  .+  ,  F ,  S ) `  k
)  .+  ( F `  ( k  +  1 ) ) )  =  ( (  seq M
(  .+  ,  F ,  S ) `  k
) Q ( F `
 ( k  +  1 ) ) ) )
6042, 49, 52, 59syl21anc 1169 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )
)  ->  ( (  seq M (  .+  ,  F ,  S ) `  k )  .+  ( F `  ( k  +  1 ) ) )  =  ( (  seq M (  .+  ,  F ,  S ) `
 k ) Q ( F `  (
k  +  1 ) ) ) )
61 simpr 108 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )
)  ->  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )
)
6261oveq1d 5578 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )
)  ->  ( (  seq M (  .+  ,  F ,  S ) `  k ) Q ( F `  ( k  +  1 ) ) )  =  ( (  seq M ( Q ,  F ,  S
) `  k ) Q ( F `  ( k  +  1 ) ) ) )
6360, 62eqtrd 2115 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )
)  ->  ( (  seq M (  .+  ,  F ,  S ) `  k )  .+  ( F `  ( k  +  1 ) ) )  =  ( (  seq M ( Q ,  F ,  S
) `  k ) Q ( F `  ( k  +  1 ) ) ) )
6436, 38, 40iseqp1 9590 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  k
)  .+  ( F `  ( k  +  1 ) ) ) )
658adantlr 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x Q y )  e.  S )
6636, 38, 65iseqp1 9590 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  (  seq M ( Q ,  F ,  S ) `  ( k  +  1 ) )  =  ( (  seq M ( Q ,  F ,  S ) `  k
) Q ( F `
 ( k  +  1 ) ) ) )
6764, 66eqeq12d 2097 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  (  seq M ( Q ,  F ,  S
) `  ( k  +  1 ) )  <-> 
( (  seq M
(  .+  ,  F ,  S ) `  k
)  .+  ( F `  ( k  +  1 ) ) )  =  ( (  seq M
( Q ,  F ,  S ) `  k
) Q ( F `
 ( k  +  1 ) ) ) ) )
6867adantr 270 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )
)  ->  ( (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  (  seq M ( Q ,  F ,  S
) `  ( k  +  1 ) )  <-> 
( (  seq M
(  .+  ,  F ,  S ) `  k
)  .+  ( F `  ( k  +  1 ) ) )  =  ( (  seq M
( Q ,  F ,  S ) `  k
) Q ( F `
 ( k  +  1 ) ) ) ) )
6963, 68mpbird 165 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ZZ>= `  M )
)  /\  (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )
)  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  (  seq M ( Q ,  F ,  S
) `  ( k  +  1 ) ) )
7069ex 113 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (  seq M (  .+  ,  F ,  S ) `  k )  =  (  seq M ( Q ,  F ,  S
) `  k )  ->  (  seq M ( 
.+  ,  F ,  S ) `  (
k  +  1 ) )  =  (  seq M ( Q ,  F ,  S ) `  ( k  +  1 ) ) ) )
7170expcom 114 . . . . 5  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( (  seq M
(  .+  ,  F ,  S ) `  k
)  =  (  seq M ( Q ,  F ,  S ) `  k )  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  (  seq M ( Q ,  F ,  S
) `  ( k  +  1 ) ) ) ) )
7271a2d 26 . . . 4  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ( ph  ->  (  seq M
(  .+  ,  F ,  S ) `  k
)  =  (  seq M ( Q ,  F ,  S ) `  k ) )  -> 
( ph  ->  (  seq M (  .+  ,  F ,  S ) `  ( k  +  1 ) )  =  (  seq M ( Q ,  F ,  S
) `  ( k  +  1 ) ) ) ) )
7315, 19, 23, 27, 35, 72uzind4 8809 . . 3  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  n
)  =  (  seq M ( Q ,  F ,  S ) `  n ) ) )
7473impcom 123 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  M )
)  ->  (  seq M (  .+  ,  F ,  S ) `  n )  =  (  seq M ( Q ,  F ,  S
) `  n )
)
757, 11, 74eqfnfvd 5320 1  |-  ( ph  ->  seq M (  .+  ,  F ,  S )  =  seq M ( Q ,  F ,  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2353    Fn wfn 4947   -->wf 4948   ` cfv 4952  (class class class)co 5563   1c1 7096    + caddc 7098   ZZcz 8484   ZZ>=cuz 8752    seqcseq 9573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753  df-iseq 9574
This theorem is referenced by: (None)
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