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Theorem subfacp1lem1 27072
Description: Lemma for subfacp1 27079. 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 3724 . . . 4  |-  ( ( K  i^i  { 1 ,  M } )  =  (/)  <->  A. x  e.  K  -.  x  e.  { 1 ,  M } )
2 eldifi 3483 . . . . . . . . 9  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  ->  x  e.  ( 2 ... ( N  + 
1 ) ) )
3 elfzle1 11459 . . . . . . . . 9  |-  ( x  e.  ( 2 ... ( N  +  1 ) )  ->  2  <_  x )
4 1lt2 10493 . . . . . . . . . . . 12  |-  1  <  2
5 1re 9390 . . . . . . . . . . . . 13  |-  1  e.  RR
6 2re 10396 . . . . . . . . . . . . 13  |-  2  e.  RR
75, 6ltnlei 9500 . . . . . . . . . . . 12  |-  ( 1  <  2  <->  -.  2  <_  1 )
84, 7mpbi 208 . . . . . . . . . . 11  |-  -.  2  <_  1
9 breq2 4301 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
2  <_  x  <->  2  <_  1 ) )
108, 9mtbiri 303 . . . . . . . . . 10  |-  ( x  =  1  ->  -.  2  <_  x )
1110necon2ai 2661 . . . . . . . . 9  |-  ( 2  <_  x  ->  x  =/=  1 )
122, 3, 113syl 20 . . . . . . . 8  |-  ( x  e.  ( ( 2 ... ( N  + 
1 ) )  \  { M } )  ->  x  =/=  1 )
13 eldifsni 4006 . . . . . . . 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 2535 . . . . . 6  |-  ( x  e.  K  ->  (
x  =/=  1  /\  x  =/=  M ) )
17 neanior 2702 . . . . . 6  |-  ( ( x  =/=  1  /\  x  =/=  M )  <->  -.  ( x  =  1  \/  x  =  M ) )
1816, 17sylib 196 . . . . 5  |-  ( x  e.  K  ->  -.  ( x  =  1  \/  x  =  M
) )
19 vex 2980 . . . . . 6  |-  x  e. 
_V
2019elpr 3900 . . . . 5  |-  ( x  e.  { 1 ,  M }  <->  ( x  =  1  \/  x  =  M ) )
2118, 20sylnibr 305 . . . 4  |-  ( x  e.  K  ->  -.  x  e.  { 1 ,  M } )
221, 21mprgbir 2791 . . 3  |-  ( K  i^i  { 1 ,  M } )  =  (/)
2322a1i 11 . 2  |-  ( ph  ->  ( K  i^i  {
1 ,  M }
)  =  (/) )
24 uncom 3505 . . . 4  |-  ( { 1 }  u.  ( K  u.  { M } ) )  =  ( ( K  u.  { M } )  u. 
{ 1 } )
25 1z 10681 . . . . . 6  |-  1  e.  ZZ
26 fzsn 11505 . . . . . 6  |-  ( 1  e.  ZZ  ->  (
1 ... 1 )  =  { 1 } )
2725, 26ax-mp 5 . . . . 5  |-  ( 1 ... 1 )  =  { 1 }
2815uneq1i 3511 . . . . . 6  |-  ( K  u.  { M }
)  =  ( ( ( 2 ... ( N  +  1 ) )  \  { M } )  u.  { M } )
29 undif1 3759 . . . . . 6  |-  ( ( ( 2 ... ( N  +  1 ) )  \  { M } )  u.  { M } )  =  ( ( 2 ... ( N  +  1 ) )  u.  { M } )
3028, 29eqtr2i 2464 . . . . 5  |-  ( ( 2 ... ( N  +  1 ) )  u.  { M }
)  =  ( K  u.  { M }
)
3127, 30uneq12i 3513 . . . 4  |-  ( ( 1 ... 1 )  u.  ( ( 2 ... ( N  + 
1 ) )  u. 
{ M } ) )  =  ( { 1 }  u.  ( K  u.  { M } ) )
32 df-pr 3885 . . . . . . 7  |-  { 1 ,  M }  =  ( { 1 }  u.  { M } )
3332equncomi 3507 . . . . . 6  |-  { 1 ,  M }  =  ( { M }  u.  { 1 } )
3433uneq2i 3512 . . . . 5  |-  ( K  u.  { 1 ,  M } )  =  ( K  u.  ( { M }  u.  {
1 } ) )
35 unass 3518 . . . . 5  |-  ( ( K  u.  { M } )  u.  {
1 } )  =  ( K  u.  ( { M }  u.  {
1 } ) )
3634, 35eqtr4i 2466 . . . 4  |-  ( K  u.  { 1 ,  M } )  =  ( ( K  u.  { M } )  u. 
{ 1 } )
3724, 31, 363eqtr4i 2473 . . 3  |-  ( ( 1 ... 1 )  u.  ( ( 2 ... ( N  + 
1 ) )  u. 
{ M } ) )  =  ( K  u.  { 1 ,  M } )
38 subfacp1lem1.m . . . . . . . 8  |-  ( ph  ->  M  e.  ( 2 ... ( N  + 
1 ) ) )
3938snssd 4023 . . . . . . 7  |-  ( ph  ->  { M }  C_  ( 2 ... ( N  +  1 ) ) )
40 ssequn2 3534 . . . . . . 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 10385 . . . . . . 7  |-  2  =  ( 1  +  1 )
4342oveq1i 6106 . . . . . 6  |-  ( 2 ... ( N  + 
1 ) )  =  ( ( 1  +  1 ) ... ( N  +  1 ) )
4441, 43syl6eq 2491 . . . . 5  |-  ( ph  ->  ( ( 2 ... ( N  +  1 ) )  u.  { M } )  =  ( ( 1  +  1 ) ... ( N  +  1 ) ) )
4544uneq2d 3515 . . . 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 10344 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  NN )
48 nnuz 10901 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
4947, 48syl6eleq 2533 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= ` 
1 ) )
50 eluzfz1 11463 . . . . 5  |-  ( ( N  +  1 )  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... ( N  +  1 ) ) )
51 fzsplit 11480 . . . . 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 2478 . . 3  |-  ( ph  ->  ( ( 1 ... 1 )  u.  (
( 2 ... ( N  +  1 ) )  u.  { M } ) )  =  ( 1 ... ( N  +  1 ) ) )
5437, 53syl5eqr 2489 . 2  |-  ( ph  ->  ( K  u.  {
1 ,  M }
)  =  ( 1 ... ( N  + 
1 ) ) )
5542oveq2i 6107 . . 3  |-  ( ( N  +  1 )  -  2 )  =  ( ( N  + 
1 )  -  (
1  +  1 ) )
56 fzfi 11799 . . . . . . . . 9  |-  ( 2 ... ( N  + 
1 ) )  e. 
Fin
57 diffi 7548 . . . . . . . . 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 2513 . . . . . . 7  |-  K  e. 
Fin
60 prfi 7591 . . . . . . 7  |-  { 1 ,  M }  e.  Fin
61 hashun 12150 . . . . . . 7  |-  ( ( K  e.  Fin  /\  { 1 ,  M }  e.  Fin  /\  ( K  i^i  { 1 ,  M } )  =  (/) )  ->  ( # `  ( K  u.  {
1 ,  M }
) )  =  ( ( # `  K
)  +  ( # `  { 1 ,  M } ) ) )
6259, 60, 22, 61mp3an 1314 . . . . . 6  |-  ( # `  ( K  u.  {
1 ,  M }
) )  =  ( ( # `  K
)  +  ( # `  { 1 ,  M } ) )
6354fveq2d 5700 . . . . . 6  |-  ( ph  ->  ( # `  ( K  u.  { 1 ,  M } ) )  =  ( # `  (
1 ... ( N  + 
1 ) ) ) )
64 neeq1 2621 . . . . . . . . . . 11  |-  ( x  =  M  ->  (
x  =/=  1  <->  M  =/=  1 ) )
653, 11syl 16 . . . . . . . . . . 11  |-  ( x  e.  ( 2 ... ( N  +  1 ) )  ->  x  =/=  1 )
6664, 65vtoclga 3041 . . . . . . . . . 10  |-  ( M  e.  ( 2 ... ( N  +  1 ) )  ->  M  =/=  1 )
6738, 66syl 16 . . . . . . . . 9  |-  ( ph  ->  M  =/=  1 )
6867necomd 2700 . . . . . . . 8  |-  ( ph  ->  1  =/=  M )
69 1ex 9386 . . . . . . . . 9  |-  1  e.  _V
70 subfacp1lem1.x . . . . . . . . 9  |-  M  e. 
_V
71 hashprg 12160 . . . . . . . . 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 6112 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  +  ( # `  { 1 ,  M } ) )  =  ( ( # `  K
)  +  2 ) )
7562, 63, 743eqtr3a 2499 . . . . 5  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( ( # `  K )  +  2 ) )
7647nnnn0d 10641 . . . . . 6  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
77 hashfz1 12122 . . . . . 6  |-  ( ( N  +  1 )  e.  NN0  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
7876, 77syl 16 . . . . 5  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( N  + 
1 ) )
7975, 78eqtr3d 2477 . . . 4  |-  ( ph  ->  ( ( # `  K
)  +  2 )  =  ( N  + 
1 ) )
8047nncnd 10343 . . . . 5  |-  ( ph  ->  ( N  +  1 )  e.  CC )
81 2cnd 10399 . . . . 5  |-  ( ph  ->  2  e.  CC )
82 hashcl 12131 . . . . . . . 8  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
8359, 82ax-mp 5 . . . . . . 7  |-  ( # `  K )  e.  NN0
8483nn0cni 10596 . . . . . 6  |-  ( # `  K )  e.  CC
8584a1i 11 . . . . 5  |-  ( ph  ->  ( # `  K
)  e.  CC )
8680, 81, 85subadd2d 9743 . . . 4  |-  ( ph  ->  ( ( ( N  +  1 )  - 
2 )  =  (
# `  K )  <->  ( ( # `  K
)  +  2 )  =  ( N  + 
1 ) ) )
8779, 86mpbird 232 . . 3  |-  ( ph  ->  ( ( N  + 
1 )  -  2 )  =  ( # `  K ) )
8846nncnd 10343 . . . 4  |-  ( ph  ->  N  e.  CC )
89 ax-1cn 9345 . . . . 5  |-  1  e.  CC
9089a1i 11 . . . 4  |-  ( ph  ->  1  e.  CC )
9188, 90, 90pnpcan2d 9762 . . 3  |-  ( ph  ->  ( ( N  + 
1 )  -  (
1  +  1 ) )  =  ( N  -  1 ) )
9255, 87, 913eqtr3a 2499 . 2  |-  ( ph  ->  ( # `  K
)  =  ( N  -  1 ) )
9323, 54, 923jca 1168 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 965    = wceq 1369    e. wcel 1756   {cab 2429    =/= wne 2611   A.wral 2720   _Vcvv 2977    \ cdif 3330    u. cun 3331    i^i cin 3332    C_ wss 3333   (/)c0 3642   {csn 3882   {cpr 3884   class class class wbr 4297    e. cmpt 4355   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   Fincfn 7315   CCcc 9285   1c1 9288    + caddc 9290    < clt 9423    <_ cle 9424    - cmin 9600   NNcn 10327   2c2 10376   NN0cn0 10584   ZZcz 10651   ZZ>=cuz 10866   ...cfz 11442   #chash 12108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-hash 12109
This theorem is referenced by:  subfacp1lem2a  27073  subfacp1lem3  27075  subfacp1lem4  27076
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