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Theorem fzennn 11790
Description: The cardinality of a finite set of sequential integers. (See om2uz0i 11770 for a description of the hypothesis.) (Contributed by Mario Carneiro, 12-Feb-2013.) (Revised by Mario Carneiro, 7-Mar-2014.)
Hypothesis
Ref Expression
fzennn.1  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
Assertion
Ref Expression
fzennn  |-  ( N  e.  NN0  ->  ( 1 ... N )  ~~  ( `' G `  N ) )

Proof of Theorem fzennn
Dummy variables  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6099 . . 3  |-  ( n  =  0  ->  (
1 ... n )  =  ( 1 ... 0
) )
2 fveq2 5691 . . 3  |-  ( n  =  0  ->  ( `' G `  n )  =  ( `' G `  0 ) )
31, 2breq12d 4305 . 2  |-  ( n  =  0  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... 0 ) 
~~  ( `' G `  0 ) ) )
4 oveq2 6099 . . 3  |-  ( n  =  m  ->  (
1 ... n )  =  ( 1 ... m
) )
5 fveq2 5691 . . 3  |-  ( n  =  m  ->  ( `' G `  n )  =  ( `' G `  m ) )
64, 5breq12d 4305 . 2  |-  ( n  =  m  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... m ) 
~~  ( `' G `  m ) ) )
7 oveq2 6099 . . 3  |-  ( n  =  ( m  + 
1 )  ->  (
1 ... n )  =  ( 1 ... (
m  +  1 ) ) )
8 fveq2 5691 . . 3  |-  ( n  =  ( m  + 
1 )  ->  ( `' G `  n )  =  ( `' G `  ( m  +  1 ) ) )
97, 8breq12d 4305 . 2  |-  ( n  =  ( m  + 
1 )  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... ( m  +  1 ) ) 
~~  ( `' G `  ( m  +  1 ) ) ) )
10 oveq2 6099 . . 3  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
11 fveq2 5691 . . 3  |-  ( n  =  N  ->  ( `' G `  n )  =  ( `' G `  N ) )
1210, 11breq12d 4305 . 2  |-  ( n  =  N  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... N ) 
~~  ( `' G `  N ) ) )
13 0ex 4422 . . . 4  |-  (/)  e.  _V
1413enref 7342 . . 3  |-  (/)  ~~  (/)
15 fz10 11470 . . 3  |-  ( 1 ... 0 )  =  (/)
16 0z 10657 . . . . . 6  |-  0  e.  ZZ
17 fzennn.1 . . . . . 6  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
1816, 17om2uzf1oi 11776 . . . . 5  |-  G : om
-1-1-onto-> ( ZZ>= `  0 )
19 peano1 6495 . . . . 5  |-  (/)  e.  om
2018, 19pm3.2i 455 . . . 4  |-  ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )
2116, 17om2uz0i 11770 . . . 4  |-  ( G `
 (/) )  =  0
22 f1ocnvfv 5985 . . . 4  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )  ->  ( ( G `  (/) )  =  0  -> 
( `' G ` 
0 )  =  (/) ) )
2320, 21, 22mp2 9 . . 3  |-  ( `' G `  0 )  =  (/)
2414, 15, 233brtr4i 4320 . 2  |-  ( 1 ... 0 )  ~~  ( `' G `  0 )
25 simpr 461 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... m )  ~~  ( `' G `  m ) )
26 ovex 6116 . . . . . . 7  |-  ( m  +  1 )  e. 
_V
27 fvex 5701 . . . . . . 7  |-  ( `' G `  m )  e.  _V
28 en2sn 7389 . . . . . . 7  |-  ( ( ( m  +  1 )  e.  _V  /\  ( `' G `  m )  e.  _V )  ->  { ( m  + 
1 ) }  ~~  { ( `' G `  m ) } )
2926, 27, 28mp2an 672 . . . . . 6  |-  { ( m  +  1 ) }  ~~  { ( `' G `  m ) }
3029a1i 11 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  { (
m  +  1 ) }  ~~  { ( `' G `  m ) } )
31 fzp1disj 11515 . . . . . 6  |-  ( ( 1 ... m )  i^i  { ( m  +  1 ) } )  =  (/)
3231a1i 11 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( (
1 ... m )  i^i 
{ ( m  + 
1 ) } )  =  (/) )
33 f1ocnvdm 5989 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  m  e.  ( ZZ>=
`  0 ) )  ->  ( `' G `  m )  e.  om )
3418, 33mpan 670 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  0
)  ->  ( `' G `  m )  e.  om )
35 nn0uz 10895 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
3634, 35eleq2s 2535 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( `' G `  m )  e.  om )
37 nnord 6484 . . . . . . . 8  |-  ( ( `' G `  m )  e.  om  ->  Ord  ( `' G `  m ) )
38 ordirr 4737 . . . . . . . 8  |-  ( Ord  ( `' G `  m )  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
3936, 37, 383syl 20 . . . . . . 7  |-  ( m  e.  NN0  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
4039adantr 465 . . . . . 6  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
41 disjsn 3936 . . . . . 6  |-  ( ( ( `' G `  m )  i^i  {
( `' G `  m ) } )  =  (/)  <->  -.  ( `' G `  m )  e.  ( `' G `  m ) )
4240, 41sylibr 212 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( ( `' G `  m )  i^i  { ( `' G `  m ) } )  =  (/) )
43 unen 7392 . . . . 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 ) } ) )
4425, 30, 32, 42, 43syl22anc 1219 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( (
1 ... m )  u. 
{ ( m  + 
1 ) } ) 
~~  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
45 1z 10676 . . . . . 6  |-  1  e.  ZZ
46 1m1e0 10390 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
4746fveq2i 5694 . . . . . . . . 9  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
4835, 47eqtr4i 2466 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
4948eleq2i 2507 . . . . . . 7  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  ( 1  -  1 ) ) )
5049biimpi 194 . . . . . 6  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  ( 1  -  1 ) ) )
51 fzsuc2 11514 . . . . . 6  |-  ( ( 1  e.  ZZ  /\  m  e.  ( ZZ>= `  ( 1  -  1 ) ) )  -> 
( 1 ... (
m  +  1 ) )  =  ( ( 1 ... m )  u.  { ( m  +  1 ) } ) )
5245, 50, 51sylancr 663 . . . . 5  |-  ( m  e.  NN0  ->  ( 1 ... ( m  + 
1 ) )  =  ( ( 1 ... m )  u.  {
( m  +  1 ) } ) )
5352adantr 465 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... ( m  + 
1 ) )  =  ( ( 1 ... m )  u.  {
( m  +  1 ) } ) )
54 peano2 6496 . . . . . . . . 9  |-  ( ( `' G `  m )  e.  om  ->  suc  ( `' G `  m )  e.  om )
5536, 54syl 16 . . . . . . . 8  |-  ( m  e.  NN0  ->  suc  ( `' G `  m )  e.  om )
5655, 18jctil 537 . . . . . . 7  |-  ( m  e.  NN0  ->  ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  suc  ( `' G `  m )  e.  om ) )
5716, 17om2uzsuci 11771 . . . . . . . . 9  |-  ( ( `' G `  m )  e.  om  ->  ( G `  suc  ( `' G `  m ) )  =  ( ( G `  ( `' G `  m ) )  +  1 ) )
5836, 57syl 16 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( G `
 suc  ( `' G `  m )
)  =  ( ( G `  ( `' G `  m ) )  +  1 ) )
5935eleq2i 2507 . . . . . . . . . . 11  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  0 ) )
6059biimpi 194 . . . . . . . . . 10  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  0 )
)
61 f1ocnvfv2 5984 . . . . . . . . . 10  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  m  e.  ( ZZ>=
`  0 ) )  ->  ( G `  ( `' G `  m ) )  =  m )
6218, 60, 61sylancr 663 . . . . . . . . 9  |-  ( m  e.  NN0  ->  ( G `
 ( `' G `  m ) )  =  m )
6362oveq1d 6106 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( ( G `  ( `' G `  m ) )  +  1 )  =  ( m  + 
1 ) )
6458, 63eqtrd 2475 . . . . . . 7  |-  ( m  e.  NN0  ->  ( G `
 suc  ( `' G `  m )
)  =  ( m  +  1 ) )
65 f1ocnvfv 5985 . . . . . . 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 ) ) )
6656, 64, 65sylc 60 . . . . . 6  |-  ( m  e.  NN0  ->  ( `' G `  ( m  +  1 ) )  =  suc  ( `' G `  m ) )
6766adantr 465 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( `' G `  ( m  +  1 ) )  =  suc  ( `' G `  m ) )
68 df-suc 4725 . . . . 5  |-  suc  ( `' G `  m )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } )
6967, 68syl6eq 2491 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( `' G `  ( m  +  1 ) )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
7044, 53, 693brtr4d 4322 . . 3  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... ( m  + 
1 ) )  ~~  ( `' G `  ( m  +  1 ) ) )
7170ex 434 . 2  |-  ( m  e.  NN0  ->  ( ( 1 ... m ) 
~~  ( `' G `  m )  ->  (
1 ... ( m  + 
1 ) )  ~~  ( `' G `  ( m  +  1 ) ) ) )
723, 6, 9, 12, 24, 71nn0ind 10738 1  |-  ( N  e.  NN0  ->  ( 1 ... N )  ~~  ( `' G `  N ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972    u. cun 3326    i^i cin 3327   (/)c0 3637   {csn 3877   class class class wbr 4292    e. cmpt 4350   Ord word 4718   suc csuc 4721   `'ccnv 4839    |` cres 4842   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091   omcom 6476   reccrdg 6865    ~~ cen 7307   0cc0 9282   1c1 9283    + caddc 9285    - cmin 9595   NN0cn0 10579   ZZcz 10646   ZZ>=cuz 10861   ...cfz 11437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438
This theorem is referenced by:  fzen2  11791  cardfz  11792
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