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Theorem frecfzennn 9894
Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 9867 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 5674 . . 3  |-  ( n  =  0  ->  (
1 ... n )  =  ( 1 ... 0
) )
2 fveq2 5318 . . 3  |-  ( n  =  0  ->  ( `' G `  n )  =  ( `' G `  0 ) )
31, 2breq12d 3864 . 2  |-  ( n  =  0  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... 0 ) 
~~  ( `' G `  0 ) ) )
4 oveq2 5674 . . 3  |-  ( n  =  m  ->  (
1 ... n )  =  ( 1 ... m
) )
5 fveq2 5318 . . 3  |-  ( n  =  m  ->  ( `' G `  n )  =  ( `' G `  m ) )
64, 5breq12d 3864 . 2  |-  ( n  =  m  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... m ) 
~~  ( `' G `  m ) ) )
7 oveq2 5674 . . 3  |-  ( n  =  ( m  + 
1 )  ->  (
1 ... n )  =  ( 1 ... (
m  +  1 ) ) )
8 fveq2 5318 . . 3  |-  ( n  =  ( m  + 
1 )  ->  ( `' G `  n )  =  ( `' G `  ( m  +  1 ) ) )
97, 8breq12d 3864 . 2  |-  ( n  =  ( m  + 
1 )  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... ( m  +  1 ) ) 
~~  ( `' G `  ( m  +  1 ) ) ) )
10 oveq2 5674 . . 3  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
11 fveq2 5318 . . 3  |-  ( n  =  N  ->  ( `' G `  n )  =  ( `' G `  N ) )
1210, 11breq12d 3864 . 2  |-  ( n  =  N  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... N ) 
~~  ( `' G `  N ) ) )
13 0ex 3972 . . . 4  |-  (/)  e.  _V
1413enref 6536 . . 3  |-  (/)  ~~  (/)
15 fz10 9521 . . 3  |-  ( 1 ... 0 )  =  (/)
16 0zd 8823 . . . . . . 7  |-  ( T. 
->  0  e.  ZZ )
17 frecfzennn.1 . . . . . . 7  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
1816, 17frec2uzf1od 9874 . . . . . 6  |-  ( T. 
->  G : om -1-1-onto-> ( ZZ>= `  0 )
)
1918mptru 1299 . . . . 5  |-  G : om
-1-1-onto-> ( ZZ>= `  0 )
20 peano1 4422 . . . . 5  |-  (/)  e.  om
2119, 20pm3.2i 267 . . . 4  |-  ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )
2216, 17frec2uz0d 9867 . . . . 5  |-  ( T. 
->  ( G `  (/) )  =  0 )
2322mptru 1299 . . . 4  |-  ( G `
 (/) )  =  0
24 f1ocnvfv 5572 . . . 4  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )  ->  ( ( G `  (/) )  =  0  -> 
( `' G ` 
0 )  =  (/) ) )
2521, 23, 24mp2 16 . . 3  |-  ( `' G `  0 )  =  (/)
2614, 15, 253brtr4i 3879 . 2  |-  ( 1 ... 0 )  ~~  ( `' G `  0 )
27 simpr 109 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... m )  ~~  ( `' G `  m ) )
28 peano2nn0 8774 . . . . . . 7  |-  ( m  e.  NN0  ->  ( m  +  1 )  e. 
NN0 )
29 zex 8820 . . . . . . . . . . . . . . 15  |-  ZZ  e.  _V
3029mptex 5537 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  |->  ( x  +  1 ) )  e.  _V
31 vex 2623 . . . . . . . . . . . . . 14  |-  z  e. 
_V
3230, 31fvex 5338 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  |->  ( x  +  1 ) ) `  z )  e.  _V
3332ax-gen 1384 . . . . . . . . . . . 12  |-  A. z
( ( x  e.  ZZ  |->  ( x  + 
1 ) ) `  z )  e.  _V
34 0z 8822 . . . . . . . . . . . 12  |-  0  e.  ZZ
35 frecfnom 6180 . . . . . . . . . . . 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 418 . . . . . . . . . . 11  |- frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  Fn  om
3717fneq1i 5121 . . . . . . . . . . 11  |-  ( G  Fn  om  <-> frec ( (
x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  Fn  om )
3836, 37mpbir 145 . . . . . . . . . 10  |-  G  Fn  om
39 omex 4421 . . . . . . . . . 10  |-  om  e.  _V
40 fnex 5533 . . . . . . . . . 10  |-  ( ( G  Fn  om  /\  om  e.  _V )  ->  G  e.  _V )
4138, 39, 40mp2an 418 . . . . . . . . 9  |-  G  e. 
_V
4241cnvex 4982 . . . . . . . 8  |-  `' G  e.  _V
43 vex 2623 . . . . . . . 8  |-  m  e. 
_V
4442, 43fvex 5338 . . . . . . 7  |-  ( `' G `  m )  e.  _V
45 en2sn 6584 . . . . . . 7  |-  ( ( ( m  +  1 )  e.  NN0  /\  ( `' G `  m )  e.  _V )  ->  { ( m  + 
1 ) }  ~~  { ( `' G `  m ) } )
4628, 44, 45sylancl 405 . . . . . 6  |-  ( m  e.  NN0  ->  { ( m  +  1 ) }  ~~  { ( `' G `  m ) } )
4746adantr 271 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  { (
m  +  1 ) }  ~~  { ( `' G `  m ) } )
48 fzp1disj 9555 . . . . . 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 5574 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  m  e.  ( ZZ>=
`  0 ) )  ->  ( `' G `  m )  e.  om )
5119, 50mpan 416 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  0
)  ->  ( `' G `  m )  e.  om )
52 nn0uz 9114 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
5351, 52eleq2s 2183 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( `' G `  m )  e.  om )
54 nnord 4439 . . . . . . . 8  |-  ( ( `' G `  m )  e.  om  ->  Ord  ( `' G `  m ) )
55 ordirr 4371 . . . . . . . 8  |-  ( Ord  ( `' G `  m )  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
5653, 54, 553syl 17 . . . . . . 7  |-  ( m  e.  NN0  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
5756adantr 271 . . . . . 6  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
58 disjsn 3508 . . . . . 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 6587 . . . . 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 1176 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( (
1 ... m )  u. 
{ ( m  + 
1 ) } ) 
~~  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
62 1z 8837 . . . . . 6  |-  1  e.  ZZ
63 1m1e0 8552 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
6463fveq2i 5321 . . . . . . . . 9  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
6552, 64eqtr4i 2112 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
6665eleq2i 2155 . . . . . . 7  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  ( 1  -  1 ) ) )
6766biimpi 119 . . . . . 6  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  ( 1  -  1 ) ) )
68 fzsuc2 9554 . . . . . 6  |-  ( ( 1  e.  ZZ  /\  m  e.  ( ZZ>= `  ( 1  -  1 ) ) )  -> 
( 1 ... (
m  +  1 ) )  =  ( ( 1 ... m )  u.  { ( m  +  1 ) } ) )
6962, 67, 68sylancr 406 . . . . 5  |-  ( m  e.  NN0  ->  ( 1 ... ( m  + 
1 ) )  =  ( ( 1 ... m )  u.  {
( m  +  1 ) } ) )
7069adantr 271 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... ( m  + 
1 ) )  =  ( ( 1 ... m )  u.  {
( m  +  1 ) } ) )
71 peano2 4423 . . . . . . . . 9  |-  ( ( `' G `  m )  e.  om  ->  suc  ( `' G `  m )  e.  om )
7253, 71syl 14 . . . . . . . 8  |-  ( m  e.  NN0  ->  suc  ( `' G `  m )  e.  om )
7372, 19jctil 306 . . . . . . 7  |-  ( m  e.  NN0  ->  ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  suc  ( `' G `  m )  e.  om ) )
74 0zd 8823 . . . . . . . . . 10  |-  ( ( `' G `  m )  e.  om  ->  0  e.  ZZ )
75 id 19 . . . . . . . . . 10  |-  ( ( `' G `  m )  e.  om  ->  ( `' G `  m )  e.  om )
7674, 17, 75frec2uzsucd 9869 . . . . . . . . 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 2155 . . . . . . . . . . 11  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  0 ) )
7978biimpi 119 . . . . . . . . . 10  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  0 )
)
80 f1ocnvfv2 5571 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  m  e.  ( ZZ>=
`  0 ) )  ->  ( G `  ( `' G `  m ) )  =  m )
8119, 79, 80sylancr 406 . . . . . . . . 9  |-  ( m  e.  NN0  ->  ( G `
 ( `' G `  m ) )  =  m )
8281oveq1d 5681 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( ( G `  ( `' G `  m ) )  +  1 )  =  ( m  + 
1 ) )
8377, 82eqtrd 2121 . . . . . . 7  |-  ( m  e.  NN0  ->  ( G `
 suc  ( `' G `  m )
)  =  ( m  +  1 ) )
84 f1ocnvfv 5572 . . . . . . 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 271 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( `' G `  ( m  +  1 ) )  =  suc  ( `' G `  m ) )
87 df-suc 4207 . . . . 5  |-  suc  ( `' G `  m )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } )
8886, 87syl6eq 2137 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( `' G `  ( m  +  1 ) )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
8961, 70, 883brtr4d 3881 . . 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 8921 1  |-  ( N  e.  NN0  ->  ( 1 ... N )  ~~  ( `' G `  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103   A.wal 1288    = wceq 1290   T. wtru 1291    e. wcel 1439   _Vcvv 2620    u. cun 2998    i^i cin 2999   (/)c0 3287   {csn 3450   class class class wbr 3851    |-> cmpt 3905   Ord word 4198   suc csuc 4201   omcom 4418   `'ccnv 4451    Fn wfn 5023   -1-1-onto->wf1o 5027   ` cfv 5028  (class class class)co 5666  freccfrec 6169    ~~ cen 6509   0cc0 7411   1c1 7412    + caddc 7414    - cmin 7714   NN0cn0 8734   ZZcz 8811   ZZ>=cuz 9080   ...cfz 9485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-recs 6084  df-frec 6170  df-1o 6195  df-er 6306  df-en 6512  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-inn 8484  df-n0 8735  df-z 8812  df-uz 9081  df-fz 9486
This theorem is referenced by:  frecfzen2  9895  hashfz1  10252
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