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Theorem fz1isolem 12229
Description: Lemma for fz1iso 12230. (Contributed by Mario Carneiro, 2-Apr-2014.)
Hypotheses
Ref Expression
fz1iso.1  |-  G  =  ( rec ( ( n  e.  _V  |->  ( n  +  1 ) ) ,  1 )  |`  om )
fz1iso.2  |-  B  =  ( NN  i^i  ( `'  <  " { ( (
# `  A )  +  1 ) } ) )
fz1iso.3  |-  C  =  ( om  i^i  ( `' G `  ( (
# `  A )  +  1 ) ) )
fz1iso.4  |-  O  = OrdIso
( R ,  A
)
Assertion
Ref Expression
fz1isolem  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E. f  f  Isom  <  ,  R  ( (
1 ... ( # `  A
) ) ,  A
) )
Distinct variable groups:    f, n, A    B, f    f, G   
f, O    R, f
Allowed substitution hints:    B( n)    C( f, n)    R( n)    G( n)    O( n)

Proof of Theorem fz1isolem
StepHypRef Expression
1 hashcl 12141 . . . . . . . . 9  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
21adantl 466 . . . . . . . 8  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( # `  A
)  e.  NN0 )
3 nnuz 10911 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
4 1z 10691 . . . . . . . . . . . . 13  |-  1  e.  ZZ
5 fz1iso.1 . . . . . . . . . . . . 13  |-  G  =  ( rec ( ( n  e.  _V  |->  ( n  +  1 ) ) ,  1 )  |`  om )
64, 5om2uzisoi 11792 . . . . . . . . . . . 12  |-  G  Isom  _E  ,  <  ( om ,  ( ZZ>= `  1
) )
7 isoeq5 6029 . . . . . . . . . . . 12  |-  ( NN  =  ( ZZ>= `  1
)  ->  ( G  Isom  _E  ,  <  ( om ,  NN )  <->  G 
Isom  _E  ,  <  ( om ,  ( ZZ>= ` 
1 ) ) ) )
86, 7mpbiri 233 . . . . . . . . . . 11  |-  ( NN  =  ( ZZ>= `  1
)  ->  G  Isom  _E  ,  <  ( om ,  NN ) )
93, 8ax-mp 5 . . . . . . . . . 10  |-  G  Isom  _E  ,  <  ( om ,  NN )
10 isocnv 6036 . . . . . . . . . 10  |-  ( G 
Isom  _E  ,  <  ( om ,  NN )  ->  `' G  Isom  <  ,  _E  ( NN ,  om ) )
119, 10ax-mp 5 . . . . . . . . 9  |-  `' G  Isom  <  ,  _E  ( NN ,  om )
12 nn0p1nn 10634 . . . . . . . . 9  |-  ( (
# `  A )  e.  NN0  ->  ( ( # `
 A )  +  1 )  e.  NN )
13 fz1iso.2 . . . . . . . . . 10  |-  B  =  ( NN  i^i  ( `'  <  " { ( (
# `  A )  +  1 ) } ) )
14 fz1iso.3 . . . . . . . . . . 11  |-  C  =  ( om  i^i  ( `' G `  ( (
# `  A )  +  1 ) ) )
15 fvex 5716 . . . . . . . . . . . . 13  |-  ( `' G `  ( (
# `  A )  +  1 ) )  e.  _V
1615epini 5214 . . . . . . . . . . . 12  |-  ( `'  _E  " { ( `' G `  ( (
# `  A )  +  1 ) ) } )  =  ( `' G `  ( (
# `  A )  +  1 ) )
1716ineq2i 3564 . . . . . . . . . . 11  |-  ( om 
i^i  ( `'  _E  " { ( `' G `  ( ( # `  A
)  +  1 ) ) } ) )  =  ( om  i^i  ( `' G `  ( (
# `  A )  +  1 ) ) )
1814, 17eqtr4i 2466 . . . . . . . . . 10  |-  C  =  ( om  i^i  ( `'  _E  " { ( `' G `  ( (
# `  A )  +  1 ) ) } ) )
1913, 18isoini2 6045 . . . . . . . . 9  |-  ( ( `' G  Isom  <  ,  _E  ( NN ,  om )  /\  ( ( # `  A )  +  1 )  e.  NN )  ->  ( `' G  |`  B )  Isom  <  ,  _E  ( B ,  C ) )
2011, 12, 19sylancr 663 . . . . . . . 8  |-  ( (
# `  A )  e.  NN0  ->  ( `' G  |`  B )  Isom  <  ,  _E  ( B ,  C ) )
212, 20syl 16 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( `' G  |`  B )  Isom  <  ,  _E  ( B ,  C ) )
22 nnz 10683 . . . . . . . . . . . . 13  |-  ( f  e.  NN  ->  f  e.  ZZ )
232nn0zd 10760 . . . . . . . . . . . . 13  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( # `  A
)  e.  ZZ )
24 eluz 10889 . . . . . . . . . . . . 13  |-  ( ( f  e.  ZZ  /\  ( # `  A )  e.  ZZ )  -> 
( ( # `  A
)  e.  ( ZZ>= `  f )  <->  f  <_  (
# `  A )
) )
2522, 23, 24syl2anr 478 . . . . . . . . . . . 12  |-  ( ( ( R  Or  A  /\  A  e.  Fin )  /\  f  e.  NN )  ->  ( ( # `  A )  e.  (
ZZ>= `  f )  <->  f  <_  (
# `  A )
) )
26 zleltp1 10710 . . . . . . . . . . . . . 14  |-  ( ( f  e.  ZZ  /\  ( # `  A )  e.  ZZ )  -> 
( f  <_  ( # `
 A )  <->  f  <  ( ( # `  A
)  +  1 ) ) )
2722, 23, 26syl2anr 478 . . . . . . . . . . . . 13  |-  ( ( ( R  Or  A  /\  A  e.  Fin )  /\  f  e.  NN )  ->  ( f  <_ 
( # `  A )  <-> 
f  <  ( ( # `
 A )  +  1 ) ) )
28 ovex 6131 . . . . . . . . . . . . . 14  |-  ( (
# `  A )  +  1 )  e. 
_V
29 vex 2990 . . . . . . . . . . . . . . 15  |-  f  e. 
_V
3029eliniseg 5213 . . . . . . . . . . . . . 14  |-  ( ( ( # `  A
)  +  1 )  e.  _V  ->  (
f  e.  ( `'  <  " { ( (
# `  A )  +  1 ) } )  <->  f  <  (
( # `  A )  +  1 ) ) )
3128, 30ax-mp 5 . . . . . . . . . . . . 13  |-  ( f  e.  ( `'  <  " { ( ( # `  A )  +  1 ) } )  <->  f  <  ( ( # `  A
)  +  1 ) )
3227, 31syl6bbr 263 . . . . . . . . . . . 12  |-  ( ( ( R  Or  A  /\  A  e.  Fin )  /\  f  e.  NN )  ->  ( f  <_ 
( # `  A )  <-> 
f  e.  ( `'  <  " { ( (
# `  A )  +  1 ) } ) ) )
3325, 32bitr2d 254 . . . . . . . . . . 11  |-  ( ( ( R  Or  A  /\  A  e.  Fin )  /\  f  e.  NN )  ->  ( f  e.  ( `'  <  " {
( ( # `  A
)  +  1 ) } )  <->  ( # `  A
)  e.  ( ZZ>= `  f ) ) )
3433pm5.32da 641 . . . . . . . . . 10  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( ( f  e.  NN  /\  f  e.  ( `'  <  " {
( ( # `  A
)  +  1 ) } ) )  <->  ( f  e.  NN  /\  ( # `  A )  e.  (
ZZ>= `  f ) ) ) )
3513elin2 3556 . . . . . . . . . 10  |-  ( f  e.  B  <->  ( f  e.  NN  /\  f  e.  ( `'  <  " {
( ( # `  A
)  +  1 ) } ) ) )
36 elfzuzb 11462 . . . . . . . . . . 11  |-  ( f  e.  ( 1 ... ( # `  A
) )  <->  ( f  e.  ( ZZ>= `  1 )  /\  ( # `  A
)  e.  ( ZZ>= `  f ) ) )
37 elnnuz 10912 . . . . . . . . . . . 12  |-  ( f  e.  NN  <->  f  e.  ( ZZ>= `  1 )
)
3837anbi1i 695 . . . . . . . . . . 11  |-  ( ( f  e.  NN  /\  ( # `  A )  e.  ( ZZ>= `  f
) )  <->  ( f  e.  ( ZZ>= `  1 )  /\  ( # `  A
)  e.  ( ZZ>= `  f ) ) )
3936, 38bitr4i 252 . . . . . . . . . 10  |-  ( f  e.  ( 1 ... ( # `  A
) )  <->  ( f  e.  NN  /\  ( # `  A )  e.  (
ZZ>= `  f ) ) )
4034, 35, 393bitr4g 288 . . . . . . . . 9  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( f  e.  B  <->  f  e.  ( 1 ... ( # `  A
) ) ) )
4140eqrdv 2441 . . . . . . . 8  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  B  =  ( 1 ... ( # `  A
) ) )
42 isoeq4 6028 . . . . . . . 8  |-  ( B  =  ( 1 ... ( # `  A
) )  ->  (
( `' G  |`  B )  Isom  <  ,  _E  ( B ,  C )  <->  ( `' G  |`  B )  Isom  <  ,  _E  ( (
1 ... ( # `  A
) ) ,  C
) ) )
4341, 42syl 16 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( ( `' G  |`  B )  Isom  <  ,  _E  ( B ,  C )  <->  ( `' G  |`  B )  Isom  <  ,  _E  ( (
1 ... ( # `  A
) ) ,  C
) ) )
4421, 43mpbid 210 . . . . . 6  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( `' G  |`  B )  Isom  <  ,  _E  ( ( 1 ... ( # `  A
) ) ,  C
) )
45 fz1iso.4 . . . . . . . . . . . . . . . . . 18  |-  O  = OrdIso
( R ,  A
)
4645oion 7765 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  Fin  ->  dom  O  e.  On )
4746adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom  O  e.  On )
48 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  A  e.  Fin )
49 wofi 7576 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  R  We  A )
5045oien 7767 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  Fin  /\  R  We  A )  ->  dom  O  ~~  A
)
5148, 49, 50syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom  O  ~~  A
)
52 enfii 7545 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  Fin  /\  dom  O  ~~  A )  ->  dom  O  e.  Fin )
5348, 51, 52syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom  O  e.  Fin )
5447, 53elind 3555 . . . . . . . . . . . . . . 15  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom  O  e.  ( On  i^i  Fin )
)
55 onfin2 7517 . . . . . . . . . . . . . . 15  |-  om  =  ( On  i^i  Fin )
5654, 55syl6eleqr 2534 . . . . . . . . . . . . . 14  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom  O  e.  om )
57 eqid 2443 . . . . . . . . . . . . . . . 16  |-  ( rec ( ( n  e. 
_V  |->  ( n  + 
1 ) ) ,  0 )  |`  om )  =  ( rec (
( n  e.  _V  |->  ( n  +  1
) ) ,  0 )  |`  om )
58 0z 10672 . . . . . . . . . . . . . . . 16  |-  0  e.  ZZ
595, 57, 4, 58uzrdgxfr 11804 . . . . . . . . . . . . . . 15  |-  ( dom 
O  e.  om  ->  ( G `  dom  O
)  =  ( ( ( rec ( ( n  e.  _V  |->  ( n  +  1 ) ) ,  0 )  |`  om ) `  dom  O )  +  ( 1  -  0 ) ) )
60 1m0e1 10447 . . . . . . . . . . . . . . . 16  |-  ( 1  -  0 )  =  1
6160oveq2i 6117 . . . . . . . . . . . . . . 15  |-  ( ( ( rec ( ( n  e.  _V  |->  ( n  +  1 ) ) ,  0 )  |`  om ) `  dom  O )  +  ( 1  -  0 ) )  =  ( ( ( rec ( ( n  e.  _V  |->  ( n  +  1 ) ) ,  0 )  |`  om ) `  dom  O
)  +  1 )
6259, 61syl6eq 2491 . . . . . . . . . . . . . 14  |-  ( dom 
O  e.  om  ->  ( G `  dom  O
)  =  ( ( ( rec ( ( n  e.  _V  |->  ( n  +  1 ) ) ,  0 )  |`  om ) `  dom  O )  +  1 ) )
6356, 62syl 16 . . . . . . . . . . . . 13  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( G `  dom  O )  =  ( ( ( rec ( ( n  e.  _V  |->  ( n  +  1 ) ) ,  0 )  |`  om ) `  dom  O )  +  1 ) )
6451ensymd 7375 . . . . . . . . . . . . . . . . 17  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  A  ~~  dom  O
)
65 cardennn 8168 . . . . . . . . . . . . . . . . 17  |-  ( ( A  ~~  dom  O  /\  dom  O  e.  om )  ->  ( card `  A
)  =  dom  O
)
6664, 56, 65syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( card `  A
)  =  dom  O
)
6766fveq2d 5710 . . . . . . . . . . . . . . 15  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( ( rec (
( n  e.  _V  |->  ( n  +  1
) ) ,  0 )  |`  om ) `  ( card `  A
) )  =  ( ( rec ( ( n  e.  _V  |->  ( n  +  1 ) ) ,  0 )  |`  om ) `  dom  O ) )
6857hashgval 12121 . . . . . . . . . . . . . . . 16  |-  ( A  e.  Fin  ->  (
( rec ( ( n  e.  _V  |->  ( n  +  1 ) ) ,  0 )  |`  om ) `  ( card `  A ) )  =  ( # `  A
) )
6968adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( ( rec (
( n  e.  _V  |->  ( n  +  1
) ) ,  0 )  |`  om ) `  ( card `  A
) )  =  (
# `  A )
)
7067, 69eqtr3d 2477 . . . . . . . . . . . . . 14  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( ( rec (
( n  e.  _V  |->  ( n  +  1
) ) ,  0 )  |`  om ) `  dom  O )  =  ( # `  A
) )
7170oveq1d 6121 . . . . . . . . . . . . 13  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( ( ( rec ( ( n  e. 
_V  |->  ( n  + 
1 ) ) ,  0 )  |`  om ) `  dom  O )  +  1 )  =  ( ( # `  A
)  +  1 ) )
7263, 71eqtrd 2475 . . . . . . . . . . . 12  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( G `  dom  O )  =  ( (
# `  A )  +  1 ) )
7372fveq2d 5710 . . . . . . . . . . 11  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( `' G `  ( G `  dom  O
) )  =  ( `' G `  ( (
# `  A )  +  1 ) ) )
74 isof1o 6031 . . . . . . . . . . . . 13  |-  ( G 
Isom  _E  ,  <  ( om ,  NN )  ->  G : om -1-1-onto-> NN )
759, 74ax-mp 5 . . . . . . . . . . . 12  |-  G : om
-1-1-onto-> NN
76 f1ocnvfv1 5998 . . . . . . . . . . . 12  |-  ( ( G : om -1-1-onto-> NN  /\  dom  O  e.  om )  ->  ( `' G `  ( G `
 dom  O )
)  =  dom  O
)
7775, 56, 76sylancr 663 . . . . . . . . . . 11  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( `' G `  ( G `  dom  O
) )  =  dom  O )
7873, 77eqtr3d 2477 . . . . . . . . . 10  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( `' G `  ( ( # `  A
)  +  1 ) )  =  dom  O
)
7978ineq2d 3567 . . . . . . . . 9  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( om  i^i  ( `' G `  ( (
# `  A )  +  1 ) ) )  =  ( om 
i^i  dom  O ) )
80 ordom 6500 . . . . . . . . . . 11  |-  Ord  om
81 ordelss 4750 . . . . . . . . . . 11  |-  ( ( Ord  om  /\  dom  O  e.  om )  ->  dom  O  C_  om )
8280, 56, 81sylancr 663 . . . . . . . . . 10  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom  O  C_  om )
83 sseqin2 3584 . . . . . . . . . 10  |-  ( dom 
O  C_  om  <->  ( om  i^i  dom  O )  =  dom  O )
8482, 83sylib 196 . . . . . . . . 9  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( om  i^i  dom  O )  =  dom  O
)
8579, 84eqtrd 2475 . . . . . . . 8  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( om  i^i  ( `' G `  ( (
# `  A )  +  1 ) ) )  =  dom  O
)
8614, 85syl5eq 2487 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  C  =  dom  O
)
87 isoeq5 6029 . . . . . . 7  |-  ( C  =  dom  O  -> 
( ( `' G  |`  B )  Isom  <  ,  _E  ( ( 1 ... ( # `  A
) ) ,  C
)  <->  ( `' G  |`  B )  Isom  <  ,  _E  ( ( 1 ... ( # `  A
) ) ,  dom  O ) ) )
8886, 87syl 16 . . . . . 6  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( ( `' G  |`  B )  Isom  <  ,  _E  ( ( 1 ... ( # `  A
) ) ,  C
)  <->  ( `' G  |`  B )  Isom  <  ,  _E  ( ( 1 ... ( # `  A
) ) ,  dom  O ) ) )
8944, 88mpbid 210 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( `' G  |`  B )  Isom  <  ,  _E  ( ( 1 ... ( # `  A
) ) ,  dom  O ) )
9045oiiso 7766 . . . . . 6  |-  ( ( A  e.  Fin  /\  R  We  A )  ->  O  Isom  _E  ,  R  ( dom  O ,  A
) )
9148, 49, 90syl2anc 661 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  O  Isom  _E  ,  R  ( dom  O ,  A
) )
92 isotr 6042 . . . . 5  |-  ( ( ( `' G  |`  B )  Isom  <  ,  _E  ( ( 1 ... ( # `  A
) ) ,  dom  O )  /\  O  Isom  _E  ,  R  ( dom 
O ,  A ) )  ->  ( O  o.  ( `' G  |`  B ) )  Isom  <  ,  R  ( (
1 ... ( # `  A
) ) ,  A
) )
9389, 91, 92syl2anc 661 . . . 4  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( O  o.  ( `' G  |`  B ) )  Isom  <  ,  R  ( ( 1 ... ( # `  A
) ) ,  A
) )
94 isof1o 6031 . . . 4  |-  ( ( O  o.  ( `' G  |`  B )
)  Isom  <  ,  R  ( ( 1 ... ( # `  A
) ) ,  A
)  ->  ( O  o.  ( `' G  |`  B ) ) : ( 1 ... ( # `
 A ) ) -1-1-onto-> A )
95 f1of 5656 . . . 4  |-  ( ( O  o.  ( `' G  |`  B )
) : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  ( O  o.  ( `' G  |`  B ) ) : ( 1 ... ( # `
 A ) ) --> A )
9693, 94, 953syl 20 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( O  o.  ( `' G  |`  B ) ) : ( 1 ... ( # `  A
) ) --> A )
97 fzfid 11810 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( 1 ... ( # `
 A ) )  e.  Fin )
98 fex 5965 . . 3  |-  ( ( ( O  o.  ( `' G  |`  B ) ) : ( 1 ... ( # `  A
) ) --> A  /\  ( 1 ... ( # `
 A ) )  e.  Fin )  -> 
( O  o.  ( `' G  |`  B ) )  e.  _V )
9996, 97, 98syl2anc 661 . 2  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( O  o.  ( `' G  |`  B ) )  e.  _V )
100 isoeq1 6025 . . 3  |-  ( f  =  ( O  o.  ( `' G  |`  B ) )  ->  ( f  Isom  <  ,  R  ( ( 1 ... ( # `
 A ) ) ,  A )  <->  ( O  o.  ( `' G  |`  B ) )  Isom  <  ,  R  ( (
1 ... ( # `  A
) ) ,  A
) ) )
101100spcegv 3073 . 2  |-  ( ( O  o.  ( `' G  |`  B )
)  e.  _V  ->  ( ( O  o.  ( `' G  |`  B ) )  Isom  <  ,  R  ( ( 1 ... ( # `  A
) ) ,  A
)  ->  E. f 
f  Isom  <  ,  R  ( ( 1 ... ( # `  A
) ) ,  A
) ) )
10299, 93, 101sylc 60 1  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E. f  f  Isom  <  ,  R  ( (
1 ... ( # `  A
) ) ,  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2987    i^i cin 3342    C_ wss 3343   {csn 3892   class class class wbr 4307    e. cmpt 4365    _E cep 4645    Or wor 4655    We wwe 4693   Ord word 4733   Oncon0 4734   `'ccnv 4854   dom cdm 4855    |` cres 4857   "cima 4858    o. ccom 4859   -->wf 5429   -1-1-onto->wf1o 5432   ` cfv 5433    Isom wiso 5434  (class class class)co 6106   omcom 6491   reccrdg 6880    ~~ cen 7322   Fincfn 7325  OrdIsocoi 7738   cardccrd 8120   0cc0 9297   1c1 9298    + caddc 9300    < clt 9433    <_ cle 9434    - cmin 9610   NNcn 10337   NN0cn0 10594   ZZcz 10661   ZZ>=cuz 10876   ...cfz 11452   #chash 12118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-oi 7739  df-card 8124  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-n0 10595  df-z 10662  df-uz 10877  df-fz 11453  df-hash 12119
This theorem is referenced by:  fz1iso  12230
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