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Theorem 2ffzoeq 32122
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 2471 . . . . . . . . . . . 12  |-  ( F  =  P  ->  ( F  =  (/)  <->  P  =  (/) ) )
21anbi1d 704 . . . . . . . . . . 11  |-  ( F  =  P  ->  (
( F  =  (/)  /\  P : ( 0..^ N ) --> Y )  <-> 
( P  =  (/)  /\  P : ( 0..^ N ) --> Y ) ) )
3 f0bi 5774 . . . . . . . . . . . . 13  |-  ( P : (/) --> Y  <->  P  =  (/) )
4 ffn 5737 . . . . . . . . . . . . . 14  |-  ( P : (/) --> Y  ->  P  Fn  (/) )
5 ffn 5737 . . . . . . . . . . . . . . 15  |-  ( P : ( 0..^ N ) --> Y  ->  P  Fn  ( 0..^ N ) )
6 fndmu 5688 . . . . . . . . . . . . . . . . 17  |-  ( ( P  Fn  ( 0..^ N )  /\  P  Fn  (/) )  ->  (
0..^ N )  =  (/) )
7 0z 10887 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  ZZ
8 nn0z 10899 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  N  e.  ZZ )
98adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  N  e.  ZZ )
10 fzon 11827 . . . . . . . . . . . . . . . . . . 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 10833 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  0  <_  N )
13 0re 9608 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  RR
1413a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  0  e.  RR )
15 nn0re 10816 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  N  e.  RR )
1614, 15letri3d 9738 . . . . . . . . . . . . . . . . . . . . 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 6303 . . . . . . . . . . . 12  |-  ( M  =  0  ->  (
0..^ M )  =  ( 0..^ 0 ) )
31 fzo0 11829 . . . . . . . . . . . 12  |-  ( 0..^ 0 )  =  (/)
3230, 31syl6eq 2524 . . . . . . . . . . 11  |-  ( M  =  0  ->  (
0..^ M )  =  (/) )
3332feq2d 5724 . . . . . . . . . 10  |-  ( M  =  0  ->  ( F : ( 0..^ M ) --> X  <->  F : (/) --> X ) )
34 f0bi 5774 . . . . . . . . . 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 2471 . . . . . . . . . 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 5724 . . . . . . . . . . . 12  |-  ( M  =  0  ->  ( F : ( 0..^ M ) --> X  <->  F :
( 0..^ 0 ) --> X ) )
4531feq2i 5730 . . . . . . . . . . . . 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 2471 . . . . . . . . . . . . 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 6303 . . . . . . . . . . . . 13  |-  ( N  =  0  ->  (
0..^ N )  =  ( 0..^ 0 ) )
5352feq2d 5724 . . . . . . . . . . . 12  |-  ( N  =  0  ->  ( P : ( 0..^ N ) --> Y  <->  P :
( 0..^ 0 ) --> Y ) )
5431feq2i 5730 . . . . . . . . . . . . 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 2495 . . . . . . . . 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 3938 . . . . . 6  |-  A. i  e.  (/)  ( F `  i )  =  ( P `  i )
6732raleqdv 3069 . . . . . 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 5737 . . . . . . 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 5983 . . . 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 2664 . . . . . 6  |-  ( M  =/=  0  <->  -.  M  =  0 )
79 elnnne0 10821 . . . . . . . 8  |-  ( M  e.  NN  <->  ( M  e.  NN0  /\  M  =/=  0 ) )
80 0zd 10888 . . . . . . . . . . . . . . 15  |-  ( M  e.  NN  ->  0  e.  ZZ )
81 nnz 10898 . . . . . . . . . . . . . . 15  |-  ( M  e.  NN  ->  M  e.  ZZ )
82 nngt0 10577 . . . . . . . . . . . . . . 15  |-  ( M  e.  NN  ->  0  <  M )
8380, 81, 823jca 1176 . . . . . . . . . . . . . 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 32121 . . . . . . . . . . . . 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 6303 . . . . . . . . . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   (/)c0 3790   class class class wbr 4453    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   RRcr 9503   0cc0 9504    < clt 9640    <_ cle 9641   NNcn 10548   NN0cn0 10807   ZZcz 10876  ..^cfzo 11804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805
This theorem is referenced by: (None)
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