Theorem subfacp1lem1 29974

Description: Lemma for subfacp1 29981. 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

Step	Hyp	Ref	Expression
1		disj 3809	. . . 4 |- ( ( K i^i { 1 , M } ) = (/) <-> A. x e. K -. x e. { 1 , M } )
2		eldifi 3544	. . . . . . . . 9 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> x e. ( 2 ... ( N + 1 ) ) )
3		elfzle1 11828	. . . . . . . . 9 |- ( x e. ( 2 ... ( N + 1 ) ) -> 2 <_ x )
4		1lt2 10799	. . . . . . . . . . . 12 |- 1 < 2
5		1re 9660	. . . . . . . . . . . . 13 |- 1 e. RR
6		2re 10701	. . . . . . . . . . . . 13 |- 2 e. RR
7	5, 6	ltnlei 9773	. . . . . . . . . . . 12 |- ( 1 < 2 <-> -. 2 <_ 1 )
8	4, 7	mpbi 213	. . . . . . . . . . 11 |- -. 2 <_ 1
9		breq2 4399	. . . . . . . . . . 11 |- ( x = 1 -> ( 2 <_ x <-> 2 <_ 1 ) )
10	8, 9	mtbiri 310	. . . . . . . . . 10 |- ( x = 1 -> -. 2 <_ x )
11	10	necon2ai 2672	. . . . . . . . 9 |- ( 2 <_ x -> x =/= 1 )
12	2, 3, 11	3syl 18	. . . . . . . 8 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> x =/= 1 )
13		eldifsni 4089	. . . . . . . 8 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> x =/= M )
14	12, 13	jca 541	. . . . . . 7 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> ( x =/= 1 /\ x =/= M ) )
15		subfacp1lem1.k	. . . . . . 7 |- K = ( ( 2 ... ( N + 1 ) ) \ { M } )
16	14, 15	eleq2s 2567	. . . . . 6 |- ( x e. K -> ( x =/= 1 /\ x =/= M ) )
17		neanior 2735	. . . . . 6 |- ( ( x =/= 1 /\ x =/= M ) <-> -. ( x = 1 \/ x = M ) )
18	16, 17	sylib 201	. . . . 5 |- ( x e. K -> -. ( x = 1 \/ x = M ) )
19		vex 3034	. . . . . 6 |- x e. _V
20	19	elpr 3977	. . . . 5 |- ( x e. { 1 , M } <-> ( x = 1 \/ x = M ) )
21	18, 20	sylnibr 312	. . . 4 |- ( x e. K -> -. x e. { 1 , M } )
22	1, 21	mprgbir 2771	. . 3 |- ( K i^i { 1 , M } ) = (/)
23	22	a1i 11	. 2 |- ( ph -> ( K i^i { 1 , M } ) = (/) )
24		uncom 3569	. . . 4 |- ( { 1 } u. ( K u. { M } ) ) = ( ( K u. { M } ) u. { 1 } )
25		1z 10991	. . . . . 6 |- 1 e. ZZ
26		fzsn 11866	. . . . . 6 |- ( 1 e. ZZ -> ( 1 ... 1 ) = { 1 } )
27	25, 26	ax-mp 5	. . . . 5 |- ( 1 ... 1 ) = { 1 }
28	15	uneq1i 3575	. . . . . 6 |- ( K u. { M } ) = ( ( ( 2 ... ( N + 1 ) ) \ { M } ) u. { M } )
29		undif1 3833	. . . . . 6 |- ( ( ( 2 ... ( N + 1 ) ) \ { M } ) u. { M } ) = ( ( 2 ... ( N + 1 ) ) u. { M } )
30	28, 29	eqtr2i 2494	. . . . 5 |- ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( K u. { M } )
31	27, 30	uneq12i 3577	. . . 4 |- ( ( 1 ... 1 ) u. ( ( 2 ... ( N + 1 ) ) u. { M } ) ) = ( { 1 } u. ( K u. { M } ) )
32		df-pr 3962	. . . . . . 7 |- { 1 , M } = ( { 1 } u. { M } )
33	32	equncomi 3571	. . . . . 6 |- { 1 , M } = ( { M } u. { 1 } )
34	33	uneq2i 3576	. . . . 5 |- ( K u. { 1 , M } ) = ( K u. ( { M } u. { 1 } ) )
35		unass 3582	. . . . 5 |- ( ( K u. { M } ) u. { 1 } ) = ( K u. ( { M } u. { 1 } ) )
36	34, 35	eqtr4i 2496	. . . 4 |- ( K u. { 1 , M } ) = ( ( K u. { M } ) u. { 1 } )
37	24, 31, 36	3eqtr4i 2503	. . 3 |- ( ( 1 ... 1 ) u. ( ( 2 ... ( N + 1 ) ) u. { M } ) ) = ( K u. { 1 , M } )
38		subfacp1lem1.m	. . . . . . . 8 |- ( ph -> M e. ( 2 ... ( N + 1 ) ) )
39	38	snssd 4108	. . . . . . 7 |- ( ph -> { M } C_ ( 2 ... ( N + 1 ) ) )
40		ssequn2 3598	. . . . . . 7 |- ( { M } C_ ( 2 ... ( N + 1 ) ) <-> ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( 2 ... ( N + 1 ) ) )
41	39, 40	sylib 201	. . . . . 6 |- ( ph -> ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( 2 ... ( N + 1 ) ) )
42		df-2 10690	. . . . . . 7 |- 2 = ( 1 + 1 )
43	42	oveq1i 6318	. . . . . 6 |- ( 2 ... ( N + 1 ) ) = ( ( 1 + 1 ) ... ( N + 1 ) )
44	41, 43	syl6eq 2521	. . . . 5 |- ( ph -> ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( ( 1 + 1 ) ... ( N + 1 ) ) )
45	44	uneq2d 3579	. . . 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 )
47	46	peano2nnd 10648	. . . . . 6 |- ( ph -> ( N + 1 ) e. NN )
48		nnuz 11218	. . . . . 6 |- NN = ( ZZ>= ` 1 )
49	47, 48	syl6eleq 2559	. . . . 5 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` 1 ) )
50		eluzfz1 11832	. . . . 5 |- ( ( N + 1 ) e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... ( N + 1 ) ) )
51		fzsplit 11851	. . . . 5 |- ( 1 e. ( 1 ... ( N + 1 ) ) -> ( 1 ... ( N + 1 ) ) = ( ( 1 ... 1 ) u. ( ( 1 + 1 ) ... ( N + 1 ) ) ) )
52	49, 50, 51	3syl 18	. . . 4 |- ( ph -> ( 1 ... ( N + 1 ) ) = ( ( 1 ... 1 ) u. ( ( 1 + 1 ) ... ( N + 1 ) ) ) )
53	45, 52	eqtr4d 2508	. . 3 |- ( ph -> ( ( 1 ... 1 ) u. ( ( 2 ... ( N + 1 ) ) u. { M } ) ) = ( 1 ... ( N + 1 ) ) )
54	37, 53	syl5eqr 2519	. 2 |- ( ph -> ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) )
55	42	oveq2i 6319	. . 3 |- ( ( N + 1 ) - 2 ) = ( ( N + 1 ) - ( 1 + 1 ) )
56		fzfi 12223	. . . . . . . . 9 |- ( 2 ... ( N + 1 ) ) e. Fin
57		diffi 7821	. . . . . . . . 9 |- ( ( 2 ... ( N + 1 ) ) e. Fin -> ( ( 2 ... ( N + 1 ) ) \ { M } ) e. Fin )
58	56, 57	ax-mp 5	. . . . . . . 8 |- ( ( 2 ... ( N + 1 ) ) \ { M } ) e. Fin
59	15, 58	eqeltri 2545	. . . . . . 7 |- K e. Fin
60		prfi 7864	. . . . . . 7 |- { 1 , M } e. Fin
61		hashun 12599	. . . . . . 7 |- ( ( K e. Fin /\ { 1 , M } e. Fin /\ ( K i^i { 1 , M } ) = (/) ) -> ( # ` ( K u. { 1 , M } ) ) = ( ( # ` K ) + ( # ` { 1 , M } ) ) )
62	59, 60, 22, 61	mp3an 1390	. . . . . 6 |- ( # ` ( K u. { 1 , M } ) ) = ( ( # ` K ) + ( # ` { 1 , M } ) )
63	54	fveq2d 5883	. . . . . 6 |- ( ph -> ( # ` ( K u. { 1 , M } ) ) = ( # ` ( 1 ... ( N + 1 ) ) ) )
64		neeq1 2705	. . . . . . . . . . 11 |- ( x = M -> ( x =/= 1 <-> M =/= 1 ) )
65	3, 11	syl 17	. . . . . . . . . . 11 |- ( x e. ( 2 ... ( N + 1 ) ) -> x =/= 1 )
66	64, 65	vtoclga 3099	. . . . . . . . . 10 |- ( M e. ( 2 ... ( N + 1 ) ) -> M =/= 1 )
67	38, 66	syl 17	. . . . . . . . 9 |- ( ph -> M =/= 1 )
68	67	necomd 2698	. . . . . . . 8 |- ( ph -> 1 =/= M )
69		1ex 9656	. . . . . . . . 9 |- 1 e. _V
70		subfacp1lem1.x	. . . . . . . . 9 |- M e. _V
71		hashprg 12610	. . . . . . . . 9 |- ( ( 1 e. _V /\ M e. _V ) -> ( 1 =/= M <-> ( # ` { 1 , M } ) = 2 ) )
72	69, 70, 71	mp2an 686	. . . . . . . 8 |- ( 1 =/= M <-> ( # ` { 1 , M } ) = 2 )
73	68, 72	sylib 201	. . . . . . 7 |- ( ph -> ( # ` { 1 , M } ) = 2 )
74	73	oveq2d 6324	. . . . . 6 |- ( ph -> ( ( # ` K ) + ( # ` { 1 , M } ) ) = ( ( # ` K ) + 2 ) )
75	62, 63, 74	3eqtr3a 2529	. . . . 5 |- ( ph -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( ( # ` K ) + 2 ) )
76	47	nnnn0d 10949	. . . . . 6 |- ( ph -> ( N + 1 ) e. NN0 )
77		hashfz1 12567	. . . . . 6 |- ( ( N + 1 ) e. NN0 -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( N + 1 ) )
78	76, 77	syl 17	. . . . 5 |- ( ph -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( N + 1 ) )
79	75, 78	eqtr3d 2507	. . . 4 |- ( ph -> ( ( # ` K ) + 2 ) = ( N + 1 ) )
80	47	nncnd 10647	. . . . 5 |- ( ph -> ( N + 1 ) e. CC )
81		2cnd 10704	. . . . 5 |- ( ph -> 2 e. CC )
82		hashcl 12576	. . . . . . . 8 |- ( K e. Fin -> ( # ` K ) e. NN0 )
83	59, 82	ax-mp 5	. . . . . . 7 |- ( # ` K ) e. NN0
84	83	nn0cni 10905	. . . . . 6 |- ( # ` K ) e. CC
85	84	a1i 11	. . . . 5 |- ( ph -> ( # ` K ) e. CC )
86	80, 81, 85	subadd2d 10024	. . . 4 |- ( ph -> ( ( ( N + 1 ) - 2 ) = ( # ` K ) <-> ( ( # ` K ) + 2 ) = ( N + 1 ) ) )
87	79, 86	mpbird 240	. . 3 |- ( ph -> ( ( N + 1 ) - 2 ) = ( # ` K ) )
88	46	nncnd 10647	. . . 4 |- ( ph -> N e. CC )
89		1cnd 9677	. . . 4 |- ( ph -> 1 e. CC )
90	88, 89, 89	pnpcan2d 10043	. . 3 |- ( ph -> ( ( N + 1 ) - ( 1 + 1 ) ) = ( N - 1 ) )
91	55, 87, 90	3eqtr3a 2529	. 2 |- ( ph -> ( # ` K ) = ( N - 1 ) )
92	23, 54, 91	3jca 1210	1 |- ( ph -> ( ( K i^i { 1 , M } ) = (/) /\ ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) /\ ( # ` K ) = ( N - 1 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   {cab 2457   =/= wne 2641   A.wral 2756   _Vcvv 3031   \ cdif 3387   u. cun 3388   i^i cin 3389   C_ wss 3390   (/)c0 3722   {csn 3959   {cpr 3961   class class class wbr 4395   |-> cmpt 4454   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   Fincfn 7587   CCcc 9555   1c1 9558   + caddc 9560   < clt 9693   <_ cle 9694   - cmin 9880   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   #chash 12553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554

This theorem is referenced by:  subfacp1lem2a  29975  subfacp1lem3  29977  subfacp1lem4  29978