Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  subfacp1lem1 Structured version   Unicode version

Theorem subfacp1lem1 28291
Description: Lemma for subfacp1 28298. The set  K together with  { 1 ,  M } partitions the set  1 ... ( N  +  1 ). (Contributed by Mario Carneiro, 23-Jan-2015.)
Hypotheses
Ref Expression
derang.d  |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y )  =/=  y
) } ) )
subfac.n  |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1 ... n ) ) )
subfacp1lem.a  |-  A  =  { f  |  ( f : ( 1 ... ( N  + 
1 ) ) -1-1-onto-> ( 1 ... ( N  + 
1 ) )  /\  A. y  e.  ( 1 ... ( N  + 
1 ) ) ( f `  y )  =/=  y ) }
subfacp1lem1.n  |-  ( ph  ->  N  e.  NN )
subfacp1lem1.m  |-  ( ph  ->  M  e.  ( 2 ... ( N  + 
1 ) ) )
subfacp1lem1.x  |-  M  e. 
_V
subfacp1lem1.k  |-  K  =  ( ( 2 ... ( N  +  1 ) )  \  { M } )
Assertion
Ref Expression
subfacp1lem1  |-  ( ph  ->  ( ( K  i^i  { 1 ,  M }
)  =  (/)  /\  ( K  u.  { 1 ,  M } )  =  ( 1 ... ( N  +  1 ) )  /\  ( # `  K )  =  ( N  -  1 ) ) )
Distinct variable groups:    f, n, x, y, A    f, N, n, x, y    ph, x, y    D, n    f, K, n, x, y    f, M, x, y    S, n, x, y
Allowed substitution hints:    ph( f, n)    D( x, y, f)    S( f)    M( n)

Proof of Theorem subfacp1lem1
StepHypRef Expression
1 disj 3867 . . . 4  |-  ( ( K  i^i  { 1 ,  M } )  =  (/)  <->  A. x  e.  K  -.  x  e.  { 1 ,  M } )
2 eldifi 3626 . . . . . . . . 9  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  ->  x  e.  ( 2 ... ( N  + 
1 ) ) )
3 elfzle1 11689 . . . . . . . . 9  |-  ( x  e.  ( 2 ... ( N  +  1 ) )  ->  2  <_  x )
4 1lt2 10702 . . . . . . . . . . . 12  |-  1  <  2
5 1re 9595 . . . . . . . . . . . . 13  |-  1  e.  RR
6 2re 10605 . . . . . . . . . . . . 13  |-  2  e.  RR
75, 6ltnlei 9705 . . . . . . . . . . . 12  |-  ( 1  <  2  <->  -.  2  <_  1 )
84, 7mpbi 208 . . . . . . . . . . 11  |-  -.  2  <_  1
9 breq2 4451 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
2  <_  x  <->  2  <_  1 ) )
108, 9mtbiri 303 . . . . . . . . . 10  |-  ( x  =  1  ->  -.  2  <_  x )
1110necon2ai 2702 . . . . . . . . 9  |-  ( 2  <_  x  ->  x  =/=  1 )
122, 3, 113syl 20 . . . . . . . 8  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  ->  x  =/=  1 )
13 eldifsni 4153 . . . . . . . 8  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  ->  x  =/=  M )
1412, 13jca 532 . . . . . . 7  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  -> 
( x  =/=  1  /\  x  =/=  M
) )
15 subfacp1lem1.k . . . . . . 7  |-  K  =  ( ( 2 ... ( N  +  1 ) )  \  { M } )
1614, 15eleq2s 2575 . . . . . 6  |-  ( x  e.  K  ->  (
x  =/=  1  /\  x  =/=  M ) )
17 neanior 2792 . . . . . 6  |-  ( ( x  =/=  1  /\  x  =/=  M )  <->  -.  ( x  =  1  \/  x  =  M ) )
1816, 17sylib 196 . . . . 5  |-  ( x  e.  K  ->  -.  ( x  =  1  \/  x  =  M
) )
19 vex 3116 . . . . . 6  |-  x  e. 
_V
2019elpr 4045 . . . . 5  |-  ( x  e.  { 1 ,  M }  <->  ( x  =  1  \/  x  =  M ) )
2118, 20sylnibr 305 . . . 4  |-  ( x  e.  K  ->  -.  x  e.  { 1 ,  M } )
221, 21mprgbir 2828 . . 3  |-  ( K  i^i  { 1 ,  M } )  =  (/)
2322a1i 11 . 2  |-  ( ph  ->  ( K  i^i  {
1 ,  M }
)  =  (/) )
24 uncom 3648 . . . 4  |-  ( { 1 }  u.  ( K  u.  { M } ) )  =  ( ( K  u.  { M } )  u. 
{ 1 } )
25 1z 10894 . . . . . 6  |-  1  e.  ZZ
26 fzsn 11725 . . . . . 6  |-  ( 1  e.  ZZ  ->  (
1 ... 1 )  =  { 1 } )
2725, 26ax-mp 5 . . . . 5  |-  ( 1 ... 1 )  =  { 1 }
2815uneq1i 3654 . . . . . 6  |-  ( K  u.  { M }
)  =  ( ( ( 2 ... ( N  +  1 ) )  \  { M } )  u.  { M } )
29 undif1 3902 . . . . . 6  |-  ( ( ( 2 ... ( N  +  1 ) )  \  { M } )  u.  { M } )  =  ( ( 2 ... ( N  +  1 ) )  u.  { M } )
3028, 29eqtr2i 2497 . . . . 5  |-  ( ( 2 ... ( N  +  1 ) )  u.  { M }
)  =  ( K  u.  { M }
)
3127, 30uneq12i 3656 . . . 4  |-  ( ( 1 ... 1 )  u.  ( ( 2 ... ( N  + 
1 ) )  u. 
{ M } ) )  =  ( { 1 }  u.  ( K  u.  { M } ) )
32 df-pr 4030 . . . . . . 7  |-  { 1 ,  M }  =  ( { 1 }  u.  { M } )
3332equncomi 3650 . . . . . 6  |-  { 1 ,  M }  =  ( { M }  u.  { 1 } )
3433uneq2i 3655 . . . . 5  |-  ( K  u.  { 1 ,  M } )  =  ( K  u.  ( { M }  u.  {
1 } ) )
35 unass 3661 . . . . 5  |-  ( ( K  u.  { M } )  u.  {
1 } )  =  ( K  u.  ( { M }  u.  {
1 } ) )
3634, 35eqtr4i 2499 . . . 4  |-  ( K  u.  { 1 ,  M } )  =  ( ( K  u.  { M } )  u. 
{ 1 } )
3724, 31, 363eqtr4i 2506 . . 3  |-  ( ( 1 ... 1 )  u.  ( ( 2 ... ( N  + 
1 ) )  u. 
{ M } ) )  =  ( K  u.  { 1 ,  M } )
38 subfacp1lem1.m . . . . . . . 8  |-  ( ph  ->  M  e.  ( 2 ... ( N  + 
1 ) ) )
3938snssd 4172 . . . . . . 7  |-  ( ph  ->  { M }  C_  ( 2 ... ( N  +  1 ) ) )
40 ssequn2 3677 . . . . . . 7  |-  ( { M }  C_  (
2 ... ( N  + 
1 ) )  <->  ( (
2 ... ( N  + 
1 ) )  u. 
{ M } )  =  ( 2 ... ( N  +  1 ) ) )
4139, 40sylib 196 . . . . . 6  |-  ( ph  ->  ( ( 2 ... ( N  +  1 ) )  u.  { M } )  =  ( 2 ... ( N  +  1 ) ) )
42 df-2 10594 . . . . . . 7  |-  2  =  ( 1  +  1 )
4342oveq1i 6294 . . . . . 6  |-  ( 2 ... ( N  + 
1 ) )  =  ( ( 1  +  1 ) ... ( N  +  1 ) )
4441, 43syl6eq 2524 . . . . 5  |-  ( ph  ->  ( ( 2 ... ( N  +  1 ) )  u.  { M } )  =  ( ( 1  +  1 ) ... ( N  +  1 ) ) )
4544uneq2d 3658 . . . 4  |-  ( ph  ->  ( ( 1 ... 1 )  u.  (
( 2 ... ( N  +  1 ) )  u.  { M } ) )  =  ( ( 1 ... 1 )  u.  (
( 1  +  1 ) ... ( N  +  1 ) ) ) )
46 subfacp1lem1.n . . . . . . 7  |-  ( ph  ->  N  e.  NN )
4746peano2nnd 10553 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  NN )
48 nnuz 11117 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
4947, 48syl6eleq 2565 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= ` 
1 ) )
50 eluzfz1 11693 . . . . 5  |-  ( ( N  +  1 )  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... ( N  +  1 ) ) )
51 fzsplit 11711 . . . . 5  |-  ( 1  e.  ( 1 ... ( N  +  1 ) )  ->  (
1 ... ( N  + 
1 ) )  =  ( ( 1 ... 1 )  u.  (
( 1  +  1 ) ... ( N  +  1 ) ) ) )
5249, 50, 513syl 20 . . . 4  |-  ( ph  ->  ( 1 ... ( N  +  1 ) )  =  ( ( 1 ... 1 )  u.  ( ( 1  +  1 ) ... ( N  +  1 ) ) ) )
5345, 52eqtr4d 2511 . . 3  |-  ( ph  ->  ( ( 1 ... 1 )  u.  (
( 2 ... ( N  +  1 ) )  u.  { M } ) )  =  ( 1 ... ( N  +  1 ) ) )
5437, 53syl5eqr 2522 . 2  |-  ( ph  ->  ( K  u.  {
1 ,  M }
)  =  ( 1 ... ( N  + 
1 ) ) )
5542oveq2i 6295 . . 3  |-  ( ( N  +  1 )  -  2 )  =  ( ( N  + 
1 )  -  (
1  +  1 ) )
56 fzfi 12050 . . . . . . . . 9  |-  ( 2 ... ( N  + 
1 ) )  e. 
Fin
57 diffi 7751 . . . . . . . . 9  |-  ( ( 2 ... ( N  +  1 ) )  e.  Fin  ->  (
( 2 ... ( N  +  1 ) )  \  { M } )  e.  Fin )
5856, 57ax-mp 5 . . . . . . . 8  |-  ( ( 2 ... ( N  +  1 ) ) 
\  { M }
)  e.  Fin
5915, 58eqeltri 2551 . . . . . . 7  |-  K  e. 
Fin
60 prfi 7795 . . . . . . 7  |-  { 1 ,  M }  e.  Fin
61 hashun 12418 . . . . . . 7  |-  ( ( K  e.  Fin  /\  { 1 ,  M }  e.  Fin  /\  ( K  i^i  { 1 ,  M } )  =  (/) )  ->  ( # `  ( K  u.  {
1 ,  M }
) )  =  ( ( # `  K
)  +  ( # `  { 1 ,  M } ) ) )
6259, 60, 22, 61mp3an 1324 . . . . . 6  |-  ( # `  ( K  u.  {
1 ,  M }
) )  =  ( ( # `  K
)  +  ( # `  { 1 ,  M } ) )
6354fveq2d 5870 . . . . . 6  |-  ( ph  ->  ( # `  ( K  u.  { 1 ,  M } ) )  =  ( # `  (
1 ... ( N  + 
1 ) ) ) )
64 neeq1 2748 . . . . . . . . . . 11  |-  ( x  =  M  ->  (
x  =/=  1  <->  M  =/=  1 ) )
653, 11syl 16 . . . . . . . . . . 11  |-  ( x  e.  ( 2 ... ( N  +  1 ) )  ->  x  =/=  1 )
6664, 65vtoclga 3177 . . . . . . . . . 10  |-  ( M  e.  ( 2 ... ( N  +  1 ) )  ->  M  =/=  1 )
6738, 66syl 16 . . . . . . . . 9  |-  ( ph  ->  M  =/=  1 )
6867necomd 2738 . . . . . . . 8  |-  ( ph  ->  1  =/=  M )
69 1ex 9591 . . . . . . . . 9  |-  1  e.  _V
70 subfacp1lem1.x . . . . . . . . 9  |-  M  e. 
_V
71 hashprg 12428 . . . . . . . . 9  |-  ( ( 1  e.  _V  /\  M  e.  _V )  ->  ( 1  =/=  M  <->  (
# `  { 1 ,  M } )  =  2 ) )
7269, 70, 71mp2an 672 . . . . . . . 8  |-  ( 1  =/=  M  <->  ( # `  {
1 ,  M }
)  =  2 )
7368, 72sylib 196 . . . . . . 7  |-  ( ph  ->  ( # `  {
1 ,  M }
)  =  2 )
7473oveq2d 6300 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  +  ( # `  { 1 ,  M } ) )  =  ( ( # `  K
)  +  2 ) )
7562, 63, 743eqtr3a 2532 . . . . 5  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( ( # `  K )  +  2 ) )
7647nnnn0d 10852 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
77 hashfz1 12387 . . . . . 6  |-  ( ( N  +  1 )  e.  NN0  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
7876, 77syl 16 . . . . 5  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( N  + 
1 ) )
7975, 78eqtr3d 2510 . . . 4  |-  ( ph  ->  ( ( # `  K
)  +  2 )  =  ( N  + 
1 ) )
8047nncnd 10552 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  CC )
81 2cnd 10608 . . . . 5  |-  ( ph  ->  2  e.  CC )
82 hashcl 12396 . . . . . . . 8  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
8359, 82ax-mp 5 . . . . . . 7  |-  ( # `  K )  e.  NN0
8483nn0cni 10807 . . . . . 6  |-  ( # `  K )  e.  CC
8584a1i 11 . . . . 5  |-  ( ph  ->  ( # `  K
)  e.  CC )
8680, 81, 85subadd2d 9949 . . . 4  |-  ( ph  ->  ( ( ( N  +  1 )  - 
2 )  =  (
# `  K )  <->  ( ( # `  K
)  +  2 )  =  ( N  + 
1 ) ) )
8779, 86mpbird 232 . . 3  |-  ( ph  ->  ( ( N  + 
1 )  -  2 )  =  ( # `  K ) )
8846nncnd 10552 . . . 4  |-  ( ph  ->  N  e.  CC )
89 ax-1cn 9550 . . . . 5  |-  1  e.  CC
9089a1i 11 . . . 4  |-  ( ph  ->  1  e.  CC )
9188, 90, 90pnpcan2d 9968 . . 3  |-  ( ph  ->  ( ( N  + 
1 )  -  (
1  +  1 ) )  =  ( N  -  1 ) )
9255, 87, 913eqtr3a 2532 . 2  |-  ( ph  ->  ( # `  K
)  =  ( N  -  1 ) )
9323, 54, 923jca 1176 1  |-  ( ph  ->  ( ( K  i^i  { 1 ,  M }
)  =  (/)  /\  ( K  u.  { 1 ,  M } )  =  ( 1 ... ( N  +  1 ) )  /\  ( # `  K )  =  ( N  -  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2814   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   {cpr 4029   class class class wbr 4447    |-> cmpt 4505   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284   Fincfn 7516   CCcc 9490   1c1 9493    + caddc 9495    < clt 9628    <_ cle 9629    - cmin 9805   NNcn 10536   2c2 10585   NN0cn0 10795   ZZcz 10864   ZZ>=cuz 11082   ...cfz 11672   #chash 12373
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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-rmo 2822  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-int 4283  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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-hash 12374
This theorem is referenced by:  subfacp1lem2a  28292  subfacp1lem3  28294  subfacp1lem4  28295
  Copyright terms: Public domain W3C validator