Theorem subfacp1lem1 28291

Description: Lemma for subfacp1 28298. 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 3867	. . . 4 |- ( ( K i^i { 1 , M } ) = (/) <-> A. x e. K -. x e. { 1 , M } )
2		eldifi 3626	. . . . . . . . 9 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> x e. ( 2 ... ( N + 1 ) ) )
3		elfzle1 11689	. . . . . . . . 9 |- ( x e. ( 2 ... ( N + 1 ) ) -> 2 <_ x )
4		1lt2 10702	. . . . . . . . . . . 12 |- 1 < 2
5		1re 9595	. . . . . . . . . . . . 13 |- 1 e. RR
6		2re 10605	. . . . . . . . . . . . 13 |- 2 e. RR
7	5, 6	ltnlei 9705	. . . . . . . . . . . 12 |- ( 1 < 2 <-> -. 2 <_ 1 )
8	4, 7	mpbi 208	. . . . . . . . . . 11 |- -. 2 <_ 1
9		breq2 4451	. . . . . . . . . . 11 |- ( x = 1 -> ( 2 <_ x <-> 2 <_ 1 ) )
10	8, 9	mtbiri 303	. . . . . . . . . 10 |- ( x = 1 -> -. 2 <_ x )
11	10	necon2ai 2702	. . . . . . . . 9 |- ( 2 <_ x -> x =/= 1 )
12	2, 3, 11	3syl 20	. . . . . . . 8 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> x =/= 1 )
13		eldifsni 4153	. . . . . . . 8 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> x =/= M )
14	12, 13	jca 532	. . . . . . 7 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> ( x =/= 1 /\ x =/= M ) )
15		subfacp1lem1.k	. . . . . . 7 |- K = ( ( 2 ... ( N + 1 ) ) \ { M } )
16	14, 15	eleq2s 2575	. . . . . 6 |- ( x e. K -> ( x =/= 1 /\ x =/= M ) )
17		neanior 2792	. . . . . 6 |- ( ( x =/= 1 /\ x =/= M ) <-> -. ( x = 1 \/ x = M ) )
18	16, 17	sylib 196	. . . . 5 |- ( x e. K -> -. ( x = 1 \/ x = M ) )
19		vex 3116	. . . . . 6 |- x e. _V
20	19	elpr 4045	. . . . 5 |- ( x e. { 1 , M } <-> ( x = 1 \/ x = M ) )
21	18, 20	sylnibr 305	. . . 4 |- ( x e. K -> -. x e. { 1 , M } )
22	1, 21	mprgbir 2828	. . 3 |- ( K i^i { 1 , M } ) = (/)
23	22	a1i 11	. 2 |- ( ph -> ( K i^i { 1 , M } ) = (/) )
24		uncom 3648	. . . 4 |- ( { 1 } u. ( K u. { M } ) ) = ( ( K u. { M } ) u. { 1 } )
25		1z 10894	. . . . . 6 |- 1 e. ZZ
26		fzsn 11725	. . . . . 6 |- ( 1 e. ZZ -> ( 1 ... 1 ) = { 1 } )
27	25, 26	ax-mp 5	. . . . 5 |- ( 1 ... 1 ) = { 1 }
28	15	uneq1i 3654	. . . . . 6 |- ( K u. { M } ) = ( ( ( 2 ... ( N + 1 ) ) \ { M } ) u. { M } )
29		undif1 3902	. . . . . 6 |- ( ( ( 2 ... ( N + 1 ) ) \ { M } ) u. { M } ) = ( ( 2 ... ( N + 1 ) ) u. { M } )
30	28, 29	eqtr2i 2497	. . . . 5 |- ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( K u. { M } )
31	27, 30	uneq12i 3656	. . . 4 |- ( ( 1 ... 1 ) u. ( ( 2 ... ( N + 1 ) ) u. { M } ) ) = ( { 1 } u. ( K u. { M } ) )
32		df-pr 4030	. . . . . . 7 |- { 1 , M } = ( { 1 } u. { M } )
33	32	equncomi 3650	. . . . . 6 |- { 1 , M } = ( { M } u. { 1 } )
34	33	uneq2i 3655	. . . . 5 |- ( K u. { 1 , M } ) = ( K u. ( { M } u. { 1 } ) )
35		unass 3661	. . . . 5 |- ( ( K u. { M } ) u. { 1 } ) = ( K u. ( { M } u. { 1 } ) )
36	34, 35	eqtr4i 2499	. . . 4 |- ( K u. { 1 , M } ) = ( ( K u. { M } ) u. { 1 } )
37	24, 31, 36	3eqtr4i 2506	. . 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 4172	. . . . . . 7 |- ( ph -> { M } C_ ( 2 ... ( N + 1 ) ) )
40		ssequn2 3677	. . . . . . 7 |- ( { M } C_ ( 2 ... ( N + 1 ) ) <-> ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( 2 ... ( N + 1 ) ) )
41	39, 40	sylib 196	. . . . . 6 |- ( ph -> ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( 2 ... ( N + 1 ) ) )
42		df-2 10594	. . . . . . 7 |- 2 = ( 1 + 1 )
43	42	oveq1i 6294	. . . . . 6 |- ( 2 ... ( N + 1 ) ) = ( ( 1 + 1 ) ... ( N + 1 ) )
44	41, 43	syl6eq 2524	. . . . 5 |- ( ph -> ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( ( 1 + 1 ) ... ( N + 1 ) ) )
45	44	uneq2d 3658	. . . 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 10553	. . . . . 6 |- ( ph -> ( N + 1 ) e. NN )
48		nnuz 11117	. . . . . 6 |- NN = ( ZZ>= ` 1 )
49	47, 48	syl6eleq 2565	. . . . 5 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` 1 ) )
50		eluzfz1 11693	. . . . 5 |- ( ( N + 1 ) e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... ( N + 1 ) ) )
51		fzsplit 11711	. . . . 5 |- ( 1 e. ( 1 ... ( N + 1 ) ) -> ( 1 ... ( N + 1 ) ) = ( ( 1 ... 1 ) u. ( ( 1 + 1 ) ... ( N + 1 ) ) ) )
52	49, 50, 51	3syl 20	. . . 4 |- ( ph -> ( 1 ... ( N + 1 ) ) = ( ( 1 ... 1 ) u. ( ( 1 + 1 ) ... ( N + 1 ) ) ) )
53	45, 52	eqtr4d 2511	. . 3 |- ( ph -> ( ( 1 ... 1 ) u. ( ( 2 ... ( N + 1 ) ) u. { M } ) ) = ( 1 ... ( N + 1 ) ) )
54	37, 53	syl5eqr 2522	. 2 |- ( ph -> ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) )
55	42	oveq2i 6295	. . 3 |- ( ( N + 1 ) - 2 ) = ( ( N + 1 ) - ( 1 + 1 ) )
56		fzfi 12050	. . . . . . . . 9 |- ( 2 ... ( N + 1 ) ) e. Fin
57		diffi 7751	. . . . . . . . 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 2551	. . . . . . 7 |- K e. Fin
60		prfi 7795	. . . . . . 7 |- { 1 , M } e. Fin
61		hashun 12418	. . . . . . 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 1324	. . . . . 6 |- ( # ` ( K u. { 1 , M } ) ) = ( ( # ` K ) + ( # ` { 1 , M } ) )
63	54	fveq2d 5870	. . . . . 6 |- ( ph -> ( # ` ( K u. { 1 , M } ) ) = ( # ` ( 1 ... ( N + 1 ) ) ) )
64		neeq1 2748	. . . . . . . . . . 11 |- ( x = M -> ( x =/= 1 <-> M =/= 1 ) )
65	3, 11	syl 16	. . . . . . . . . . 11 |- ( x e. ( 2 ... ( N + 1 ) ) -> x =/= 1 )
66	64, 65	vtoclga 3177	. . . . . . . . . 10 |- ( M e. ( 2 ... ( N + 1 ) ) -> M =/= 1 )
67	38, 66	syl 16	. . . . . . . . 9 |- ( ph -> M =/= 1 )
68	67	necomd 2738	. . . . . . . 8 |- ( ph -> 1 =/= M )
69		1ex 9591	. . . . . . . . 9 |- 1 e. _V
70		subfacp1lem1.x	. . . . . . . . 9 |- M e. _V
71		hashprg 12428	. . . . . . . . 9 |- ( ( 1 e. _V /\ M e. _V ) -> ( 1 =/= M <-> ( # ` { 1 , M } ) = 2 ) )
72	69, 70, 71	mp2an 672	. . . . . . . 8 |- ( 1 =/= M <-> ( # ` { 1 , M } ) = 2 )
73	68, 72	sylib 196	. . . . . . 7 |- ( ph -> ( # ` { 1 , M } ) = 2 )
74	73	oveq2d 6300	. . . . . 6 |- ( ph -> ( ( # ` K ) + ( # ` { 1 , M } ) ) = ( ( # ` K ) + 2 ) )
75	62, 63, 74	3eqtr3a 2532	. . . . 5 |- ( ph -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( ( # ` K ) + 2 ) )
76	47	nnnn0d 10852	. . . . . 6 |- ( ph -> ( N + 1 ) e. NN0 )
77		hashfz1 12387	. . . . . 6 |- ( ( N + 1 ) e. NN0 -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( N + 1 ) )
78	76, 77	syl 16	. . . . 5 |- ( ph -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( N + 1 ) )
79	75, 78	eqtr3d 2510	. . . 4 |- ( ph -> ( ( # ` K ) + 2 ) = ( N + 1 ) )
80	47	nncnd 10552	. . . . 5 |- ( ph -> ( N + 1 ) e. CC )
81		2cnd 10608	. . . . 5 |- ( ph -> 2 e. CC )
82		hashcl 12396	. . . . . . . 8 |- ( K e. Fin -> ( # ` K ) e. NN0 )
83	59, 82	ax-mp 5	. . . . . . 7 |- ( # ` K ) e. NN0
84	83	nn0cni 10807	. . . . . 6 |- ( # ` K ) e. CC
85	84	a1i 11	. . . . 5 |- ( ph -> ( # ` K ) e. CC )
86	80, 81, 85	subadd2d 9949	. . . 4 |- ( ph -> ( ( ( N + 1 ) - 2 ) = ( # ` K ) <-> ( ( # ` K ) + 2 ) = ( N + 1 ) ) )
87	79, 86	mpbird 232	. . 3 |- ( ph -> ( ( N + 1 ) - 2 ) = ( # ` K ) )
88	46	nncnd 10552	. . . 4 |- ( ph -> N e. CC )
89		ax-1cn 9550	. . . . 5 |- 1 e. CC
90	89	a1i 11	. . . 4 |- ( ph -> 1 e. CC )
91	88, 90, 90	pnpcan2d 9968	. . 3 |- ( ph -> ( ( N + 1 ) - ( 1 + 1 ) ) = ( N - 1 ) )
92	55, 87, 91	3eqtr3a 2532	. 2 |- ( ph -> ( # ` K ) = ( N - 1 ) )
93	23, 54, 92	3jca 1176	1 |- ( ph -> ( ( K i^i { 1 , M } ) = (/) /\ ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) /\ ( # ` K ) = ( N - 1 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   {cab 2452   =/= wne 2662   A.wral 2814   _Vcvv 3113   \ cdif 3473   u. cun 3474   i^i cin 3475   C_ wss 3476   (/)c0 3785   {csn 4027   {cpr 4029   class class class wbr 4447   |-> cmpt 4505   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284   Fincfn 7516   CCcc 9490   1c1 9493   + caddc 9495   < clt 9628   <_ cle 9629   - cmin 9805   NNcn 10536   2c2 10585   NN0cn0 10795   ZZcz 10864   ZZ>=cuz 11082   ...cfz 11672   #chash 12373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-hash 12374

This theorem is referenced by:  subfacp1lem2a  28292  subfacp1lem3  28294  subfacp1lem4  28295