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Theorem subfacp1lem1 23994
Description: Lemma for subfacp1 24001. 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 3529 . . . 4  |-  ( ( K  i^i  { 1 ,  M } )  =  (/)  <->  A. x  e.  K  -.  x  e.  { 1 ,  M } )
2 eldifi 3332 . . . . . . . . 9  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  ->  x  e.  ( 2 ... ( N  + 
1 ) ) )
3 elfzle1 10846 . . . . . . . . 9  |-  ( x  e.  ( 2 ... ( N  +  1 ) )  ->  2  <_  x )
4 1lt2 9933 . . . . . . . . . . . 12  |-  1  <  2
5 1re 8882 . . . . . . . . . . . . 13  |-  1  e.  RR
6 2re 9860 . . . . . . . . . . . . 13  |-  2  e.  RR
75, 6ltnlei 8984 . . . . . . . . . . . 12  |-  ( 1  <  2  <->  -.  2  <_  1 )
84, 7mpbi 199 . . . . . . . . . . 11  |-  -.  2  <_  1
9 breq2 4064 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
2  <_  x  <->  2  <_  1 ) )
108, 9mtbiri 294 . . . . . . . . . 10  |-  ( x  =  1  ->  -.  2  <_  x )
1110necon2ai 2524 . . . . . . . . 9  |-  ( 2  <_  x  ->  x  =/=  1 )
122, 3, 113syl 18 . . . . . . . 8  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  ->  x  =/=  1 )
13 eldifsni 3784 . . . . . . . 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 2408 . . . . . 6  |-  ( x  e.  K  ->  (
x  =/=  1  /\  x  =/=  M ) )
17 neanior 2564 . . . . . 6  |-  ( ( x  =/=  1  /\  x  =/=  M )  <->  -.  ( x  =  1  \/  x  =  M ) )
1816, 17sylib 188 . . . . 5  |-  ( x  e.  K  ->  -.  ( x  =  1  \/  x  =  M
) )
19 vex 2825 . . . . . 6  |-  x  e. 
_V
2019elpr 3692 . . . . 5  |-  ( x  e.  { 1 ,  M }  <->  ( x  =  1  \/  x  =  M ) )
2118, 20sylnibr 296 . . . 4  |-  ( x  e.  K  ->  -.  x  e.  { 1 ,  M } )
221, 21mprgbir 2647 . . 3  |-  ( K  i^i  { 1 ,  M } )  =  (/)
2322a1i 10 . 2  |-  ( ph  ->  ( K  i^i  {
1 ,  M }
)  =  (/) )
24 uncom 3353 . . . 4  |-  ( { 1 }  u.  ( K  u.  { M } ) )  =  ( ( K  u.  { M } )  u. 
{ 1 } )
25 1z 10100 . . . . . 6  |-  1  e.  ZZ
26 fzsn 10880 . . . . . 6  |-  ( 1  e.  ZZ  ->  (
1 ... 1 )  =  { 1 } )
2725, 26ax-mp 8 . . . . 5  |-  ( 1 ... 1 )  =  { 1 }
2815uneq1i 3359 . . . . . 6  |-  ( K  u.  { M }
)  =  ( ( ( 2 ... ( N  +  1 ) )  \  { M } )  u.  { M } )
29 undif1 3563 . . . . . 6  |-  ( ( ( 2 ... ( N  +  1 ) )  \  { M } )  u.  { M } )  =  ( ( 2 ... ( N  +  1 ) )  u.  { M } )
3028, 29eqtr2i 2337 . . . . 5  |-  ( ( 2 ... ( N  +  1 ) )  u.  { M }
)  =  ( K  u.  { M }
)
3127, 30uneq12i 3361 . . . 4  |-  ( ( 1 ... 1 )  u.  ( ( 2 ... ( N  + 
1 ) )  u. 
{ M } ) )  =  ( { 1 }  u.  ( K  u.  { M } ) )
32 df-pr 3681 . . . . . . 7  |-  { 1 ,  M }  =  ( { 1 }  u.  { M } )
3332equncomi 3355 . . . . . 6  |-  { 1 ,  M }  =  ( { M }  u.  { 1 } )
3433uneq2i 3360 . . . . 5  |-  ( K  u.  { 1 ,  M } )  =  ( K  u.  ( { M }  u.  {
1 } ) )
35 unass 3366 . . . . 5  |-  ( ( K  u.  { M } )  u.  {
1 } )  =  ( K  u.  ( { M }  u.  {
1 } ) )
3634, 35eqtr4i 2339 . . . 4  |-  ( K  u.  { 1 ,  M } )  =  ( ( K  u.  { M } )  u. 
{ 1 } )
3724, 31, 363eqtr4i 2346 . . 3  |-  ( ( 1 ... 1 )  u.  ( ( 2 ... ( N  + 
1 ) )  u. 
{ M } ) )  =  ( K  u.  { 1 ,  M } )
38 subfacp1lem1.m . . . . . . . 8  |-  ( ph  ->  M  e.  ( 2 ... ( N  + 
1 ) ) )
3938snssd 3797 . . . . . . 7  |-  ( ph  ->  { M }  C_  ( 2 ... ( N  +  1 ) ) )
40 ssequn2 3382 . . . . . . 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 9849 . . . . . . 7  |-  2  =  ( 1  +  1 )
4342oveq1i 5910 . . . . . 6  |-  ( 2 ... ( N  + 
1 ) )  =  ( ( 1  +  1 ) ... ( N  +  1 ) )
4441, 43syl6eq 2364 . . . . 5  |-  ( ph  ->  ( ( 2 ... ( N  +  1 ) )  u.  { M } )  =  ( ( 1  +  1 ) ... ( N  +  1 ) ) )
4544uneq2d 3363 . . . 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 9808 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  NN )
48 nnuz 10310 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
4947, 48syl6eleq 2406 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= ` 
1 ) )
50 eluzfz1 10850 . . . . 5  |-  ( ( N  +  1 )  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... ( N  +  1 ) ) )
51 fzsplit 10863 . . . . 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 2351 . . 3  |-  ( ph  ->  ( ( 1 ... 1 )  u.  (
( 2 ... ( N  +  1 ) )  u.  { M } ) )  =  ( 1 ... ( N  +  1 ) ) )
5437, 53syl5eqr 2362 . 2  |-  ( ph  ->  ( K  u.  {
1 ,  M }
)  =  ( 1 ... ( N  + 
1 ) ) )
5542oveq2i 5911 . . 3  |-  ( ( N  +  1 )  -  2 )  =  ( ( N  + 
1 )  -  (
1  +  1 ) )
56 fzfi 11081 . . . . . . . . 9  |-  ( 2 ... ( N  + 
1 ) )  e. 
Fin
57 diffi 7134 . . . . . . . . 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 2386 . . . . . . 7  |-  K  e. 
Fin
60 prfi 7176 . . . . . . 7  |-  { 1 ,  M }  e.  Fin
61 hashun 11411 . . . . . . 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 5567 . . . . . 6  |-  ( ph  ->  ( # `  ( K  u.  { 1 ,  M } ) )  =  ( # `  (
1 ... ( N  + 
1 ) ) ) )
64 neeq1 2487 . . . . . . . . . . 11  |-  ( x  =  M  ->  (
x  =/=  1  <->  M  =/=  1 ) )
653, 11syl 15 . . . . . . . . . . 11  |-  ( x  e.  ( 2 ... ( N  +  1 ) )  ->  x  =/=  1 )
6664, 65vtoclga 2883 . . . . . . . . . 10  |-  ( M  e.  ( 2 ... ( N  +  1 ) )  ->  M  =/=  1 )
6738, 66syl 15 . . . . . . . . 9  |-  ( ph  ->  M  =/=  1 )
6867necomd 2562 . . . . . . . 8  |-  ( ph  ->  1  =/=  M )
69 1ex 8878 . . . . . . . . 9  |-  1  e.  _V
70 subfacp1lem1.x . . . . . . . . 9  |-  M  e. 
_V
71 hashprg 11415 . . . . . . . . 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 5916 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  +  ( # `  { 1 ,  M } ) )  =  ( ( # `  K
)  +  2 ) )
7562, 63, 743eqtr3a 2372 . . . . 5  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( ( # `  K )  +  2 ) )
7647nnnn0d 10065 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
77 hashfz1 11392 . . . . . 6  |-  ( ( N  +  1 )  e.  NN0  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
7876, 77syl 15 . . . . 5  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( N  + 
1 ) )
7975, 78eqtr3d 2350 . . . 4  |-  ( ph  ->  ( ( # `  K
)  +  2 )  =  ( N  + 
1 ) )
8047nncnd 9807 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  CC )
81 2cn 9861 . . . . . 6  |-  2  e.  CC
8281a1i 10 . . . . 5  |-  ( ph  ->  2  e.  CC )
83 hashcl 11397 . . . . . . . 8  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
8459, 83ax-mp 8 . . . . . . 7  |-  ( # `  K )  e.  NN0
8584nn0cni 10024 . . . . . 6  |-  ( # `  K )  e.  CC
8685a1i 10 . . . . 5  |-  ( ph  ->  ( # `  K
)  e.  CC )
8780, 82, 86subadd2d 9221 . . . 4  |-  ( ph  ->  ( ( ( N  +  1 )  - 
2 )  =  (
# `  K )  <->  ( ( # `  K
)  +  2 )  =  ( N  + 
1 ) ) )
8879, 87mpbird 223 . . 3  |-  ( ph  ->  ( ( N  + 
1 )  -  2 )  =  ( # `  K ) )
8946nncnd 9807 . . . 4  |-  ( ph  ->  N  e.  CC )
90 ax-1cn 8840 . . . . 5  |-  1  e.  CC
9190a1i 10 . . . 4  |-  ( ph  ->  1  e.  CC )
9289, 91, 91pnpcan2d 9240 . . 3  |-  ( ph  ->  ( ( N  + 
1 )  -  (
1  +  1 ) )  =  ( N  -  1 ) )
9355, 88, 923eqtr3a 2372 . 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 1633    e. wcel 1701   {cab 2302    =/= wne 2479   A.wral 2577   _Vcvv 2822    \ cdif 3183    u. cun 3184    i^i cin 3185    C_ wss 3186   (/)c0 3489   {csn 3674   {cpr 3675   class class class wbr 4060    e. cmpt 4114   -1-1-onto->wf1o 5291   ` cfv 5292  (class class class)co 5900   Fincfn 6906   CCcc 8780   1c1 8783    + caddc 8785    < clt 8912    <_ cle 8913    - cmin 9082   NNcn 9791   2c2 9840   NN0cn0 10012   ZZcz 10071   ZZ>=cuz 10277   ...cfz 10829   #chash 11384
This theorem is referenced by:  subfacp1lem2a  23995  subfacp1lem3  23997  subfacp1lem4  23998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-card 7617  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-hash 11385
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