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Theorem frecfzennn 10192
Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 10165 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)
Hypothesis
Ref Expression
frecfzennn.1  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
Assertion
Ref Expression
frecfzennn  |-  ( N  e.  NN0  ->  ( 1 ... N )  ~~  ( `' G `  N ) )

Proof of Theorem frecfzennn
Dummy variables  m  n  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5775 . . 3  |-  ( n  =  0  ->  (
1 ... n )  =  ( 1 ... 0
) )
2 fveq2 5414 . . 3  |-  ( n  =  0  ->  ( `' G `  n )  =  ( `' G `  0 ) )
31, 2breq12d 3937 . 2  |-  ( n  =  0  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... 0 ) 
~~  ( `' G `  0 ) ) )
4 oveq2 5775 . . 3  |-  ( n  =  m  ->  (
1 ... n )  =  ( 1 ... m
) )
5 fveq2 5414 . . 3  |-  ( n  =  m  ->  ( `' G `  n )  =  ( `' G `  m ) )
64, 5breq12d 3937 . 2  |-  ( n  =  m  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... m ) 
~~  ( `' G `  m ) ) )
7 oveq2 5775 . . 3  |-  ( n  =  ( m  + 
1 )  ->  (
1 ... n )  =  ( 1 ... (
m  +  1 ) ) )
8 fveq2 5414 . . 3  |-  ( n  =  ( m  + 
1 )  ->  ( `' G `  n )  =  ( `' G `  ( m  +  1 ) ) )
97, 8breq12d 3937 . 2  |-  ( n  =  ( m  + 
1 )  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... ( m  +  1 ) ) 
~~  ( `' G `  ( m  +  1 ) ) ) )
10 oveq2 5775 . . 3  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
11 fveq2 5414 . . 3  |-  ( n  =  N  ->  ( `' G `  n )  =  ( `' G `  N ) )
1210, 11breq12d 3937 . 2  |-  ( n  =  N  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... N ) 
~~  ( `' G `  N ) ) )
13 0ex 4050 . . . 4  |-  (/)  e.  _V
1413enref 6652 . . 3  |-  (/)  ~~  (/)
15 fz10 9819 . . 3  |-  ( 1 ... 0 )  =  (/)
16 0zd 9059 . . . . . . 7  |-  ( T. 
->  0  e.  ZZ )
17 frecfzennn.1 . . . . . . 7  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
1816, 17frec2uzf1od 10172 . . . . . 6  |-  ( T. 
->  G : om -1-1-onto-> ( ZZ>= `  0 )
)
1918mptru 1340 . . . . 5  |-  G : om
-1-1-onto-> ( ZZ>= `  0 )
20 peano1 4503 . . . . 5  |-  (/)  e.  om
2119, 20pm3.2i 270 . . . 4  |-  ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )
2216, 17frec2uz0d 10165 . . . . 5  |-  ( T. 
->  ( G `  (/) )  =  0 )
2322mptru 1340 . . . 4  |-  ( G `
 (/) )  =  0
24 f1ocnvfv 5673 . . . 4  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )  ->  ( ( G `  (/) )  =  0  -> 
( `' G ` 
0 )  =  (/) ) )
2521, 23, 24mp2 16 . . 3  |-  ( `' G `  0 )  =  (/)
2614, 15, 253brtr4i 3953 . 2  |-  ( 1 ... 0 )  ~~  ( `' G `  0 )
27 simpr 109 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... m )  ~~  ( `' G `  m ) )
28 peano2nn0 9010 . . . . . . 7  |-  ( m  e.  NN0  ->  ( m  +  1 )  e. 
NN0 )
29 zex 9056 . . . . . . . . . . . . . . 15  |-  ZZ  e.  _V
3029mptex 5639 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  |->  ( x  +  1 ) )  e.  _V
31 vex 2684 . . . . . . . . . . . . . 14  |-  z  e. 
_V
3230, 31fvex 5434 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  |->  ( x  +  1 ) ) `  z )  e.  _V
3332ax-gen 1425 . . . . . . . . . . . 12  |-  A. z
( ( x  e.  ZZ  |->  ( x  + 
1 ) ) `  z )  e.  _V
34 0z 9058 . . . . . . . . . . . 12  |-  0  e.  ZZ
35 frecfnom 6291 . . . . . . . . . . . 12  |-  ( ( A. z ( ( x  e.  ZZ  |->  ( x  +  1 ) ) `  z )  e.  _V  /\  0  e.  ZZ )  -> frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  Fn  om )
3633, 34, 35mp2an 422 . . . . . . . . . . 11  |- frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  Fn  om
3717fneq1i 5212 . . . . . . . . . . 11  |-  ( G  Fn  om  <-> frec ( (
x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  Fn  om )
3836, 37mpbir 145 . . . . . . . . . 10  |-  G  Fn  om
39 omex 4502 . . . . . . . . . 10  |-  om  e.  _V
40 fnex 5635 . . . . . . . . . 10  |-  ( ( G  Fn  om  /\  om  e.  _V )  ->  G  e.  _V )
4138, 39, 40mp2an 422 . . . . . . . . 9  |-  G  e. 
_V
4241cnvex 5072 . . . . . . . 8  |-  `' G  e.  _V
43 vex 2684 . . . . . . . 8  |-  m  e. 
_V
4442, 43fvex 5434 . . . . . . 7  |-  ( `' G `  m )  e.  _V
45 en2sn 6700 . . . . . . 7  |-  ( ( ( m  +  1 )  e.  NN0  /\  ( `' G `  m )  e.  _V )  ->  { ( m  + 
1 ) }  ~~  { ( `' G `  m ) } )
4628, 44, 45sylancl 409 . . . . . 6  |-  ( m  e.  NN0  ->  { ( m  +  1 ) }  ~~  { ( `' G `  m ) } )
4746adantr 274 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  { (
m  +  1 ) }  ~~  { ( `' G `  m ) } )
48 fzp1disj 9853 . . . . . 6  |-  ( ( 1 ... m )  i^i  { ( m  +  1 ) } )  =  (/)
4948a1i 9 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( (
1 ... m )  i^i 
{ ( m  + 
1 ) } )  =  (/) )
50 f1ocnvdm 5675 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  m  e.  ( ZZ>=
`  0 ) )  ->  ( `' G `  m )  e.  om )
5119, 50mpan 420 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  0
)  ->  ( `' G `  m )  e.  om )
52 nn0uz 9353 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
5351, 52eleq2s 2232 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( `' G `  m )  e.  om )
54 nnord 4520 . . . . . . . 8  |-  ( ( `' G `  m )  e.  om  ->  Ord  ( `' G `  m ) )
55 ordirr 4452 . . . . . . . 8  |-  ( Ord  ( `' G `  m )  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
5653, 54, 553syl 17 . . . . . . 7  |-  ( m  e.  NN0  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
5756adantr 274 . . . . . 6  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
58 disjsn 3580 . . . . . 6  |-  ( ( ( `' G `  m )  i^i  {
( `' G `  m ) } )  =  (/)  <->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
5957, 58sylibr 133 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( ( `' G `  m )  i^i  { ( `' G `  m ) } )  =  (/) )
60 unen 6703 . . . . 5  |-  ( ( ( ( 1 ... m )  ~~  ( `' G `  m )  /\  { ( m  +  1 ) } 
~~  { ( `' G `  m ) } )  /\  (
( ( 1 ... m )  i^i  {
( m  +  1 ) } )  =  (/)  /\  ( ( `' G `  m )  i^i  { ( `' G `  m ) } )  =  (/) ) )  ->  (
( 1 ... m
)  u.  { ( m  +  1 ) } )  ~~  (
( `' G `  m )  u.  {
( `' G `  m ) } ) )
6127, 47, 49, 59, 60syl22anc 1217 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( (
1 ... m )  u. 
{ ( m  + 
1 ) } ) 
~~  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
62 1z 9073 . . . . . 6  |-  1  e.  ZZ
63 1m1e0 8782 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
6463fveq2i 5417 . . . . . . . . 9  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
6552, 64eqtr4i 2161 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
6665eleq2i 2204 . . . . . . 7  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  ( 1  -  1 ) ) )
6766biimpi 119 . . . . . 6  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  ( 1  -  1 ) ) )
68 fzsuc2 9852 . . . . . 6  |-  ( ( 1  e.  ZZ  /\  m  e.  ( ZZ>= `  ( 1  -  1 ) ) )  -> 
( 1 ... (
m  +  1 ) )  =  ( ( 1 ... m )  u.  { ( m  +  1 ) } ) )
6962, 67, 68sylancr 410 . . . . 5  |-  ( m  e.  NN0  ->  ( 1 ... ( m  + 
1 ) )  =  ( ( 1 ... m )  u.  {
( m  +  1 ) } ) )
7069adantr 274 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... ( m  + 
1 ) )  =  ( ( 1 ... m )  u.  {
( m  +  1 ) } ) )
71 peano2 4504 . . . . . . . . 9  |-  ( ( `' G `  m )  e.  om  ->  suc  ( `' G `  m )  e.  om )
7253, 71syl 14 . . . . . . . 8  |-  ( m  e.  NN0  ->  suc  ( `' G `  m )  e.  om )
7372, 19jctil 310 . . . . . . 7  |-  ( m  e.  NN0  ->  ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  suc  ( `' G `  m )  e.  om ) )
74 0zd 9059 . . . . . . . . . 10  |-  ( ( `' G `  m )  e.  om  ->  0  e.  ZZ )
75 id 19 . . . . . . . . . 10  |-  ( ( `' G `  m )  e.  om  ->  ( `' G `  m )  e.  om )
7674, 17, 75frec2uzsucd 10167 . . . . . . . . 9  |-  ( ( `' G `  m )  e.  om  ->  ( G `  suc  ( `' G `  m ) )  =  ( ( G `  ( `' G `  m ) )  +  1 ) )
7753, 76syl 14 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( G `
 suc  ( `' G `  m )
)  =  ( ( G `  ( `' G `  m ) )  +  1 ) )
7852eleq2i 2204 . . . . . . . . . . 11  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  0 ) )
7978biimpi 119 . . . . . . . . . 10  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  0 )
)
80 f1ocnvfv2 5672 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  m  e.  ( ZZ>=
`  0 ) )  ->  ( G `  ( `' G `  m ) )  =  m )
8119, 79, 80sylancr 410 . . . . . . . . 9  |-  ( m  e.  NN0  ->  ( G `
 ( `' G `  m ) )  =  m )
8281oveq1d 5782 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( ( G `  ( `' G `  m ) )  +  1 )  =  ( m  + 
1 ) )
8377, 82eqtrd 2170 . . . . . . 7  |-  ( m  e.  NN0  ->  ( G `
 suc  ( `' G `  m )
)  =  ( m  +  1 ) )
84 f1ocnvfv 5673 . . . . . . 7  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  suc  ( `' G `  m )  e.  om )  ->  ( ( G `
 suc  ( `' G `  m )
)  =  ( m  +  1 )  -> 
( `' G `  ( m  +  1
) )  =  suc  ( `' G `  m ) ) )
8573, 83, 84sylc 62 . . . . . 6  |-  ( m  e.  NN0  ->  ( `' G `  ( m  +  1 ) )  =  suc  ( `' G `  m ) )
8685adantr 274 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( `' G `  ( m  +  1 ) )  =  suc  ( `' G `  m ) )
87 df-suc 4288 . . . . 5  |-  suc  ( `' G `  m )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } )
8886, 87syl6eq 2186 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( `' G `  ( m  +  1 ) )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
8961, 70, 883brtr4d 3955 . . 3  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... ( m  + 
1 ) )  ~~  ( `' G `  ( m  +  1 ) ) )
9089ex 114 . 2  |-  ( m  e.  NN0  ->  ( ( 1 ... m ) 
~~  ( `' G `  m )  ->  (
1 ... ( m  + 
1 ) )  ~~  ( `' G `  ( m  +  1 ) ) ) )
913, 6, 9, 12, 26, 90nn0ind 9158 1  |-  ( N  e.  NN0  ->  ( 1 ... N )  ~~  ( `' G `  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103   A.wal 1329    = wceq 1331   T. wtru 1332    e. wcel 1480   _Vcvv 2681    u. cun 3064    i^i cin 3065   (/)c0 3358   {csn 3522   class class class wbr 3924    |-> cmpt 3984   Ord word 4279   suc csuc 4282   omcom 4499   `'ccnv 4533    Fn wfn 5113   -1-1-onto->wf1o 5117   ` cfv 5118  (class class class)co 5767  freccfrec 6280    ~~ cen 6625   0cc0 7613   1c1 7614    + caddc 7616    - cmin 7926   NN0cn0 8970   ZZcz 9047   ZZ>=cuz 9319   ...cfz 9783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-recs 6195  df-frec 6281  df-1o 6306  df-er 6422  df-en 6628  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-fz 9784
This theorem is referenced by:  frecfzen2  10193  hashfz1  10522
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