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Theorem frecfzennn 10167
Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 10140 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 5750 . . 3  |-  ( n  =  0  ->  (
1 ... n )  =  ( 1 ... 0
) )
2 fveq2 5389 . . 3  |-  ( n  =  0  ->  ( `' G `  n )  =  ( `' G `  0 ) )
31, 2breq12d 3912 . 2  |-  ( n  =  0  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... 0 ) 
~~  ( `' G `  0 ) ) )
4 oveq2 5750 . . 3  |-  ( n  =  m  ->  (
1 ... n )  =  ( 1 ... m
) )
5 fveq2 5389 . . 3  |-  ( n  =  m  ->  ( `' G `  n )  =  ( `' G `  m ) )
64, 5breq12d 3912 . 2  |-  ( n  =  m  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... m ) 
~~  ( `' G `  m ) ) )
7 oveq2 5750 . . 3  |-  ( n  =  ( m  + 
1 )  ->  (
1 ... n )  =  ( 1 ... (
m  +  1 ) ) )
8 fveq2 5389 . . 3  |-  ( n  =  ( m  + 
1 )  ->  ( `' G `  n )  =  ( `' G `  ( m  +  1 ) ) )
97, 8breq12d 3912 . 2  |-  ( n  =  ( m  + 
1 )  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... ( m  +  1 ) ) 
~~  ( `' G `  ( m  +  1 ) ) ) )
10 oveq2 5750 . . 3  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
11 fveq2 5389 . . 3  |-  ( n  =  N  ->  ( `' G `  n )  =  ( `' G `  N ) )
1210, 11breq12d 3912 . 2  |-  ( n  =  N  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... N ) 
~~  ( `' G `  N ) ) )
13 0ex 4025 . . . 4  |-  (/)  e.  _V
1413enref 6627 . . 3  |-  (/)  ~~  (/)
15 fz10 9794 . . 3  |-  ( 1 ... 0 )  =  (/)
16 0zd 9034 . . . . . . 7  |-  ( T. 
->  0  e.  ZZ )
17 frecfzennn.1 . . . . . . 7  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
1816, 17frec2uzf1od 10147 . . . . . 6  |-  ( T. 
->  G : om -1-1-onto-> ( ZZ>= `  0 )
)
1918mptru 1325 . . . . 5  |-  G : om
-1-1-onto-> ( ZZ>= `  0 )
20 peano1 4478 . . . . 5  |-  (/)  e.  om
2119, 20pm3.2i 270 . . . 4  |-  ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )
2216, 17frec2uz0d 10140 . . . . 5  |-  ( T. 
->  ( G `  (/) )  =  0 )
2322mptru 1325 . . . 4  |-  ( G `
 (/) )  =  0
24 f1ocnvfv 5648 . . . 4  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )  ->  ( ( G `  (/) )  =  0  -> 
( `' G ` 
0 )  =  (/) ) )
2521, 23, 24mp2 16 . . 3  |-  ( `' G `  0 )  =  (/)
2614, 15, 253brtr4i 3928 . 2  |-  ( 1 ... 0 )  ~~  ( `' G `  0 )
27 simpr 109 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... m )  ~~  ( `' G `  m ) )
28 peano2nn0 8985 . . . . . . 7  |-  ( m  e.  NN0  ->  ( m  +  1 )  e. 
NN0 )
29 zex 9031 . . . . . . . . . . . . . . 15  |-  ZZ  e.  _V
3029mptex 5614 . . . . . . . . . . . . . 14  |-  ( x  e.  ZZ  |->  ( x  +  1 ) )  e.  _V
31 vex 2663 . . . . . . . . . . . . . 14  |-  z  e. 
_V
3230, 31fvex 5409 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  |->  ( x  +  1 ) ) `  z )  e.  _V
3332ax-gen 1410 . . . . . . . . . . . 12  |-  A. z
( ( x  e.  ZZ  |->  ( x  + 
1 ) ) `  z )  e.  _V
34 0z 9033 . . . . . . . . . . . 12  |-  0  e.  ZZ
35 frecfnom 6266 . . . . . . . . . . . 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 5187 . . . . . . . . . . 11  |-  ( G  Fn  om  <-> frec ( (
x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  Fn  om )
3836, 37mpbir 145 . . . . . . . . . 10  |-  G  Fn  om
39 omex 4477 . . . . . . . . . 10  |-  om  e.  _V
40 fnex 5610 . . . . . . . . . 10  |-  ( ( G  Fn  om  /\  om  e.  _V )  ->  G  e.  _V )
4138, 39, 40mp2an 422 . . . . . . . . 9  |-  G  e. 
_V
4241cnvex 5047 . . . . . . . 8  |-  `' G  e.  _V
43 vex 2663 . . . . . . . 8  |-  m  e. 
_V
4442, 43fvex 5409 . . . . . . 7  |-  ( `' G `  m )  e.  _V
45 en2sn 6675 . . . . . . 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 9828 . . . . . 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 5650 . . . . . . . . . 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 9328 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
5351, 52eleq2s 2212 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( `' G `  m )  e.  om )
54 nnord 4495 . . . . . . . 8  |-  ( ( `' G `  m )  e.  om  ->  Ord  ( `' G `  m ) )
55 ordirr 4427 . . . . . . . 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 3555 . . . . . 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 6678 . . . . 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 1202 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( (
1 ... m )  u. 
{ ( m  + 
1 ) } ) 
~~  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
62 1z 9048 . . . . . 6  |-  1  e.  ZZ
63 1m1e0 8757 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
6463fveq2i 5392 . . . . . . . . 9  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
6552, 64eqtr4i 2141 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
6665eleq2i 2184 . . . . . . 7  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  ( 1  -  1 ) ) )
6766biimpi 119 . . . . . 6  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  ( 1  -  1 ) ) )
68 fzsuc2 9827 . . . . . 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 4479 . . . . . . . . 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 9034 . . . . . . . . . 10  |-  ( ( `' G `  m )  e.  om  ->  0  e.  ZZ )
75 id 19 . . . . . . . . . 10  |-  ( ( `' G `  m )  e.  om  ->  ( `' G `  m )  e.  om )
7674, 17, 75frec2uzsucd 10142 . . . . . . . . 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 2184 . . . . . . . . . . 11  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  0 ) )
7978biimpi 119 . . . . . . . . . 10  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  0 )
)
80 f1ocnvfv2 5647 . . . . . . . . . 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 5757 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( ( G `  ( `' G `  m ) )  +  1 )  =  ( m  + 
1 ) )
8377, 82eqtrd 2150 . . . . . . 7  |-  ( m  e.  NN0  ->  ( G `
 suc  ( `' G `  m )
)  =  ( m  +  1 ) )
84 f1ocnvfv 5648 . . . . . . 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 4263 . . . . 5  |-  suc  ( `' G `  m )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } )
8886, 87syl6eq 2166 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( `' G `  ( m  +  1 ) )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
8961, 70, 883brtr4d 3930 . . 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 9133 1  |-  ( N  e.  NN0  ->  ( 1 ... N )  ~~  ( `' G `  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103   A.wal 1314    = wceq 1316   T. wtru 1317    e. wcel 1465   _Vcvv 2660    u. cun 3039    i^i cin 3040   (/)c0 3333   {csn 3497   class class class wbr 3899    |-> cmpt 3959   Ord word 4254   suc csuc 4257   omcom 4474   `'ccnv 4508    Fn wfn 5088   -1-1-onto->wf1o 5092   ` cfv 5093  (class class class)co 5742  freccfrec 6255    ~~ cen 6600   0cc0 7588   1c1 7589    + caddc 7591    - cmin 7901   NN0cn0 8945   ZZcz 9022   ZZ>=cuz 9294   ...cfz 9758
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-recs 6170  df-frec 6256  df-1o 6281  df-er 6397  df-en 6603  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-n0 8946  df-z 9023  df-uz 9295  df-fz 9759
This theorem is referenced by:  frecfzen2  10168  hashfz1  10497
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