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Theorem fzennn 12041
Description: The cardinality of a finite set of sequential integers. (See om2uz0i 12021 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 6290 . . 3  |-  ( n  =  0  ->  (
1 ... n )  =  ( 1 ... 0
) )
2 fveq2 5864 . . 3  |-  ( n  =  0  ->  ( `' G `  n )  =  ( `' G `  0 ) )
31, 2breq12d 4460 . 2  |-  ( n  =  0  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... 0 ) 
~~  ( `' G `  0 ) ) )
4 oveq2 6290 . . 3  |-  ( n  =  m  ->  (
1 ... n )  =  ( 1 ... m
) )
5 fveq2 5864 . . 3  |-  ( n  =  m  ->  ( `' G `  n )  =  ( `' G `  m ) )
64, 5breq12d 4460 . 2  |-  ( n  =  m  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... m ) 
~~  ( `' G `  m ) ) )
7 oveq2 6290 . . 3  |-  ( n  =  ( m  + 
1 )  ->  (
1 ... n )  =  ( 1 ... (
m  +  1 ) ) )
8 fveq2 5864 . . 3  |-  ( n  =  ( m  + 
1 )  ->  ( `' G `  n )  =  ( `' G `  ( m  +  1 ) ) )
97, 8breq12d 4460 . 2  |-  ( n  =  ( m  + 
1 )  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... ( m  +  1 ) ) 
~~  ( `' G `  ( m  +  1 ) ) ) )
10 oveq2 6290 . . 3  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
11 fveq2 5864 . . 3  |-  ( n  =  N  ->  ( `' G `  n )  =  ( `' G `  N ) )
1210, 11breq12d 4460 . 2  |-  ( n  =  N  ->  (
( 1 ... n
)  ~~  ( `' G `  n )  <->  ( 1 ... N ) 
~~  ( `' G `  N ) ) )
13 0ex 4577 . . . 4  |-  (/)  e.  _V
1413enref 7545 . . 3  |-  (/)  ~~  (/)
15 fz10 11702 . . 3  |-  ( 1 ... 0 )  =  (/)
16 0z 10871 . . . . . 6  |-  0  e.  ZZ
17 fzennn.1 . . . . . 6  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
1816, 17om2uzf1oi 12027 . . . . 5  |-  G : om
-1-1-onto-> ( ZZ>= `  0 )
19 peano1 6697 . . . . 5  |-  (/)  e.  om
2018, 19pm3.2i 455 . . . 4  |-  ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )
2116, 17om2uz0i 12021 . . . 4  |-  ( G `
 (/) )  =  0
22 f1ocnvfv 6170 . . . 4  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  0 )  /\  (/)  e.  om )  ->  ( ( G `  (/) )  =  0  -> 
( `' G ` 
0 )  =  (/) ) )
2320, 21, 22mp2 9 . . 3  |-  ( `' G `  0 )  =  (/)
2414, 15, 233brtr4i 4475 . 2  |-  ( 1 ... 0 )  ~~  ( `' G `  0 )
25 simpr 461 . . . . 5  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( 1 ... m )  ~~  ( `' G `  m ) )
26 ovex 6307 . . . . . . 7  |-  ( m  +  1 )  e. 
_V
27 fvex 5874 . . . . . . 7  |-  ( `' G `  m )  e.  _V
28 en2sn 7592 . . . . . . 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 11734 . . . . . 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 6174 . . . . . . . . . 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 11112 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
3634, 35eleq2s 2575 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( `' G `  m )  e.  om )
37 nnord 6686 . . . . . . . 8  |-  ( ( `' G `  m )  e.  om  ->  Ord  ( `' G `  m ) )
38 ordirr 4896 . . . . . . . 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 4088 . . . . . 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 7595 . . . . 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 1229 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( (
1 ... m )  u. 
{ ( m  + 
1 ) } ) 
~~  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
45 1z 10890 . . . . . 6  |-  1  e.  ZZ
46 1m1e0 10600 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
4746fveq2i 5867 . . . . . . . . 9  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
4835, 47eqtr4i 2499 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
4948eleq2i 2545 . . . . . . 7  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  ( 1  -  1 ) ) )
5049biimpi 194 . . . . . 6  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  ( 1  -  1 ) ) )
51 fzsuc2 11733 . . . . . 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 6698 . . . . . . . . 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 12022 . . . . . . . . 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 2545 . . . . . . . . . . 11  |-  ( m  e.  NN0  <->  m  e.  ( ZZ>=
`  0 ) )
6059biimpi 194 . . . . . . . . . 10  |-  ( m  e.  NN0  ->  m  e.  ( ZZ>= `  0 )
)
61 f1ocnvfv2 6169 . . . . . . . . . 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 6297 . . . . . . . 8  |-  ( m  e.  NN0  ->  ( ( G `  ( `' G `  m ) )  +  1 )  =  ( m  + 
1 ) )
6458, 63eqtrd 2508 . . . . . . 7  |-  ( m  e.  NN0  ->  ( G `
 suc  ( `' G `  m )
)  =  ( m  +  1 ) )
65 f1ocnvfv 6170 . . . . . . 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 4884 . . . . 5  |-  suc  ( `' G `  m )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } )
6967, 68syl6eq 2524 . . . 4  |-  ( ( m  e.  NN0  /\  ( 1 ... m
)  ~~  ( `' G `  m )
)  ->  ( `' G `  ( m  +  1 ) )  =  ( ( `' G `  m )  u.  { ( `' G `  m ) } ) )
7044, 53, 693brtr4d 4477 . . 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 10953 1  |-  ( N  e.  NN0  ->  ( 1 ... N )  ~~  ( `' G `  N ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474    i^i cin 3475   (/)c0 3785   {csn 4027   class class class wbr 4447    |-> cmpt 4505   Ord word 4877   suc csuc 4880   `'ccnv 4998    |` cres 5001   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   omcom 6678   reccrdg 7072    ~~ cen 7510   0cc0 9488   1c1 9489    + caddc 9491    - cmin 9801   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669
This theorem is referenced by:  fzen2  12042  cardfz  12043
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