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Theorem subfacp1lem1 29902
Description: Lemma for subfacp1 29909. 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 3805 . . . 4  |-  ( ( K  i^i  { 1 ,  M } )  =  (/)  <->  A. x  e.  K  -.  x  e.  { 1 ,  M } )
2 eldifi 3555 . . . . . . . . 9  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  ->  x  e.  ( 2 ... ( N  + 
1 ) ) )
3 elfzle1 11802 . . . . . . . . 9  |-  ( x  e.  ( 2 ... ( N  +  1 ) )  ->  2  <_  x )
4 1lt2 10776 . . . . . . . . . . . 12  |-  1  <  2
5 1re 9642 . . . . . . . . . . . . 13  |-  1  e.  RR
6 2re 10679 . . . . . . . . . . . . 13  |-  2  e.  RR
75, 6ltnlei 9755 . . . . . . . . . . . 12  |-  ( 1  <  2  <->  -.  2  <_  1 )
84, 7mpbi 212 . . . . . . . . . . 11  |-  -.  2  <_  1
9 breq2 4406 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
2  <_  x  <->  2  <_  1 ) )
108, 9mtbiri 305 . . . . . . . . . 10  |-  ( x  =  1  ->  -.  2  <_  x )
1110necon2ai 2653 . . . . . . . . 9  |-  ( 2  <_  x  ->  x  =/=  1 )
122, 3, 113syl 18 . . . . . . . 8  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  ->  x  =/=  1 )
13 eldifsni 4098 . . . . . . . 8  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  ->  x  =/=  M )
1412, 13jca 535 . . . . . . 7  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  -> 
( x  =/=  1  /\  x  =/=  M
) )
15 subfacp1lem1.k . . . . . . 7  |-  K  =  ( ( 2 ... ( N  +  1 ) )  \  { M } )
1614, 15eleq2s 2547 . . . . . 6  |-  ( x  e.  K  ->  (
x  =/=  1  /\  x  =/=  M ) )
17 neanior 2716 . . . . . 6  |-  ( ( x  =/=  1  /\  x  =/=  M )  <->  -.  ( x  =  1  \/  x  =  M ) )
1816, 17sylib 200 . . . . 5  |-  ( x  e.  K  ->  -.  ( x  =  1  \/  x  =  M
) )
19 vex 3048 . . . . . 6  |-  x  e. 
_V
2019elpr 3986 . . . . 5  |-  ( x  e.  { 1 ,  M }  <->  ( x  =  1  \/  x  =  M ) )
2118, 20sylnibr 307 . . . 4  |-  ( x  e.  K  ->  -.  x  e.  { 1 ,  M } )
221, 21mprgbir 2752 . . 3  |-  ( K  i^i  { 1 ,  M } )  =  (/)
2322a1i 11 . 2  |-  ( ph  ->  ( K  i^i  {
1 ,  M }
)  =  (/) )
24 uncom 3578 . . . 4  |-  ( { 1 }  u.  ( K  u.  { M } ) )  =  ( ( K  u.  { M } )  u. 
{ 1 } )
25 1z 10967 . . . . . 6  |-  1  e.  ZZ
26 fzsn 11840 . . . . . 6  |-  ( 1  e.  ZZ  ->  (
1 ... 1 )  =  { 1 } )
2725, 26ax-mp 5 . . . . 5  |-  ( 1 ... 1 )  =  { 1 }
2815uneq1i 3584 . . . . . 6  |-  ( K  u.  { M }
)  =  ( ( ( 2 ... ( N  +  1 ) )  \  { M } )  u.  { M } )
29 undif1 3842 . . . . . 6  |-  ( ( ( 2 ... ( N  +  1 ) )  \  { M } )  u.  { M } )  =  ( ( 2 ... ( N  +  1 ) )  u.  { M } )
3028, 29eqtr2i 2474 . . . . 5  |-  ( ( 2 ... ( N  +  1 ) )  u.  { M }
)  =  ( K  u.  { M }
)
3127, 30uneq12i 3586 . . . 4  |-  ( ( 1 ... 1 )  u.  ( ( 2 ... ( N  + 
1 ) )  u. 
{ M } ) )  =  ( { 1 }  u.  ( K  u.  { M } ) )
32 df-pr 3971 . . . . . . 7  |-  { 1 ,  M }  =  ( { 1 }  u.  { M } )
3332equncomi 3580 . . . . . 6  |-  { 1 ,  M }  =  ( { M }  u.  { 1 } )
3433uneq2i 3585 . . . . 5  |-  ( K  u.  { 1 ,  M } )  =  ( K  u.  ( { M }  u.  {
1 } ) )
35 unass 3591 . . . . 5  |-  ( ( K  u.  { M } )  u.  {
1 } )  =  ( K  u.  ( { M }  u.  {
1 } ) )
3634, 35eqtr4i 2476 . . . 4  |-  ( K  u.  { 1 ,  M } )  =  ( ( K  u.  { M } )  u. 
{ 1 } )
3724, 31, 363eqtr4i 2483 . . 3  |-  ( ( 1 ... 1 )  u.  ( ( 2 ... ( N  + 
1 ) )  u. 
{ M } ) )  =  ( K  u.  { 1 ,  M } )
38 subfacp1lem1.m . . . . . . . 8  |-  ( ph  ->  M  e.  ( 2 ... ( N  + 
1 ) ) )
3938snssd 4117 . . . . . . 7  |-  ( ph  ->  { M }  C_  ( 2 ... ( N  +  1 ) ) )
40 ssequn2 3607 . . . . . . 7  |-  ( { M }  C_  (
2 ... ( N  + 
1 ) )  <->  ( (
2 ... ( N  + 
1 ) )  u. 
{ M } )  =  ( 2 ... ( N  +  1 ) ) )
4139, 40sylib 200 . . . . . 6  |-  ( ph  ->  ( ( 2 ... ( N  +  1 ) )  u.  { M } )  =  ( 2 ... ( N  +  1 ) ) )
42 df-2 10668 . . . . . . 7  |-  2  =  ( 1  +  1 )
4342oveq1i 6300 . . . . . 6  |-  ( 2 ... ( N  + 
1 ) )  =  ( ( 1  +  1 ) ... ( N  +  1 ) )
4441, 43syl6eq 2501 . . . . 5  |-  ( ph  ->  ( ( 2 ... ( N  +  1 ) )  u.  { M } )  =  ( ( 1  +  1 ) ... ( N  +  1 ) ) )
4544uneq2d 3588 . . . 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 10626 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  NN )
48 nnuz 11194 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
4947, 48syl6eleq 2539 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= ` 
1 ) )
50 eluzfz1 11806 . . . . 5  |-  ( ( N  +  1 )  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... ( N  +  1 ) ) )
51 fzsplit 11825 . . . . 5  |-  ( 1  e.  ( 1 ... ( N  +  1 ) )  ->  (
1 ... ( N  + 
1 ) )  =  ( ( 1 ... 1 )  u.  (
( 1  +  1 ) ... ( N  +  1 ) ) ) )
5249, 50, 513syl 18 . . . 4  |-  ( ph  ->  ( 1 ... ( N  +  1 ) )  =  ( ( 1 ... 1 )  u.  ( ( 1  +  1 ) ... ( N  +  1 ) ) ) )
5345, 52eqtr4d 2488 . . 3  |-  ( ph  ->  ( ( 1 ... 1 )  u.  (
( 2 ... ( N  +  1 ) )  u.  { M } ) )  =  ( 1 ... ( N  +  1 ) ) )
5437, 53syl5eqr 2499 . 2  |-  ( ph  ->  ( K  u.  {
1 ,  M }
)  =  ( 1 ... ( N  + 
1 ) ) )
5542oveq2i 6301 . . 3  |-  ( ( N  +  1 )  -  2 )  =  ( ( N  + 
1 )  -  (
1  +  1 ) )
56 fzfi 12185 . . . . . . . . 9  |-  ( 2 ... ( N  + 
1 ) )  e. 
Fin
57 diffi 7803 . . . . . . . . 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 2525 . . . . . . 7  |-  K  e. 
Fin
60 prfi 7846 . . . . . . 7  |-  { 1 ,  M }  e.  Fin
61 hashun 12561 . . . . . . 7  |-  ( ( K  e.  Fin  /\  { 1 ,  M }  e.  Fin  /\  ( K  i^i  { 1 ,  M } )  =  (/) )  ->  ( # `  ( K  u.  {
1 ,  M }
) )  =  ( ( # `  K
)  +  ( # `  { 1 ,  M } ) ) )
6259, 60, 22, 61mp3an 1364 . . . . . 6  |-  ( # `  ( K  u.  {
1 ,  M }
) )  =  ( ( # `  K
)  +  ( # `  { 1 ,  M } ) )
6354fveq2d 5869 . . . . . 6  |-  ( ph  ->  ( # `  ( K  u.  { 1 ,  M } ) )  =  ( # `  (
1 ... ( N  + 
1 ) ) ) )
64 neeq1 2686 . . . . . . . . . . 11  |-  ( x  =  M  ->  (
x  =/=  1  <->  M  =/=  1 ) )
653, 11syl 17 . . . . . . . . . . 11  |-  ( x  e.  ( 2 ... ( N  +  1 ) )  ->  x  =/=  1 )
6664, 65vtoclga 3113 . . . . . . . . . 10  |-  ( M  e.  ( 2 ... ( N  +  1 ) )  ->  M  =/=  1 )
6738, 66syl 17 . . . . . . . . 9  |-  ( ph  ->  M  =/=  1 )
6867necomd 2679 . . . . . . . 8  |-  ( ph  ->  1  =/=  M )
69 1ex 9638 . . . . . . . . 9  |-  1  e.  _V
70 subfacp1lem1.x . . . . . . . . 9  |-  M  e. 
_V
71 hashprg 12572 . . . . . . . . 9  |-  ( ( 1  e.  _V  /\  M  e.  _V )  ->  ( 1  =/=  M  <->  (
# `  { 1 ,  M } )  =  2 ) )
7269, 70, 71mp2an 678 . . . . . . . 8  |-  ( 1  =/=  M  <->  ( # `  {
1 ,  M }
)  =  2 )
7368, 72sylib 200 . . . . . . 7  |-  ( ph  ->  ( # `  {
1 ,  M }
)  =  2 )
7473oveq2d 6306 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  +  ( # `  { 1 ,  M } ) )  =  ( ( # `  K
)  +  2 ) )
7562, 63, 743eqtr3a 2509 . . . . 5  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( ( # `  K )  +  2 ) )
7647nnnn0d 10925 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
77 hashfz1 12529 . . . . . 6  |-  ( ( N  +  1 )  e.  NN0  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
7876, 77syl 17 . . . . 5  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( N  + 
1 ) )
7975, 78eqtr3d 2487 . . . 4  |-  ( ph  ->  ( ( # `  K
)  +  2 )  =  ( N  + 
1 ) )
8047nncnd 10625 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  CC )
81 2cnd 10682 . . . . 5  |-  ( ph  ->  2  e.  CC )
82 hashcl 12538 . . . . . . . 8  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
8359, 82ax-mp 5 . . . . . . 7  |-  ( # `  K )  e.  NN0
8483nn0cni 10881 . . . . . 6  |-  ( # `  K )  e.  CC
8584a1i 11 . . . . 5  |-  ( ph  ->  ( # `  K
)  e.  CC )
8680, 81, 85subadd2d 10005 . . . 4  |-  ( ph  ->  ( ( ( N  +  1 )  - 
2 )  =  (
# `  K )  <->  ( ( # `  K
)  +  2 )  =  ( N  + 
1 ) ) )
8779, 86mpbird 236 . . 3  |-  ( ph  ->  ( ( N  + 
1 )  -  2 )  =  ( # `  K ) )
8846nncnd 10625 . . . 4  |-  ( ph  ->  N  e.  CC )
89 1cnd 9659 . . . 4  |-  ( ph  ->  1  e.  CC )
9088, 89, 89pnpcan2d 10024 . . 3  |-  ( ph  ->  ( ( N  + 
1 )  -  (
1  +  1 ) )  =  ( N  -  1 ) )
9155, 87, 903eqtr3a 2509 . 2  |-  ( ph  ->  ( # `  K
)  =  ( N  -  1 ) )
9223, 54, 913jca 1188 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 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   {cab 2437    =/= wne 2622   A.wral 2737   _Vcvv 3045    \ cdif 3401    u. cun 3402    i^i cin 3403    C_ wss 3404   (/)c0 3731   {csn 3968   {cpr 3970   class class class wbr 4402    |-> cmpt 4461   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290   Fincfn 7569   CCcc 9537   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676    - cmin 9860   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784   #chash 12515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-hash 12516
This theorem is referenced by:  subfacp1lem2a  29903  subfacp1lem3  29905  subfacp1lem4  29906
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