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Theorem 2ffzoeq 30382
Description: Two functions over a half-open range of nonnegative integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
Assertion
Ref Expression
2ffzoeq  |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) )  ->  ( F  =  P  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) )
Distinct variable groups:    i, F    i, M    P, i
Allowed substitution hints:    N( i)    X( i)    Y( i)

Proof of Theorem 2ffzoeq
StepHypRef Expression
1 eqeq1 2458 . . . . . . . . . . . 12  |-  ( F  =  P  ->  ( F  =  (/)  <->  P  =  (/) ) )
21anbi1d 704 . . . . . . . . . . 11  |-  ( F  =  P  ->  (
( F  =  (/)  /\  P : ( 0..^ N ) --> Y )  <-> 
( P  =  (/)  /\  P : ( 0..^ N ) --> Y ) ) )
3 f0bi 5705 . . . . . . . . . . . . 13  |-  ( P : (/) --> Y  <->  P  =  (/) )
4 ffn 5670 . . . . . . . . . . . . . 14  |-  ( P : (/) --> Y  ->  P  Fn  (/) )
5 ffn 5670 . . . . . . . . . . . . . . 15  |-  ( P : ( 0..^ N ) --> Y  ->  P  Fn  ( 0..^ N ) )
6 fndmu 5623 . . . . . . . . . . . . . . . . 17  |-  ( ( P  Fn  ( 0..^ N )  /\  P  Fn  (/) )  ->  (
0..^ N )  =  (/) )
7 0z 10771 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  ZZ
8 nn0z 10783 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  N  e.  ZZ )
98adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  N  e.  ZZ )
10 fzon 11691 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <_  0  <->  ( 0..^ N )  =  (/) ) )
117, 9, 10sylancr 663 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( N  <_  0  <->  ( 0..^ N )  =  (/) ) )
12 nn0ge0 10719 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  0  <_  N )
13 0re 9500 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  RR
1413a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  0  e.  RR )
15 nn0re 10702 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  N  e.  RR )
1614, 15letri3d 9630 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  ( 0  =  N  <->  ( 0  <_  N  /\  N  <_  0 ) ) )
1716biimprd 223 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( ( 0  <_  N  /\  N  <_  0 )  -> 
0  =  N ) )
1812, 17mpand 675 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  ( N  <_  0  ->  0  =  N ) )
1918adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( N  <_  0  ->  0  =  N ) )
2011, 19sylbird 235 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( 0..^ N )  =  (/)  ->  0  =  N ) )
216, 20syl5com 30 . . . . . . . . . . . . . . . 16  |-  ( ( P  Fn  ( 0..^ N )  /\  P  Fn  (/) )  ->  (
( M  e.  NN0  /\  N  e.  NN0 )  ->  0  =  N ) )
2221ex 434 . . . . . . . . . . . . . . 15  |-  ( P  Fn  ( 0..^ N )  ->  ( P  Fn  (/)  ->  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  0  =  N ) ) )
235, 22syl 16 . . . . . . . . . . . . . 14  |-  ( P : ( 0..^ N ) --> Y  ->  ( P  Fn  (/)  ->  (
( M  e.  NN0  /\  N  e.  NN0 )  ->  0  =  N ) ) )
244, 23syl5com 30 . . . . . . . . . . . . 13  |-  ( P : (/) --> Y  ->  ( P : ( 0..^ N ) --> Y  ->  (
( M  e.  NN0  /\  N  e.  NN0 )  ->  0  =  N ) ) )
253, 24sylbir 213 . . . . . . . . . . . 12  |-  ( P  =  (/)  ->  ( P : ( 0..^ N ) --> Y  ->  (
( M  e.  NN0  /\  N  e.  NN0 )  ->  0  =  N ) ) )
2625imp 429 . . . . . . . . . . 11  |-  ( ( P  =  (/)  /\  P : ( 0..^ N ) --> Y )  -> 
( ( M  e. 
NN0  /\  N  e.  NN0 )  ->  0  =  N ) )
272, 26syl6bi 228 . . . . . . . . . 10  |-  ( F  =  P  ->  (
( F  =  (/)  /\  P : ( 0..^ N ) --> Y )  ->  ( ( M  e.  NN0  /\  N  e. 
NN0 )  ->  0  =  N ) ) )
2827com3l 81 . . . . . . . . 9  |-  ( ( F  =  (/)  /\  P : ( 0..^ N ) --> Y )  -> 
( ( M  e. 
NN0  /\  N  e.  NN0 )  ->  ( F  =  P  ->  0  =  N ) ) )
2928a1i 11 . . . . . . . 8  |-  ( M  =  0  ->  (
( F  =  (/)  /\  P : ( 0..^ N ) --> Y )  ->  ( ( M  e.  NN0  /\  N  e. 
NN0 )  ->  ( F  =  P  ->  0  =  N ) ) ) )
30 oveq2 6211 . . . . . . . . . . . 12  |-  ( M  =  0  ->  (
0..^ M )  =  ( 0..^ 0 ) )
31 fzo0 11693 . . . . . . . . . . . 12  |-  ( 0..^ 0 )  =  (/)
3230, 31syl6eq 2511 . . . . . . . . . . 11  |-  ( M  =  0  ->  (
0..^ M )  =  (/) )
3332feq2d 5658 . . . . . . . . . 10  |-  ( M  =  0  ->  ( F : ( 0..^ M ) --> X  <->  F : (/) --> X ) )
34 f0bi 5705 . . . . . . . . . 10  |-  ( F : (/) --> X  <->  F  =  (/) )
3533, 34syl6bb 261 . . . . . . . . 9  |-  ( M  =  0  ->  ( F : ( 0..^ M ) --> X  <->  F  =  (/) ) )
3635anbi1d 704 . . . . . . . 8  |-  ( M  =  0  ->  (
( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  <->  ( F  =  (/)  /\  P : ( 0..^ N ) --> Y ) ) )
37 eqeq1 2458 . . . . . . . . . 10  |-  ( M  =  0  ->  ( M  =  N  <->  0  =  N ) )
3837imbi2d 316 . . . . . . . . 9  |-  ( M  =  0  ->  (
( F  =  P  ->  M  =  N )  <->  ( F  =  P  ->  0  =  N ) ) )
3938imbi2d 316 . . . . . . . 8  |-  ( M  =  0  ->  (
( ( M  e. 
NN0  /\  N  e.  NN0 )  ->  ( F  =  P  ->  M  =  N ) )  <->  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( F  =  P  ->  0  =  N ) ) ) )
4029, 36, 393imtr4d 268 . . . . . . 7  |-  ( M  =  0  ->  (
( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  ->  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( F  =  P  ->  M  =  N ) ) ) )
4140com3l 81 . . . . . 6  |-  ( ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  -> 
( ( M  e. 
NN0  /\  N  e.  NN0 )  ->  ( M  =  0  ->  ( F  =  P  ->  M  =  N ) ) ) )
4241impcom 430 . . . . 5  |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) )  ->  ( M  =  0  ->  ( F  =  P  ->  M  =  N )
) )
4342impcom 430 . . . 4  |-  ( ( M  =  0  /\  ( ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) ) )  ->  ( F  =  P  ->  M  =  N ) )
4430feq2d 5658 . . . . . . . . . . . 12  |-  ( M  =  0  ->  ( F : ( 0..^ M ) --> X  <->  F :
( 0..^ 0 ) --> X ) )
4531feq2i 5663 . . . . . . . . . . . . 13  |-  ( F : ( 0..^ 0 ) --> X  <->  F : (/) --> X )
4645, 34bitri 249 . . . . . . . . . . . 12  |-  ( F : ( 0..^ 0 ) --> X  <->  F  =  (/) )
4744, 46syl6bb 261 . . . . . . . . . . 11  |-  ( M  =  0  ->  ( F : ( 0..^ M ) --> X  <->  F  =  (/) ) )
4847adantr 465 . . . . . . . . . 10  |-  ( ( M  =  0  /\  M  =  N )  ->  ( F :
( 0..^ M ) --> X  <->  F  =  (/) ) )
49 eqeq1 2458 . . . . . . . . . . . . 13  |-  ( M  =  N  ->  ( M  =  0  <->  N  = 
0 ) )
5049biimpd 207 . . . . . . . . . . . 12  |-  ( M  =  N  ->  ( M  =  0  ->  N  =  0 ) )
5150impcom 430 . . . . . . . . . . 11  |-  ( ( M  =  0  /\  M  =  N )  ->  N  =  0 )
52 oveq2 6211 . . . . . . . . . . . . 13  |-  ( N  =  0  ->  (
0..^ N )  =  ( 0..^ 0 ) )
5352feq2d 5658 . . . . . . . . . . . 12  |-  ( N  =  0  ->  ( P : ( 0..^ N ) --> Y  <->  P :
( 0..^ 0 ) --> Y ) )
5431feq2i 5663 . . . . . . . . . . . . 13  |-  ( P : ( 0..^ 0 ) --> Y  <->  P : (/) --> Y )
5554, 3bitri 249 . . . . . . . . . . . 12  |-  ( P : ( 0..^ 0 ) --> Y  <->  P  =  (/) )
5653, 55syl6bb 261 . . . . . . . . . . 11  |-  ( N  =  0  ->  ( P : ( 0..^ N ) --> Y  <->  P  =  (/) ) )
5751, 56syl 16 . . . . . . . . . 10  |-  ( ( M  =  0  /\  M  =  N )  ->  ( P :
( 0..^ N ) --> Y  <->  P  =  (/) ) )
5848, 57anbi12d 710 . . . . . . . . 9  |-  ( ( M  =  0  /\  M  =  N )  ->  ( ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  <->  ( F  =  (/)  /\  P  =  (/) ) ) )
59 eqtr3 2482 . . . . . . . . 9  |-  ( ( F  =  (/)  /\  P  =  (/) )  ->  F  =  P )
6058, 59syl6bi 228 . . . . . . . 8  |-  ( ( M  =  0  /\  M  =  N )  ->  ( ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  ->  F  =  P )
)
6160com12 31 . . . . . . 7  |-  ( ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  -> 
( ( M  =  0  /\  M  =  N )  ->  F  =  P ) )
6261expd 436 . . . . . 6  |-  ( ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  -> 
( M  =  0  ->  ( M  =  N  ->  F  =  P ) ) )
6362adantl 466 . . . . 5  |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) )  ->  ( M  =  0  ->  ( M  =  N  ->  F  =  P )
) )
6463impcom 430 . . . 4  |-  ( ( M  =  0  /\  ( ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) ) )  ->  ( M  =  N  ->  F  =  P ) )
6543, 64impbid 191 . . 3  |-  ( ( M  =  0  /\  ( ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) ) )  ->  ( F  =  P  <->  M  =  N
) )
66 ral0 3895 . . . . . 6  |-  A. i  e.  (/)  ( F `  i )  =  ( P `  i )
6732raleqdv 3029 . . . . . 6  |-  ( M  =  0  ->  ( A. i  e.  (
0..^ M ) ( F `  i )  =  ( P `  i )  <->  A. i  e.  (/)  ( F `  i )  =  ( P `  i ) ) )
6866, 67mpbiri 233 . . . . 5  |-  ( M  =  0  ->  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) )
6968biantrud 507 . . . 4  |-  ( M  =  0  ->  ( M  =  N  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) )
7069adantr 465 . . 3  |-  ( ( M  =  0  /\  ( ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) ) )  ->  ( M  =  N  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) )
7165, 70bitrd 253 . 2  |-  ( ( M  =  0  /\  ( ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) ) )  ->  ( F  =  P  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) )
72 ffn 5670 . . . . . . 7  |-  ( F : ( 0..^ M ) --> X  ->  F  Fn  ( 0..^ M ) )
7372, 5anim12i 566 . . . . . 6  |-  ( ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  -> 
( F  Fn  (
0..^ M )  /\  P  Fn  ( 0..^ N ) ) )
7473adantl 466 . . . . 5  |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) )  ->  ( F  Fn  ( 0..^ M )  /\  P  Fn  ( 0..^ N ) ) )
7574adantl 466 . . . 4  |-  ( ( -.  M  =  0  /\  ( ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) ) )  ->  ( F  Fn  ( 0..^ M )  /\  P  Fn  (
0..^ N ) ) )
76 eqfnfv2 5910 . . . 4  |-  ( ( F  Fn  ( 0..^ M )  /\  P  Fn  ( 0..^ N ) )  ->  ( F  =  P  <->  ( ( 0..^ M )  =  ( 0..^ N )  /\  A. i  e.  ( 0..^ M ) ( F `
 i )  =  ( P `  i
) ) ) )
7775, 76syl 16 . . 3  |-  ( ( -.  M  =  0  /\  ( ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) ) )  ->  ( F  =  P  <->  ( ( 0..^ M )  =  ( 0..^ N )  /\  A. i  e.  ( 0..^ M ) ( F `
 i )  =  ( P `  i
) ) ) )
78 df-ne 2650 . . . . . 6  |-  ( M  =/=  0  <->  -.  M  =  0 )
79 elnnne0 10707 . . . . . . . 8  |-  ( M  e.  NN  <->  ( M  e.  NN0  /\  M  =/=  0 ) )
80 0zd 10772 . . . . . . . . . . . . . . 15  |-  ( M  e.  NN  ->  0  e.  ZZ )
81 nnz 10782 . . . . . . . . . . . . . . 15  |-  ( M  e.  NN  ->  M  e.  ZZ )
82 nngt0 10465 . . . . . . . . . . . . . . 15  |-  ( M  e.  NN  ->  0  <  M )
8380, 81, 823jca 1168 . . . . . . . . . . . . . 14  |-  ( M  e.  NN  ->  (
0  e.  ZZ  /\  M  e.  ZZ  /\  0  <  M ) )
8483adantr 465 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN  /\  N  e.  NN0 )  -> 
( 0  e.  ZZ  /\  M  e.  ZZ  /\  0  <  M ) )
85 fzoopth 30381 . . . . . . . . . . . . 13  |-  ( ( 0  e.  ZZ  /\  M  e.  ZZ  /\  0  <  M )  ->  (
( 0..^ M )  =  ( 0..^ N )  <->  ( 0  =  0  /\  M  =  N ) ) )
8684, 85syl 16 . . . . . . . . . . . 12  |-  ( ( M  e.  NN  /\  N  e.  NN0 )  -> 
( ( 0..^ M )  =  ( 0..^ N )  <->  ( 0  =  0  /\  M  =  N ) ) )
87 simpr 461 . . . . . . . . . . . 12  |-  ( ( 0  =  0  /\  M  =  N )  ->  M  =  N )
8886, 87syl6bi 228 . . . . . . . . . . 11  |-  ( ( M  e.  NN  /\  N  e.  NN0 )  -> 
( ( 0..^ M )  =  ( 0..^ N )  ->  M  =  N ) )
8988anim1d 564 . . . . . . . . . 10  |-  ( ( M  e.  NN  /\  N  e.  NN0 )  -> 
( ( ( 0..^ M )  =  ( 0..^ N )  /\  A. i  e.  ( 0..^ M ) ( F `
 i )  =  ( P `  i
) )  ->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `
 i )  =  ( P `  i
) ) ) )
90 oveq2 6211 . . . . . . . . . . 11  |-  ( M  =  N  ->  (
0..^ M )  =  ( 0..^ N ) )
9190anim1i 568 . . . . . . . . . 10  |-  ( ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `
 i )  =  ( P `  i
) )  ->  (
( 0..^ M )  =  ( 0..^ N )  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) )
9289, 91impbid1 203 . . . . . . . . 9  |-  ( ( M  e.  NN  /\  N  e.  NN0 )  -> 
( ( ( 0..^ M )  =  ( 0..^ N )  /\  A. i  e.  ( 0..^ M ) ( F `
 i )  =  ( P `  i
) )  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) )
9392ex 434 . . . . . . . 8  |-  ( M  e.  NN  ->  ( N  e.  NN0  ->  (
( ( 0..^ M )  =  ( 0..^ N )  /\  A. i  e.  ( 0..^ M ) ( F `
 i )  =  ( P `  i
) )  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) ) )
9479, 93sylbir 213 . . . . . . 7  |-  ( ( M  e.  NN0  /\  M  =/=  0 )  -> 
( N  e.  NN0  ->  ( ( ( 0..^ M )  =  ( 0..^ N )  /\  A. i  e.  ( 0..^ M ) ( F `
 i )  =  ( P `  i
) )  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) ) )
9594impancom 440 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  =/=  0  ->  ( ( ( 0..^ M )  =  ( 0..^ N )  /\  A. i  e.  ( 0..^ M ) ( F `
 i )  =  ( P `  i
) )  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) ) )
9678, 95syl5bir 218 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( -.  M  =  0  ->  ( (
( 0..^ M )  =  ( 0..^ N )  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) )  <-> 
( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) ) )
9796adantr 465 . . . 4  |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) )  ->  ( -.  M  =  0  ->  ( ( ( 0..^ M )  =  ( 0..^ N )  /\  A. i  e.  ( 0..^ M ) ( F `
 i )  =  ( P `  i
) )  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) ) )
9897impcom 430 . . 3  |-  ( ( -.  M  =  0  /\  ( ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) ) )  ->  ( (
( 0..^ M )  =  ( 0..^ N )  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) )  <-> 
( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) )
9977, 98bitrd 253 . 2  |-  ( ( -.  M  =  0  /\  ( ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) ) )  ->  ( F  =  P  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) )
10071, 99pm2.61ian 788 1  |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) )  ->  ( F  =  P  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   (/)c0 3748   class class class wbr 4403    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203   RRcr 9395   0cc0 9396    < clt 9532    <_ cle 9533   NNcn 10436   NN0cn0 10693   ZZcz 10760  ..^cfzo 11668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669
This theorem is referenced by: (None)
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