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Theorem subfacp1lem1 23725
Description: Lemma for subfacp1 23732. 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 3508 . . . 4  |-  ( ( K  i^i  { 1 ,  M } )  =  (/)  <->  A. x  e.  K  -.  x  e.  { 1 ,  M } )
2 eldifi 3311 . . . . . . . . 9  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  ->  x  e.  ( 2 ... ( N  + 
1 ) ) )
3 elfzle1 10815 . . . . . . . . 9  |-  ( x  e.  ( 2 ... ( N  +  1 ) )  ->  2  <_  x )
4 1lt2 9902 . . . . . . . . . . . 12  |-  1  <  2
5 1re 8853 . . . . . . . . . . . . 13  |-  1  e.  RR
6 2re 9831 . . . . . . . . . . . . 13  |-  2  e.  RR
75, 6ltnlei 8955 . . . . . . . . . . . 12  |-  ( 1  <  2  <->  -.  2  <_  1 )
84, 7mpbi 199 . . . . . . . . . . 11  |-  -.  2  <_  1
9 breq2 4043 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
2  <_  x  <->  2  <_  1 ) )
108, 9mtbiri 294 . . . . . . . . . 10  |-  ( x  =  1  ->  -.  2  <_  x )
1110necon2ai 2504 . . . . . . . . 9  |-  ( 2  <_  x  ->  x  =/=  1 )
122, 3, 113syl 18 . . . . . . . 8  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  ->  x  =/=  1 )
13 eldifsni 3763 . . . . . . . 8  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  ->  x  =/=  M )
1412, 13jca 518 . . . . . . 7  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  -> 
( x  =/=  1  /\  x  =/=  M
) )
15 subfacp1lem1.k . . . . . . 7  |-  K  =  ( ( 2 ... ( N  +  1 ) )  \  { M } )
1614, 15eleq2s 2388 . . . . . 6  |-  ( x  e.  K  ->  (
x  =/=  1  /\  x  =/=  M ) )
17 neanior 2544 . . . . . 6  |-  ( ( x  =/=  1  /\  x  =/=  M )  <->  -.  ( x  =  1  \/  x  =  M ) )
1816, 17sylib 188 . . . . 5  |-  ( x  e.  K  ->  -.  ( x  =  1  \/  x  =  M
) )
19 vex 2804 . . . . . 6  |-  x  e. 
_V
2019elpr 3671 . . . . 5  |-  ( x  e.  { 1 ,  M }  <->  ( x  =  1  \/  x  =  M ) )
2118, 20sylnibr 296 . . . 4  |-  ( x  e.  K  ->  -.  x  e.  { 1 ,  M } )
221, 21mprgbir 2626 . . 3  |-  ( K  i^i  { 1 ,  M } )  =  (/)
2322a1i 10 . 2  |-  ( ph  ->  ( K  i^i  {
1 ,  M }
)  =  (/) )
24 uncom 3332 . . . 4  |-  ( { 1 }  u.  ( K  u.  { M } ) )  =  ( ( K  u.  { M } )  u. 
{ 1 } )
25 1z 10069 . . . . . 6  |-  1  e.  ZZ
26 fzsn 10849 . . . . . 6  |-  ( 1  e.  ZZ  ->  (
1 ... 1 )  =  { 1 } )
2725, 26ax-mp 8 . . . . 5  |-  ( 1 ... 1 )  =  { 1 }
2815uneq1i 3338 . . . . . 6  |-  ( K  u.  { M }
)  =  ( ( ( 2 ... ( N  +  1 ) )  \  { M } )  u.  { M } )
29 undif1 3542 . . . . . 6  |-  ( ( ( 2 ... ( N  +  1 ) )  \  { M } )  u.  { M } )  =  ( ( 2 ... ( N  +  1 ) )  u.  { M } )
3028, 29eqtr2i 2317 . . . . 5  |-  ( ( 2 ... ( N  +  1 ) )  u.  { M }
)  =  ( K  u.  { M }
)
3127, 30uneq12i 3340 . . . 4  |-  ( ( 1 ... 1 )  u.  ( ( 2 ... ( N  + 
1 ) )  u. 
{ M } ) )  =  ( { 1 }  u.  ( K  u.  { M } ) )
32 df-pr 3660 . . . . . . 7  |-  { 1 ,  M }  =  ( { 1 }  u.  { M } )
3332equncomi 3334 . . . . . 6  |-  { 1 ,  M }  =  ( { M }  u.  { 1 } )
3433uneq2i 3339 . . . . 5  |-  ( K  u.  { 1 ,  M } )  =  ( K  u.  ( { M }  u.  {
1 } ) )
35 unass 3345 . . . . 5  |-  ( ( K  u.  { M } )  u.  {
1 } )  =  ( K  u.  ( { M }  u.  {
1 } ) )
3634, 35eqtr4i 2319 . . . 4  |-  ( K  u.  { 1 ,  M } )  =  ( ( K  u.  { M } )  u. 
{ 1 } )
3724, 31, 363eqtr4i 2326 . . 3  |-  ( ( 1 ... 1 )  u.  ( ( 2 ... ( N  + 
1 ) )  u. 
{ M } ) )  =  ( K  u.  { 1 ,  M } )
38 subfacp1lem1.m . . . . . . . 8  |-  ( ph  ->  M  e.  ( 2 ... ( N  + 
1 ) ) )
3938snssd 3776 . . . . . . 7  |-  ( ph  ->  { M }  C_  ( 2 ... ( N  +  1 ) ) )
40 ssequn2 3361 . . . . . . 7  |-  ( { M }  C_  (
2 ... ( N  + 
1 ) )  <->  ( (
2 ... ( N  + 
1 ) )  u. 
{ M } )  =  ( 2 ... ( N  +  1 ) ) )
4139, 40sylib 188 . . . . . 6  |-  ( ph  ->  ( ( 2 ... ( N  +  1 ) )  u.  { M } )  =  ( 2 ... ( N  +  1 ) ) )
42 df-2 9820 . . . . . . 7  |-  2  =  ( 1  +  1 )
4342oveq1i 5884 . . . . . 6  |-  ( 2 ... ( N  + 
1 ) )  =  ( ( 1  +  1 ) ... ( N  +  1 ) )
4441, 43syl6eq 2344 . . . . 5  |-  ( ph  ->  ( ( 2 ... ( N  +  1 ) )  u.  { M } )  =  ( ( 1  +  1 ) ... ( N  +  1 ) ) )
4544uneq2d 3342 . . . 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 9779 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  NN )
48 nnuz 10279 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
4947, 48syl6eleq 2386 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= ` 
1 ) )
50 eluzfz1 10819 . . . . 5  |-  ( ( N  +  1 )  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... ( N  +  1 ) ) )
51 fzsplit 10832 . . . . 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 2331 . . 3  |-  ( ph  ->  ( ( 1 ... 1 )  u.  (
( 2 ... ( N  +  1 ) )  u.  { M } ) )  =  ( 1 ... ( N  +  1 ) ) )
5437, 53syl5eqr 2342 . 2  |-  ( ph  ->  ( K  u.  {
1 ,  M }
)  =  ( 1 ... ( N  + 
1 ) ) )
5542oveq2i 5885 . . 3  |-  ( ( N  +  1 )  -  2 )  =  ( ( N  + 
1 )  -  (
1  +  1 ) )
56 fzfi 11050 . . . . . . . . 9  |-  ( 2 ... ( N  + 
1 ) )  e. 
Fin
57 diffi 7105 . . . . . . . . 9  |-  ( ( 2 ... ( N  +  1 ) )  e.  Fin  ->  (
( 2 ... ( N  +  1 ) )  \  { M } )  e.  Fin )
5856, 57ax-mp 8 . . . . . . . 8  |-  ( ( 2 ... ( N  +  1 ) ) 
\  { M }
)  e.  Fin
5915, 58eqeltri 2366 . . . . . . 7  |-  K  e. 
Fin
60 prfi 7147 . . . . . . 7  |-  { 1 ,  M }  e.  Fin
61 hashun 11380 . . . . . . 7  |-  ( ( K  e.  Fin  /\  { 1 ,  M }  e.  Fin  /\  ( K  i^i  { 1 ,  M } )  =  (/) )  ->  ( # `  ( K  u.  {
1 ,  M }
) )  =  ( ( # `  K
)  +  ( # `  { 1 ,  M } ) ) )
6259, 60, 22, 61mp3an 1277 . . . . . 6  |-  ( # `  ( K  u.  {
1 ,  M }
) )  =  ( ( # `  K
)  +  ( # `  { 1 ,  M } ) )
6354fveq2d 5545 . . . . . 6  |-  ( ph  ->  ( # `  ( K  u.  { 1 ,  M } ) )  =  ( # `  (
1 ... ( N  + 
1 ) ) ) )
64 neeq1 2467 . . . . . . . . . . 11  |-  ( x  =  M  ->  (
x  =/=  1  <->  M  =/=  1 ) )
653, 11syl 15 . . . . . . . . . . 11  |-  ( x  e.  ( 2 ... ( N  +  1 ) )  ->  x  =/=  1 )
6664, 65vtoclga 2862 . . . . . . . . . 10  |-  ( M  e.  ( 2 ... ( N  +  1 ) )  ->  M  =/=  1 )
6738, 66syl 15 . . . . . . . . 9  |-  ( ph  ->  M  =/=  1 )
6867necomd 2542 . . . . . . . 8  |-  ( ph  ->  1  =/=  M )
69 1ex 8849 . . . . . . . . 9  |-  1  e.  _V
70 subfacp1lem1.x . . . . . . . . 9  |-  M  e. 
_V
71 hashprg 11384 . . . . . . . . 9  |-  ( ( 1  e.  _V  /\  M  e.  _V )  ->  ( 1  =/=  M  <->  (
# `  { 1 ,  M } )  =  2 ) )
7269, 70, 71mp2an 653 . . . . . . . 8  |-  ( 1  =/=  M  <->  ( # `  {
1 ,  M }
)  =  2 )
7368, 72sylib 188 . . . . . . 7  |-  ( ph  ->  ( # `  {
1 ,  M }
)  =  2 )
7473oveq2d 5890 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  +  ( # `  { 1 ,  M } ) )  =  ( ( # `  K
)  +  2 ) )
7562, 63, 743eqtr3a 2352 . . . . 5  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( ( # `  K )  +  2 ) )
7647nnnn0d 10034 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
77 hashfz1 11361 . . . . . 6  |-  ( ( N  +  1 )  e.  NN0  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
7876, 77syl 15 . . . . 5  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( N  + 
1 ) )
7975, 78eqtr3d 2330 . . . 4  |-  ( ph  ->  ( ( # `  K
)  +  2 )  =  ( N  + 
1 ) )
8047nncnd 9778 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  CC )
81 2cn 9832 . . . . . 6  |-  2  e.  CC
8281a1i 10 . . . . 5  |-  ( ph  ->  2  e.  CC )
83 hashcl 11366 . . . . . . . 8  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
8459, 83ax-mp 8 . . . . . . 7  |-  ( # `  K )  e.  NN0
8584nn0cni 9993 . . . . . 6  |-  ( # `  K )  e.  CC
8685a1i 10 . . . . 5  |-  ( ph  ->  ( # `  K
)  e.  CC )
8780, 82, 86subadd2d 9192 . . . 4  |-  ( ph  ->  ( ( ( N  +  1 )  - 
2 )  =  (
# `  K )  <->  ( ( # `  K
)  +  2 )  =  ( N  + 
1 ) ) )
8879, 87mpbird 223 . . 3  |-  ( ph  ->  ( ( N  + 
1 )  -  2 )  =  ( # `  K ) )
8946nncnd 9778 . . . 4  |-  ( ph  ->  N  e.  CC )
90 ax-1cn 8811 . . . . 5  |-  1  e.  CC
9190a1i 10 . . . 4  |-  ( ph  ->  1  e.  CC )
9289, 91, 91pnpcan2d 9211 . . 3  |-  ( ph  ->  ( ( N  + 
1 )  -  (
1  +  1 ) )  =  ( N  -  1 ) )
9355, 88, 923eqtr3a 2352 . 2  |-  ( ph  ->  ( # `  K
)  =  ( N  -  1 ) )
9423, 54, 933jca 1132 1  |-  ( ph  ->  ( ( K  i^i  { 1 ,  M }
)  =  (/)  /\  ( K  u.  { 1 ,  M } )  =  ( 1 ... ( N  +  1 ) )  /\  ( # `  K )  =  ( N  -  1 ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   {cpr 3654   class class class wbr 4039    e. cmpt 4093   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   #chash 11353
This theorem is referenced by:  subfacp1lem2a  23726  subfacp1lem3  23728  subfacp1lem4  23729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354
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