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Theorem frecuzrdgfn 9362
Description: The recursive definition generator on upper integers is a function. See comment in frec2uz0d 9349 for the description of  G as the mapping from  om to  ( ZZ>= `  C
). (Contributed by Jim Kingdon, 26-May-2020.)
Hypotheses
Ref Expression
frec2uz.1  |-  ( ph  ->  C  e.  ZZ )
frec2uz.2  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
uzrdg.s  |-  ( ph  ->  S  e.  V )
uzrdg.a  |-  ( ph  ->  A  e.  S )
uzrdg.f  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
uzrdg.2  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
frecuzrdgfn.3  |-  ( ph  ->  T  =  ran  R
)
Assertion
Ref Expression
frecuzrdgfn  |-  ( ph  ->  T  Fn  ( ZZ>= `  C ) )
Distinct variable groups:    y, A    x, C, y    y, G    x, F, y    x, S, y    ph, x, y
Allowed substitution hints:    A( x)    R( x, y)    T( x, y)    G( x)    V( x, y)

Proof of Theorem frecuzrdgfn
Dummy variables  w  z  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frecuzrdgfn.3 . . . . . . . . 9  |-  ( ph  ->  T  =  ran  R
)
21eleq2d 2123 . . . . . . . 8  |-  ( ph  ->  ( z  e.  T  <->  z  e.  ran  R ) )
3 frec2uz.1 . . . . . . . . . 10  |-  ( ph  ->  C  e.  ZZ )
4 frec2uz.2 . . . . . . . . . 10  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  C )
5 uzrdg.s . . . . . . . . . 10  |-  ( ph  ->  S  e.  V )
6 uzrdg.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  S )
7 uzrdg.f . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
8 uzrdg.2 . . . . . . . . . 10  |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )
93, 4, 5, 6, 7, 8frecuzrdgrom 9360 . . . . . . . . 9  |-  ( ph  ->  R  Fn  om )
10 fvelrnb 5249 . . . . . . . . 9  |-  ( R  Fn  om  ->  (
z  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  z ) )
119, 10syl 14 . . . . . . . 8  |-  ( ph  ->  ( z  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  z ) )
122, 11bitrd 181 . . . . . . 7  |-  ( ph  ->  ( z  e.  T  <->  E. w  e.  om  ( R `  w )  =  z ) )
133, 4, 5, 6, 7, 8frecuzrdgrrn 9358 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  om )  ->  ( R `  w )  e.  ( ( ZZ>= `  C )  X.  S ) )
14 eleq1 2116 . . . . . . . . 9  |-  ( ( R `  w )  =  z  ->  (
( R `  w
)  e.  ( (
ZZ>= `  C )  X.  S )  <->  z  e.  ( ( ZZ>= `  C
)  X.  S ) ) )
1513, 14syl5ibcom 148 . . . . . . . 8  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  z  ->  z  e.  ( ( ZZ>= `  C
)  X.  S ) ) )
1615rexlimdva 2450 . . . . . . 7  |-  ( ph  ->  ( E. w  e. 
om  ( R `  w )  =  z  ->  z  e.  ( ( ZZ>= `  C )  X.  S ) ) )
1712, 16sylbid 143 . . . . . 6  |-  ( ph  ->  ( z  e.  T  ->  z  e.  ( (
ZZ>= `  C )  X.  S ) ) )
1817ssrdv 2979 . . . . 5  |-  ( ph  ->  T  C_  ( ( ZZ>=
`  C )  X.  S ) )
19 xpss 4474 . . . . 5  |-  ( (
ZZ>= `  C )  X.  S )  C_  ( _V  X.  _V )
2018, 19syl6ss 2985 . . . 4  |-  ( ph  ->  T  C_  ( _V  X.  _V ) )
21 df-rel 4380 . . . 4  |-  ( Rel 
T  <->  T  C_  ( _V 
X.  _V ) )
2220, 21sylibr 141 . . 3  |-  ( ph  ->  Rel  T )
233, 4frec2uzf1od 9356 . . . . . . . . . 10  |-  ( ph  ->  G : om -1-1-onto-> ( ZZ>= `  C )
)
24 f1ocnvdm 5449 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  v  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  v )  e.  om )
2523, 24sylan 271 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( `' G `  v )  e.  om )
263, 4, 5, 6, 7, 8frecuzrdgrrn 9358 . . . . . . . . 9  |-  ( (
ph  /\  ( `' G `  v )  e.  om )  ->  ( R `  ( `' G `  v )
)  e.  ( (
ZZ>= `  C )  X.  S ) )
2725, 26syldan 270 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( R `  ( `' G `  v ) )  e.  ( ( ZZ>= `  C
)  X.  S ) )
28 xp2nd 5821 . . . . . . . 8  |-  ( ( R `  ( `' G `  v ) )  e.  ( (
ZZ>= `  C )  X.  S )  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  e.  S
)
2927, 28syl 14 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  e.  S
)
301eleq2d 2123 . . . . . . . . . . 11  |-  ( ph  ->  ( <. v ,  z
>.  e.  T  <->  <. v ,  z >.  e.  ran  R ) )
31 fvelrnb 5249 . . . . . . . . . . . 12  |-  ( R  Fn  om  ->  ( <. v ,  z >.  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  <. v ,  z >. )
)
329, 31syl 14 . . . . . . . . . . 11  |-  ( ph  ->  ( <. v ,  z
>.  e.  ran  R  <->  E. w  e.  om  ( R `  w )  =  <. v ,  z >. )
)
3330, 32bitrd 181 . . . . . . . . . 10  |-  ( ph  ->  ( <. v ,  z
>.  e.  T  <->  E. w  e.  om  ( R `  w )  =  <. v ,  z >. )
)
343adantr 265 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  om )  ->  C  e.  ZZ )
355adantr 265 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  om )  ->  S  e.  V )
366adantr 265 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  om )  ->  A  e.  S )
377adantlr 454 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  w  e.  om )  /\  (
x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
38 simpr 107 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  w  e.  om )  ->  w  e.  om )
3934, 4, 35, 36, 37, 8, 38frec2uzrdg 9359 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  w  e.  om )  ->  ( R `  w )  =  <. ( G `  w ) ,  ( 2nd `  ( R `  w )
) >. )
4039eqeq1d 2064 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>. 
<-> 
<. ( G `  w
) ,  ( 2nd `  ( R `  w
) ) >.  =  <. v ,  z >. )
)
41 vex 2577 . . . . . . . . . . . . . . . . . . 19  |-  v  e. 
_V
42 vex 2577 . . . . . . . . . . . . . . . . . . 19  |-  z  e. 
_V
4341, 42opth2 4005 . . . . . . . . . . . . . . . . . 18  |-  ( <.
( G `  w
) ,  ( 2nd `  ( R `  w
) ) >.  =  <. v ,  z >.  <->  ( ( G `  w )  =  v  /\  ( 2nd `  ( R `  w ) )  =  z ) )
4443simplbi 263 . . . . . . . . . . . . . . . . 17  |-  ( <.
( G `  w
) ,  ( 2nd `  ( R `  w
) ) >.  =  <. v ,  z >.  ->  ( G `  w )  =  v )
4540, 44syl6bi 156 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>.  ->  ( G `  w )  =  v ) )
46 f1ocnvfv 5447 . . . . . . . . . . . . . . . . 17  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  w  e.  om )  ->  ( ( G `
 w )  =  v  ->  ( `' G `  v )  =  w ) )
4723, 46sylan 271 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  w  e.  om )  ->  ( ( G `  w )  =  v  ->  ( `' G `  v )  =  w ) )
4845, 47syld 44 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>.  ->  ( `' G `  v )  =  w ) )
49 fveq2 5206 . . . . . . . . . . . . . . . 16  |-  ( ( `' G `  v )  =  w  ->  ( R `  ( `' G `  v )
)  =  ( R `
 w ) )
5049fveq2d 5210 . . . . . . . . . . . . . . 15  |-  ( ( `' G `  v )  =  w  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  =  ( 2nd `  ( R `
 w ) ) )
5148, 50syl6 33 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>.  ->  ( 2nd `  ( R `  ( `' G `  v )
) )  =  ( 2nd `  ( R `
 w ) ) ) )
5251imp 119 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  e.  om )  /\  ( R `  w )  =  <. v ,  z
>. )  ->  ( 2nd `  ( R `  ( `' G `  v ) ) )  =  ( 2nd `  ( R `
 w ) ) )
5341, 42op2ndd 5804 . . . . . . . . . . . . . 14  |-  ( ( R `  w )  =  <. v ,  z
>.  ->  ( 2nd `  ( R `  w )
)  =  z )
5453adantl 266 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  e.  om )  /\  ( R `  w )  =  <. v ,  z
>. )  ->  ( 2nd `  ( R `  w
) )  =  z )
5552, 54eqtr2d 2089 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  e.  om )  /\  ( R `  w )  =  <. v ,  z
>. )  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) )
5655ex 112 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  om )  ->  ( ( R `  w )  =  <. v ,  z
>.  ->  z  =  ( 2nd `  ( R `
 ( `' G `  v ) ) ) ) )
5756rexlimdva 2450 . . . . . . . . . 10  |-  ( ph  ->  ( E. w  e. 
om  ( R `  w )  =  <. v ,  z >.  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) ) )
5833, 57sylbid 143 . . . . . . . . 9  |-  ( ph  ->  ( <. v ,  z
>.  e.  T  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) ) )
5958alrimiv 1770 . . . . . . . 8  |-  ( ph  ->  A. z ( <.
v ,  z >.  e.  T  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) ) )
6059adantr 265 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  A. z
( <. v ,  z
>.  e.  T  ->  z  =  ( 2nd `  ( R `  ( `' G `  v )
) ) ) )
61 eqeq2 2065 . . . . . . . . . 10  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  v )
) )  ->  (
z  =  w  <->  z  =  ( 2nd `  ( R `
 ( `' G `  v ) ) ) ) )
6261imbi2d 223 . . . . . . . . 9  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  v )
) )  ->  (
( <. v ,  z
>.  e.  T  ->  z  =  w )  <->  ( <. v ,  z >.  e.  T  ->  z  =  ( 2nd `  ( R `  ( `' G `  v ) ) ) ) ) )
6362albidv 1721 . . . . . . . 8  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  v )
) )  ->  ( A. z ( <. v ,  z >.  e.  T  ->  z  =  w )  <->  A. z ( <. v ,  z >.  e.  T  ->  z  =  ( 2nd `  ( R `  ( `' G `  v ) ) ) ) ) )
6463spcegv 2658 . . . . . . 7  |-  ( ( 2nd `  ( R `
 ( `' G `  v ) ) )  e.  S  ->  ( A. z ( <. v ,  z >.  e.  T  ->  z  =  ( 2nd `  ( R `  ( `' G `  v ) ) ) )  ->  E. w A. z (
<. v ,  z >.  e.  T  ->  z  =  w ) ) )
6529, 60, 64sylc 60 . . . . . 6  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  E. w A. z ( <. v ,  z >.  e.  T  ->  z  =  w ) )
66 nfv 1437 . . . . . . 7  |-  F/ w <. v ,  z >.  e.  T
6766mo2r 1968 . . . . . 6  |-  ( E. w A. z (
<. v ,  z >.  e.  T  ->  z  =  w )  ->  E* z <. v ,  z
>.  e.  T )
6865, 67syl 14 . . . . 5  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  E* z <. v ,  z >.  e.  T )
69 dmss 4562 . . . . . . . . . 10  |-  ( T 
C_  ( ( ZZ>= `  C )  X.  S
)  ->  dom  T  C_  dom  ( ( ZZ>= `  C
)  X.  S ) )
7018, 69syl 14 . . . . . . . . 9  |-  ( ph  ->  dom  T  C_  dom  ( ( ZZ>= `  C
)  X.  S ) )
71 dmxpss 4781 . . . . . . . . 9  |-  dom  (
( ZZ>= `  C )  X.  S )  C_  ( ZZ>=
`  C )
7270, 71syl6ss 2985 . . . . . . . 8  |-  ( ph  ->  dom  T  C_  ( ZZ>=
`  C ) )
733adantr 265 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  C  e.  ZZ )
745adantr 265 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  S  e.  V )
756adantr 265 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  A  e.  S )
767adantlr 454 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  ( ZZ>= `  C )
)  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S
) )  ->  (
x F y )  e.  S )
77 simpr 107 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  v  e.  ( ZZ>= `  C )
)
7873, 4, 74, 75, 76, 8, 77frecuzrdglem 9361 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  ran  R )
791eleq2d 2123 . . . . . . . . . . . . 13  |-  ( ph  ->  ( <. v ,  ( 2nd `  ( R `
 ( `' G `  v ) ) )
>.  e.  T  <->  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  ran  R ) )
8079adantr 265 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  ( <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  T  <->  <.
v ,  ( 2nd `  ( R `  ( `' G `  v ) ) ) >.  e.  ran  R ) )
8178, 80mpbird 160 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  <. v ,  ( 2nd `  ( R `  ( `' G `  v )
) ) >.  e.  T
)
82 opeldmg 4568 . . . . . . . . . . . 12  |-  ( ( v  e.  _V  /\  ( 2nd `  ( R `
 ( `' G `  v ) ) )  e.  S )  -> 
( <. v ,  ( 2nd `  ( R `
 ( `' G `  v ) ) )
>.  e.  T  ->  v  e.  dom  T ) )
8341, 82mpan 408 . . . . . . . . . . 11  |-  ( ( 2nd `  ( R `
 ( `' G `  v ) ) )  e.  S  ->  ( <. v ,  ( 2nd `  ( R `  ( `' G `  v ) ) ) >.  e.  T  ->  v  e.  dom  T
) )
8429, 81, 83sylc 60 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( ZZ>= `  C )
)  ->  v  e.  dom  T )
8584ex 112 . . . . . . . . 9  |-  ( ph  ->  ( v  e.  (
ZZ>= `  C )  -> 
v  e.  dom  T
) )
8685ssrdv 2979 . . . . . . . 8  |-  ( ph  ->  ( ZZ>= `  C )  C_ 
dom  T )
8772, 86eqssd 2990 . . . . . . 7  |-  ( ph  ->  dom  T  =  (
ZZ>= `  C ) )
8887eleq2d 2123 . . . . . 6  |-  ( ph  ->  ( v  e.  dom  T  <-> 
v  e.  ( ZZ>= `  C ) ) )
8988pm5.32i 435 . . . . 5  |-  ( (
ph  /\  v  e.  dom  T )  <->  ( ph  /\  v  e.  ( ZZ>= `  C ) ) )
90 df-br 3793 . . . . . 6  |-  ( v T z  <->  <. v ,  z >.  e.  T
)
9190mobii 1953 . . . . 5  |-  ( E* z  v T z  <->  E* z <. v ,  z
>.  e.  T )
9268, 89, 913imtr4i 194 . . . 4  |-  ( (
ph  /\  v  e.  dom  T )  ->  E* z  v T z )
9392ralrimiva 2409 . . 3  |-  ( ph  ->  A. v  e.  dom  T E* z  v T z )
94 dffun7 4956 . . 3  |-  ( Fun 
T  <->  ( Rel  T  /\  A. v  e.  dom  T E* z  v T z ) )
9522, 93, 94sylanbrc 402 . 2  |-  ( ph  ->  Fun  T )
96 df-fn 4933 . 2  |-  ( T  Fn  ( ZZ>= `  C
)  <->  ( Fun  T  /\  dom  T  =  (
ZZ>= `  C ) ) )
9795, 87, 96sylanbrc 402 1  |-  ( ph  ->  T  Fn  ( ZZ>= `  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   E*wmo 1917   A.wral 2323   E.wrex 2324   _Vcvv 2574    C_ wss 2945   <.cop 3406   class class class wbr 3792    |-> cmpt 3846   omcom 4341    X. cxp 4371   `'ccnv 4372   dom cdm 4373   ran crn 4374   Rel wrel 4378   Fun wfun 4924    Fn wfn 4925   -1-1-onto->wf1o 4929   ` cfv 4930  (class class class)co 5540    |-> cmpt2 5542   2ndc2nd 5794  freccfrec 6008   1c1 6948    + caddc 6950   ZZcz 8302   ZZ>=cuz 8569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-ltadd 7058
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-frec 6009  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-inn 7991  df-n0 8240  df-z 8303  df-uz 8570
This theorem is referenced by:  frecuzrdgcl  9363  frecuzrdg0  9364  frecuzrdgsuc  9365  iseqfn  9385
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