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Theorem 2ffzoeq 39051
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 2454 . . . . . . . . . . . 12  |-  ( F  =  P  ->  ( F  =  (/)  <->  P  =  (/) ) )
21anbi1d 710 . . . . . . . . . . 11  |-  ( F  =  P  ->  (
( F  =  (/)  /\  P : ( 0..^ N ) --> Y )  <-> 
( P  =  (/)  /\  P : ( 0..^ N ) --> Y ) ) )
3 f0bi 5764 . . . . . . . . . . . . 13  |-  ( P : (/) --> Y  <->  P  =  (/) )
4 ffn 5726 . . . . . . . . . . . . . 14  |-  ( P : (/) --> Y  ->  P  Fn  (/) )
5 ffn 5726 . . . . . . . . . . . . . . 15  |-  ( P : ( 0..^ N ) --> Y  ->  P  Fn  ( 0..^ N ) )
6 fndmu 5675 . . . . . . . . . . . . . . . . 17  |-  ( ( P  Fn  ( 0..^ N )  /\  P  Fn  (/) )  ->  (
0..^ N )  =  (/) )
7 0z 10945 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  ZZ
8 nn0z 10957 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  N  e.  ZZ )
98adantl 468 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  N  e.  ZZ )
10 fzon 11936 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <_  0  <->  ( 0..^ N )  =  (/) ) )
117, 9, 10sylancr 668 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( N  <_  0  <->  ( 0..^ N )  =  (/) ) )
12 nn0ge0 10892 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  0  <_  N )
13 0red 9641 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  0  e.  RR )
14 nn0re 10875 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  N  e.  RR )
1513, 14letri3d 9774 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  ( 0  =  N  <->  ( 0  <_  N  /\  N  <_  0 ) ) )
1615biimprd 227 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( ( 0  <_  N  /\  N  <_  0 )  -> 
0  =  N ) )
1712, 16mpand 680 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  ( N  <_  0  ->  0  =  N ) )
1817adantl 468 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( N  <_  0  ->  0  =  N ) )
1911, 18sylbird 239 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( 0..^ N )  =  (/)  ->  0  =  N ) )
206, 19syl5com 31 . . . . . . . . . . . . . . . 16  |-  ( ( P  Fn  ( 0..^ N )  /\  P  Fn  (/) )  ->  (
( M  e.  NN0  /\  N  e.  NN0 )  ->  0  =  N ) )
2120ex 436 . . . . . . . . . . . . . . 15  |-  ( P  Fn  ( 0..^ N )  ->  ( P  Fn  (/)  ->  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  0  =  N ) ) )
225, 21syl 17 . . . . . . . . . . . . . 14  |-  ( P : ( 0..^ N ) --> Y  ->  ( P  Fn  (/)  ->  (
( M  e.  NN0  /\  N  e.  NN0 )  ->  0  =  N ) ) )
234, 22syl5com 31 . . . . . . . . . . . . 13  |-  ( P : (/) --> Y  ->  ( P : ( 0..^ N ) --> Y  ->  (
( M  e.  NN0  /\  N  e.  NN0 )  ->  0  =  N ) ) )
243, 23sylbir 217 . . . . . . . . . . . 12  |-  ( P  =  (/)  ->  ( P : ( 0..^ N ) --> Y  ->  (
( M  e.  NN0  /\  N  e.  NN0 )  ->  0  =  N ) ) )
2524imp 431 . . . . . . . . . . 11  |-  ( ( P  =  (/)  /\  P : ( 0..^ N ) --> Y )  -> 
( ( M  e. 
NN0  /\  N  e.  NN0 )  ->  0  =  N ) )
262, 25syl6bi 232 . . . . . . . . . 10  |-  ( F  =  P  ->  (
( F  =  (/)  /\  P : ( 0..^ N ) --> Y )  ->  ( ( M  e.  NN0  /\  N  e. 
NN0 )  ->  0  =  N ) ) )
2726com3l 84 . . . . . . . . 9  |-  ( ( F  =  (/)  /\  P : ( 0..^ N ) --> Y )  -> 
( ( M  e. 
NN0  /\  N  e.  NN0 )  ->  ( F  =  P  ->  0  =  N ) ) )
2827a1i 11 . . . . . . . 8  |-  ( M  =  0  ->  (
( F  =  (/)  /\  P : ( 0..^ N ) --> Y )  ->  ( ( M  e.  NN0  /\  N  e. 
NN0 )  ->  ( F  =  P  ->  0  =  N ) ) ) )
29 oveq2 6296 . . . . . . . . . . . 12  |-  ( M  =  0  ->  (
0..^ M )  =  ( 0..^ 0 ) )
30 fzo0 11939 . . . . . . . . . . . 12  |-  ( 0..^ 0 )  =  (/)
3129, 30syl6eq 2500 . . . . . . . . . . 11  |-  ( M  =  0  ->  (
0..^ M )  =  (/) )
3231feq2d 5713 . . . . . . . . . 10  |-  ( M  =  0  ->  ( F : ( 0..^ M ) --> X  <->  F : (/) --> X ) )
33 f0bi 5764 . . . . . . . . . 10  |-  ( F : (/) --> X  <->  F  =  (/) )
3432, 33syl6bb 265 . . . . . . . . 9  |-  ( M  =  0  ->  ( F : ( 0..^ M ) --> X  <->  F  =  (/) ) )
3534anbi1d 710 . . . . . . . 8  |-  ( M  =  0  ->  (
( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  <->  ( F  =  (/)  /\  P : ( 0..^ N ) --> Y ) ) )
36 eqeq1 2454 . . . . . . . . . 10  |-  ( M  =  0  ->  ( M  =  N  <->  0  =  N ) )
3736imbi2d 318 . . . . . . . . 9  |-  ( M  =  0  ->  (
( F  =  P  ->  M  =  N )  <->  ( F  =  P  ->  0  =  N ) ) )
3837imbi2d 318 . . . . . . . 8  |-  ( M  =  0  ->  (
( ( M  e. 
NN0  /\  N  e.  NN0 )  ->  ( F  =  P  ->  M  =  N ) )  <->  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( F  =  P  ->  0  =  N ) ) ) )
3928, 35, 383imtr4d 272 . . . . . . 7  |-  ( M  =  0  ->  (
( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  ->  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( F  =  P  ->  M  =  N ) ) ) )
4039com3l 84 . . . . . 6  |-  ( ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  -> 
( ( M  e. 
NN0  /\  N  e.  NN0 )  ->  ( M  =  0  ->  ( F  =  P  ->  M  =  N ) ) ) )
4140impcom 432 . . . . 5  |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) )  ->  ( M  =  0  ->  ( F  =  P  ->  M  =  N )
) )
4241impcom 432 . . . 4  |-  ( ( M  =  0  /\  ( ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) ) )  ->  ( F  =  P  ->  M  =  N ) )
4329feq2d 5713 . . . . . . . . . . . 12  |-  ( M  =  0  ->  ( F : ( 0..^ M ) --> X  <->  F :
( 0..^ 0 ) --> X ) )
4430feq2i 5719 . . . . . . . . . . . . 13  |-  ( F : ( 0..^ 0 ) --> X  <->  F : (/) --> X )
4544, 33bitri 253 . . . . . . . . . . . 12  |-  ( F : ( 0..^ 0 ) --> X  <->  F  =  (/) )
4643, 45syl6bb 265 . . . . . . . . . . 11  |-  ( M  =  0  ->  ( F : ( 0..^ M ) --> X  <->  F  =  (/) ) )
4746adantr 467 . . . . . . . . . 10  |-  ( ( M  =  0  /\  M  =  N )  ->  ( F :
( 0..^ M ) --> X  <->  F  =  (/) ) )
48 eqeq1 2454 . . . . . . . . . . . 12  |-  ( M  =  N  ->  ( M  =  0  <->  N  = 
0 ) )
4948biimpac 489 . . . . . . . . . . 11  |-  ( ( M  =  0  /\  M  =  N )  ->  N  =  0 )
50 oveq2 6296 . . . . . . . . . . . . 13  |-  ( N  =  0  ->  (
0..^ N )  =  ( 0..^ 0 ) )
5150feq2d 5713 . . . . . . . . . . . 12  |-  ( N  =  0  ->  ( P : ( 0..^ N ) --> Y  <->  P :
( 0..^ 0 ) --> Y ) )
5230feq2i 5719 . . . . . . . . . . . . 13  |-  ( P : ( 0..^ 0 ) --> Y  <->  P : (/) --> Y )
5352, 3bitri 253 . . . . . . . . . . . 12  |-  ( P : ( 0..^ 0 ) --> Y  <->  P  =  (/) )
5451, 53syl6bb 265 . . . . . . . . . . 11  |-  ( N  =  0  ->  ( P : ( 0..^ N ) --> Y  <->  P  =  (/) ) )
5549, 54syl 17 . . . . . . . . . 10  |-  ( ( M  =  0  /\  M  =  N )  ->  ( P :
( 0..^ N ) --> Y  <->  P  =  (/) ) )
5647, 55anbi12d 716 . . . . . . . . 9  |-  ( ( M  =  0  /\  M  =  N )  ->  ( ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  <->  ( F  =  (/)  /\  P  =  (/) ) ) )
57 eqtr3 2471 . . . . . . . . 9  |-  ( ( F  =  (/)  /\  P  =  (/) )  ->  F  =  P )
5856, 57syl6bi 232 . . . . . . . 8  |-  ( ( M  =  0  /\  M  =  N )  ->  ( ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  ->  F  =  P )
)
5958com12 32 . . . . . . 7  |-  ( ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  -> 
( ( M  =  0  /\  M  =  N )  ->  F  =  P ) )
6059expd 438 . . . . . 6  |-  ( ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  -> 
( M  =  0  ->  ( M  =  N  ->  F  =  P ) ) )
6160adantl 468 . . . . 5  |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) )  ->  ( M  =  0  ->  ( M  =  N  ->  F  =  P )
) )
6261impcom 432 . . . 4  |-  ( ( M  =  0  /\  ( ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) ) )  ->  ( M  =  N  ->  F  =  P ) )
6342, 62impbid 194 . . 3  |-  ( ( M  =  0  /\  ( ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) ) )  ->  ( F  =  P  <->  M  =  N
) )
64 ral0 3873 . . . . . 6  |-  A. i  e.  (/)  ( F `  i )  =  ( P `  i )
6531raleqdv 2992 . . . . . 6  |-  ( M  =  0  ->  ( A. i  e.  (
0..^ M ) ( F `  i )  =  ( P `  i )  <->  A. i  e.  (/)  ( F `  i )  =  ( P `  i ) ) )
6664, 65mpbiri 237 . . . . 5  |-  ( M  =  0  ->  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) )
6766biantrud 510 . . . 4  |-  ( M  =  0  ->  ( M  =  N  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `  i
)  =  ( P `
 i ) ) ) )
6867adantr 467 . . 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 ) ) ) )
6963, 68bitrd 257 . 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 ) ) ) )
70 ffn 5726 . . . . . . 7  |-  ( F : ( 0..^ M ) --> X  ->  F  Fn  ( 0..^ M ) )
7170, 5anim12i 569 . . . . . 6  |-  ( ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y )  -> 
( F  Fn  (
0..^ M )  /\  P  Fn  ( 0..^ N ) ) )
7271adantl 468 . . . . 5  |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N ) --> Y ) )  ->  ( F  Fn  ( 0..^ M )  /\  P  Fn  ( 0..^ N ) ) )
7372adantl 468 . . . 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 ) ) )
74 eqfnfv2 5975 . . . 4  |-  ( ( F  Fn  ( 0..^ M )  /\  P  Fn  ( 0..^ N ) )  ->  ( F  =  P  <->  ( ( 0..^ M )  =  ( 0..^ N )  /\  A. i  e.  ( 0..^ M ) ( F `
 i )  =  ( P `  i
) ) ) )
7573, 74syl 17 . . 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
) ) ) )
76 df-ne 2623 . . . . . 6  |-  ( M  =/=  0  <->  -.  M  =  0 )
77 elnnne0 10880 . . . . . . . 8  |-  ( M  e.  NN  <->  ( M  e.  NN0  /\  M  =/=  0 ) )
78 0zd 10946 . . . . . . . . . . . . . . 15  |-  ( M  e.  NN  ->  0  e.  ZZ )
79 nnz 10956 . . . . . . . . . . . . . . 15  |-  ( M  e.  NN  ->  M  e.  ZZ )
80 nngt0 10635 . . . . . . . . . . . . . . 15  |-  ( M  e.  NN  ->  0  <  M )
8178, 79, 803jca 1187 . . . . . . . . . . . . . 14  |-  ( M  e.  NN  ->  (
0  e.  ZZ  /\  M  e.  ZZ  /\  0  <  M ) )
8281adantr 467 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN  /\  N  e.  NN0 )  -> 
( 0  e.  ZZ  /\  M  e.  ZZ  /\  0  <  M ) )
83 fzoopth 39050 . . . . . . . . . . . . 13  |-  ( ( 0  e.  ZZ  /\  M  e.  ZZ  /\  0  <  M )  ->  (
( 0..^ M )  =  ( 0..^ N )  <->  ( 0  =  0  /\  M  =  N ) ) )
8482, 83syl 17 . . . . . . . . . . . 12  |-  ( ( M  e.  NN  /\  N  e.  NN0 )  -> 
( ( 0..^ M )  =  ( 0..^ N )  <->  ( 0  =  0  /\  M  =  N ) ) )
85 simpr 463 . . . . . . . . . . . 12  |-  ( ( 0  =  0  /\  M  =  N )  ->  M  =  N )
8684, 85syl6bi 232 . . . . . . . . . . 11  |-  ( ( M  e.  NN  /\  N  e.  NN0 )  -> 
( ( 0..^ M )  =  ( 0..^ N )  ->  M  =  N ) )
8786anim1d 567 . . . . . . . . . 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
) ) ) )
88 oveq2 6296 . . . . . . . . . . 11  |-  ( M  =  N  ->  (
0..^ M )  =  ( 0..^ N ) )
8988anim1i 571 . . . . . . . . . 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 ) ) )
9087, 89impbid1 207 . . . . . . . . 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 ) ) ) )
9190ex 436 . . . . . . . 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 ) ) ) ) )
9277, 91sylbir 217 . . . . . . 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 ) ) ) ) )
9392impancom 442 . . . . . 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 ) ) ) ) )
9476, 93syl5bir 222 . . . . 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 ) ) ) ) )
9594adantr 467 . . . 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 ) ) ) ) )
9695impcom 432 . . 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 ) ) ) )
9775, 96bitrd 257 . 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 ) ) ) )
9869, 97pm2.61ian 798 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 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   (/)c0 3730   class class class wbr 4401    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6288   0cc0 9536    < clt 9672    <_ cle 9673   NNcn 10606   NN0cn0 10866   ZZcz 10934  ..^cfzo 11912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-fzo 11913
This theorem is referenced by: (None)
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