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Theorem fz1isolem 12210
Description: Lemma for fz1iso 12211. (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 12122 . . . . . . . . 9  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
21adantl 463 . . . . . . . 8  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( # `  A
)  e.  NN0 )
3 nnuz 10892 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
4 1z 10672 . . . . . . . . . . . . 13  |-  1  e.  ZZ
5 fz1iso.1 . . . . . . . . . . . . 13  |-  G  =  ( rec ( ( n  e.  _V  |->  ( n  +  1 ) ) ,  1 )  |`  om )
64, 5om2uzisoi 11773 . . . . . . . . . . . 12  |-  G  Isom  _E  ,  <  ( om ,  ( ZZ>= `  1
) )
7 isoeq5 6011 . . . . . . . . . . . 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 6018 . . . . . . . . . 10  |-  ( G 
Isom  _E  ,  <  ( om ,  NN )  ->  `' G  Isom  <  ,  _E  ( NN ,  om ) )
119, 10ax-mp 5 . . . . . . . . 9  |-  `' G  Isom  <  ,  _E  ( NN ,  om )
12 nn0p1nn 10615 . . . . . . . . 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 5698 . . . . . . . . . . . . 13  |-  ( `' G `  ( (
# `  A )  +  1 ) )  e.  _V
1615epini 5196 . . . . . . . . . . . 12  |-  ( `'  _E  " { ( `' G `  ( (
# `  A )  +  1 ) ) } )  =  ( `' G `  ( (
# `  A )  +  1 ) )
1716ineq2i 3546 . . . . . . . . . . 11  |-  ( om 
i^i  ( `'  _E  " { ( `' G `  ( ( # `  A
)  +  1 ) ) } ) )  =  ( om  i^i  ( `' G `  ( (
# `  A )  +  1 ) ) )
1814, 17eqtr4i 2464 . . . . . . . . . 10  |-  C  =  ( om  i^i  ( `'  _E  " { ( `' G `  ( (
# `  A )  +  1 ) ) } ) )
1913, 18isoini2 6027 . . . . . . . . 9  |-  ( ( `' G  Isom  <  ,  _E  ( NN ,  om )  /\  ( ( # `  A )  +  1 )  e.  NN )  ->  ( `' G  |`  B )  Isom  <  ,  _E  ( B ,  C ) )
2011, 12, 19sylancr 658 . . . . . . . 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 10664 . . . . . . . . . . . . 13  |-  ( f  e.  NN  ->  f  e.  ZZ )
232nn0zd 10741 . . . . . . . . . . . . 13  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( # `  A
)  e.  ZZ )
24 eluz 10870 . . . . . . . . . . . . 13  |-  ( ( f  e.  ZZ  /\  ( # `  A )  e.  ZZ )  -> 
( ( # `  A
)  e.  ( ZZ>= `  f )  <->  f  <_  (
# `  A )
) )
2522, 23, 24syl2anr 475 . . . . . . . . . . . 12  |-  ( ( ( R  Or  A  /\  A  e.  Fin )  /\  f  e.  NN )  ->  ( ( # `  A )  e.  (
ZZ>= `  f )  <->  f  <_  (
# `  A )
) )
26 zleltp1 10691 . . . . . . . . . . . . . 14  |-  ( ( f  e.  ZZ  /\  ( # `  A )  e.  ZZ )  -> 
( f  <_  ( # `
 A )  <->  f  <  ( ( # `  A
)  +  1 ) ) )
2722, 23, 26syl2anr 475 . . . . . . . . . . . . 13  |-  ( ( ( R  Or  A  /\  A  e.  Fin )  /\  f  e.  NN )  ->  ( f  <_ 
( # `  A )  <-> 
f  <  ( ( # `
 A )  +  1 ) ) )
28 ovex 6115 . . . . . . . . . . . . . 14  |-  ( (
# `  A )  +  1 )  e. 
_V
29 vex 2973 . . . . . . . . . . . . . . 15  |-  f  e. 
_V
3029eliniseg 5195 . . . . . . . . . . . . . 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 636 . . . . . . . . . 10  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( ( f  e.  NN  /\  f  e.  ( `'  <  " {
( ( # `  A
)  +  1 ) } ) )  <->  ( f  e.  NN  /\  ( # `  A )  e.  (
ZZ>= `  f ) ) ) )
3513elin2 3538 . . . . . . . . . 10  |-  ( f  e.  B  <->  ( f  e.  NN  /\  f  e.  ( `'  <  " {
( ( # `  A
)  +  1 ) } ) ) )
36 elfzuzb 11443 . . . . . . . . . . 11  |-  ( f  e.  ( 1 ... ( # `  A
) )  <->  ( f  e.  ( ZZ>= `  1 )  /\  ( # `  A
)  e.  ( ZZ>= `  f ) ) )
37 elnnuz 10893 . . . . . . . . . . . 12  |-  ( f  e.  NN  <->  f  e.  ( ZZ>= `  1 )
)
3837anbi1i 690 . . . . . . . . . . 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 2439 . . . . . . . 8  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  B  =  ( 1 ... ( # `  A
) ) )
42 isoeq4 6010 . . . . . . . 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 7746 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  Fin  ->  dom  O  e.  On )
4746adantl 463 . . . . . . . . . . . . . . . 16  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom  O  e.  On )
48 simpr 458 . . . . . . . . . . . . . . . . 17  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  A  e.  Fin )
49 wofi 7557 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  R  We  A )
5045oien 7748 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  Fin  /\  R  We  A )  ->  dom  O  ~~  A
)
5148, 49, 50syl2anc 656 . . . . . . . . . . . . . . . . 17  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom  O  ~~  A
)
52 enfii 7526 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  Fin  /\  dom  O  ~~  A )  ->  dom  O  e.  Fin )
5348, 51, 52syl2anc 656 . . . . . . . . . . . . . . . 16  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom  O  e.  Fin )
5447, 53elind 3537 . . . . . . . . . . . . . . 15  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom  O  e.  ( On  i^i  Fin )
)
55 onfin2 7498 . . . . . . . . . . . . . . 15  |-  om  =  ( On  i^i  Fin )
5654, 55syl6eleqr 2532 . . . . . . . . . . . . . 14  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom  O  e.  om )
57 eqid 2441 . . . . . . . . . . . . . . . 16  |-  ( rec ( ( n  e. 
_V  |->  ( n  + 
1 ) ) ,  0 )  |`  om )  =  ( rec (
( n  e.  _V  |->  ( n  +  1
) ) ,  0 )  |`  om )
58 0z 10653 . . . . . . . . . . . . . . . 16  |-  0  e.  ZZ
595, 57, 4, 58uzrdgxfr 11785 . . . . . . . . . . . . . . 15  |-  ( dom 
O  e.  om  ->  ( G `  dom  O
)  =  ( ( ( rec ( ( n  e.  _V  |->  ( n  +  1 ) ) ,  0 )  |`  om ) `  dom  O )  +  ( 1  -  0 ) ) )
60 1m0e1 10428 . . . . . . . . . . . . . . . 16  |-  ( 1  -  0 )  =  1
6160oveq2i 6101 . . . . . . . . . . . . . . 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 2489 . . . . . . . . . . . . . 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 7356 . . . . . . . . . . . . . . . . 17  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  A  ~~  dom  O
)
65 cardennn 8149 . . . . . . . . . . . . . . . . 17  |-  ( ( A  ~~  dom  O  /\  dom  O  e.  om )  ->  ( card `  A
)  =  dom  O
)
6664, 56, 65syl2anc 656 . . . . . . . . . . . . . . . 16  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( card `  A
)  =  dom  O
)
6766fveq2d 5692 . . . . . . . . . . . . . . 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 12102 . . . . . . . . . . . . . . . 16  |-  ( A  e.  Fin  ->  (
( rec ( ( n  e.  _V  |->  ( n  +  1 ) ) ,  0 )  |`  om ) `  ( card `  A ) )  =  ( # `  A
) )
6968adantl 463 . . . . . . . . . . . . . . 15  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( ( rec (
( n  e.  _V  |->  ( n  +  1
) ) ,  0 )  |`  om ) `  ( card `  A
) )  =  (
# `  A )
)
7067, 69eqtr3d 2475 . . . . . . . . . . . . . 14  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( ( rec (
( n  e.  _V  |->  ( n  +  1
) ) ,  0 )  |`  om ) `  dom  O )  =  ( # `  A
) )
7170oveq1d 6105 . . . . . . . . . . . . 13  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( ( ( rec ( ( n  e. 
_V  |->  ( n  + 
1 ) ) ,  0 )  |`  om ) `  dom  O )  +  1 )  =  ( ( # `  A
)  +  1 ) )
7263, 71eqtrd 2473 . . . . . . . . . . . 12  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( G `  dom  O )  =  ( (
# `  A )  +  1 ) )
7372fveq2d 5692 . . . . . . . . . . 11  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( `' G `  ( G `  dom  O
) )  =  ( `' G `  ( (
# `  A )  +  1 ) ) )
74 isof1o 6013 . . . . . . . . . . . . 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 5980 . . . . . . . . . . . 12  |-  ( ( G : om -1-1-onto-> NN  /\  dom  O  e.  om )  ->  ( `' G `  ( G `
 dom  O )
)  =  dom  O
)
7775, 56, 76sylancr 658 . . . . . . . . . . 11  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( `' G `  ( G `  dom  O
) )  =  dom  O )
7873, 77eqtr3d 2475 . . . . . . . . . 10  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( `' G `  ( ( # `  A
)  +  1 ) )  =  dom  O
)
7978ineq2d 3549 . . . . . . . . 9  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( om  i^i  ( `' G `  ( (
# `  A )  +  1 ) ) )  =  ( om 
i^i  dom  O ) )
80 ordom 6484 . . . . . . . . . . 11  |-  Ord  om
81 ordelss 4731 . . . . . . . . . . 11  |-  ( ( Ord  om  /\  dom  O  e.  om )  ->  dom  O  C_  om )
8280, 56, 81sylancr 658 . . . . . . . . . 10  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  dom  O  C_  om )
83 sseqin2 3566 . . . . . . . . . 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 2473 . . . . . . . 8  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( om  i^i  ( `' G `  ( (
# `  A )  +  1 ) ) )  =  dom  O
)
8614, 85syl5eq 2485 . . . . . . 7  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  C  =  dom  O
)
87 isoeq5 6011 . . . . . . 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 7747 . . . . . 6  |-  ( ( A  e.  Fin  /\  R  We  A )  ->  O  Isom  _E  ,  R  ( dom  O ,  A
) )
9148, 49, 90syl2anc 656 . . . . 5  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  O  Isom  _E  ,  R  ( dom  O ,  A
) )
92 isotr 6024 . . . . 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 656 . . . 4  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( O  o.  ( `' G  |`  B ) )  Isom  <  ,  R  ( ( 1 ... ( # `  A
) ) ,  A
) )
94 isof1o 6013 . . . 4  |-  ( ( O  o.  ( `' G  |`  B )
)  Isom  <  ,  R  ( ( 1 ... ( # `  A
) ) ,  A
)  ->  ( O  o.  ( `' G  |`  B ) ) : ( 1 ... ( # `
 A ) ) -1-1-onto-> A )
95 f1of 5638 . . . 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 11791 . . 3  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( 1 ... ( # `
 A ) )  e.  Fin )
98 fex 5947 . . 3  |-  ( ( ( O  o.  ( `' G  |`  B ) ) : ( 1 ... ( # `  A
) ) --> A  /\  ( 1 ... ( # `
 A ) )  e.  Fin )  -> 
( O  o.  ( `' G  |`  B ) )  e.  _V )
9996, 97, 98syl2anc 656 . 2  |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  ( O  o.  ( `' G  |`  B ) )  e.  _V )
100 isoeq1 6007 . . 3  |-  ( f  =  ( O  o.  ( `' G  |`  B ) )  ->  ( f  Isom  <  ,  R  ( ( 1 ... ( # `
 A ) ) ,  A )  <->  ( O  o.  ( `' G  |`  B ) )  Isom  <  ,  R  ( (
1 ... ( # `  A
) ) ,  A
) ) )
101100spcegv 3055 . 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 1364   E.wex 1591    e. wcel 1761   _Vcvv 2970    i^i cin 3324    C_ wss 3325   {csn 3874   class class class wbr 4289    e. cmpt 4347    _E cep 4626    Or wor 4636    We wwe 4674   Ord word 4714   Oncon0 4715   `'ccnv 4835   dom cdm 4836    |` cres 4838   "cima 4839    o. ccom 4840   -->wf 5411   -1-1-onto->wf1o 5414   ` cfv 5415    Isom wiso 5416  (class class class)co 6090   omcom 6475   reccrdg 6861    ~~ cen 7303   Fincfn 7306  OrdIsocoi 7719   cardccrd 8101   0cc0 9278   1c1 9279    + caddc 9281    < clt 9414    <_ cle 9415    - cmin 9591   NNcn 10318   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433   #chash 12099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-hash 12100
This theorem is referenced by:  fz1iso  12211
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