Theorem subfacp1lem1 28601

Description: Lemma for subfacp1 28608. 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 3853	. . . 4 |- ( ( K i^i { 1 , M } ) = (/) <-> A. x e. K -. x e. { 1 , M } )
2		eldifi 3611	. . . . . . . . 9 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> x e. ( 2 ... ( N + 1 ) ) )
3		elfzle1 11700	. . . . . . . . 9 |- ( x e. ( 2 ... ( N + 1 ) ) -> 2 <_ x )
4		1lt2 10709	. . . . . . . . . . . 12 |- 1 < 2
5		1re 9598	. . . . . . . . . . . . 13 |- 1 e. RR
6		2re 10612	. . . . . . . . . . . . 13 |- 2 e. RR
7	5, 6	ltnlei 9708	. . . . . . . . . . . 12 |- ( 1 < 2 <-> -. 2 <_ 1 )
8	4, 7	mpbi 208	. . . . . . . . . . 11 |- -. 2 <_ 1
9		breq2 4441	. . . . . . . . . . 11 |- ( x = 1 -> ( 2 <_ x <-> 2 <_ 1 ) )
10	8, 9	mtbiri 303	. . . . . . . . . 10 |- ( x = 1 -> -. 2 <_ x )
11	10	necon2ai 2678	. . . . . . . . 9 |- ( 2 <_ x -> x =/= 1 )
12	2, 3, 11	3syl 20	. . . . . . . 8 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> x =/= 1 )
13		eldifsni 4141	. . . . . . . 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 2551	. . . . . 6 |- ( x e. K -> ( x =/= 1 /\ x =/= M ) )
17		neanior 2768	. . . . . 6 |- ( ( x =/= 1 /\ x =/= M ) <-> -. ( x = 1 \/ x = M ) )
18	16, 17	sylib 196	. . . . 5 |- ( x e. K -> -. ( x = 1 \/ x = M ) )
19		vex 3098	. . . . . 6 |- x e. _V
20	19	elpr 4032	. . . . 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 2807	. . 3 |- ( K i^i { 1 , M } ) = (/)
23	22	a1i 11	. 2 |- ( ph -> ( K i^i { 1 , M } ) = (/) )
24		uncom 3633	. . . 4 |- ( { 1 } u. ( K u. { M } ) ) = ( ( K u. { M } ) u. { 1 } )
25		1z 10901	. . . . . 6 |- 1 e. ZZ
26		fzsn 11736	. . . . . 6 |- ( 1 e. ZZ -> ( 1 ... 1 ) = { 1 } )
27	25, 26	ax-mp 5	. . . . 5 |- ( 1 ... 1 ) = { 1 }
28	15	uneq1i 3639	. . . . . 6 |- ( K u. { M } ) = ( ( ( 2 ... ( N + 1 ) ) \ { M } ) u. { M } )
29		undif1 3889	. . . . . 6 |- ( ( ( 2 ... ( N + 1 ) ) \ { M } ) u. { M } ) = ( ( 2 ... ( N + 1 ) ) u. { M } )
30	28, 29	eqtr2i 2473	. . . . 5 |- ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( K u. { M } )
31	27, 30	uneq12i 3641	. . . 4 |- ( ( 1 ... 1 ) u. ( ( 2 ... ( N + 1 ) ) u. { M } ) ) = ( { 1 } u. ( K u. { M } ) )
32		df-pr 4017	. . . . . . 7 |- { 1 , M } = ( { 1 } u. { M } )
33	32	equncomi 3635	. . . . . 6 |- { 1 , M } = ( { M } u. { 1 } )
34	33	uneq2i 3640	. . . . 5 |- ( K u. { 1 , M } ) = ( K u. ( { M } u. { 1 } ) )
35		unass 3646	. . . . 5 |- ( ( K u. { M } ) u. { 1 } ) = ( K u. ( { M } u. { 1 } ) )
36	34, 35	eqtr4i 2475	. . . 4 |- ( K u. { 1 , M } ) = ( ( K u. { M } ) u. { 1 } )
37	24, 31, 36	3eqtr4i 2482	. . 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 4160	. . . . . . 7 |- ( ph -> { M } C_ ( 2 ... ( N + 1 ) ) )
40		ssequn2 3662	. . . . . . 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 10601	. . . . . . 7 |- 2 = ( 1 + 1 )
43	42	oveq1i 6291	. . . . . 6 |- ( 2 ... ( N + 1 ) ) = ( ( 1 + 1 ) ... ( N + 1 ) )
44	41, 43	syl6eq 2500	. . . . 5 |- ( ph -> ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( ( 1 + 1 ) ... ( N + 1 ) ) )
45	44	uneq2d 3643	. . . 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 10560	. . . . . 6 |- ( ph -> ( N + 1 ) e. NN )
48		nnuz 11127	. . . . . 6 |- NN = ( ZZ>= ` 1 )
49	47, 48	syl6eleq 2541	. . . . 5 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` 1 ) )
50		eluzfz1 11704	. . . . 5 |- ( ( N + 1 ) e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... ( N + 1 ) ) )
51		fzsplit 11722	. . . . 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 2487	. . 3 |- ( ph -> ( ( 1 ... 1 ) u. ( ( 2 ... ( N + 1 ) ) u. { M } ) ) = ( 1 ... ( N + 1 ) ) )
54	37, 53	syl5eqr 2498	. 2 |- ( ph -> ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) )
55	42	oveq2i 6292	. . 3 |- ( ( N + 1 ) - 2 ) = ( ( N + 1 ) - ( 1 + 1 ) )
56		fzfi 12064	. . . . . . . . 9 |- ( 2 ... ( N + 1 ) ) e. Fin
57		diffi 7753	. . . . . . . . 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 2527	. . . . . . 7 |- K e. Fin
60		prfi 7797	. . . . . . 7 |- { 1 , M } e. Fin
61		hashun 12432	. . . . . . 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 1325	. . . . . 6 |- ( # ` ( K u. { 1 , M } ) ) = ( ( # ` K ) + ( # ` { 1 , M } ) )
63	54	fveq2d 5860	. . . . . 6 |- ( ph -> ( # ` ( K u. { 1 , M } ) ) = ( # ` ( 1 ... ( N + 1 ) ) ) )
64		neeq1 2724	. . . . . . . . . . 11 |- ( x = M -> ( x =/= 1 <-> M =/= 1 ) )
65	3, 11	syl 16	. . . . . . . . . . 11 |- ( x e. ( 2 ... ( N + 1 ) ) -> x =/= 1 )
66	64, 65	vtoclga 3159	. . . . . . . . . 10 |- ( M e. ( 2 ... ( N + 1 ) ) -> M =/= 1 )
67	38, 66	syl 16	. . . . . . . . 9 |- ( ph -> M =/= 1 )
68	67	necomd 2714	. . . . . . . 8 |- ( ph -> 1 =/= M )
69		1ex 9594	. . . . . . . . 9 |- 1 e. _V
70		subfacp1lem1.x	. . . . . . . . 9 |- M e. _V
71		hashprg 12442	. . . . . . . . 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 6297	. . . . . 6 |- ( ph -> ( ( # ` K ) + ( # ` { 1 , M } ) ) = ( ( # ` K ) + 2 ) )
75	62, 63, 74	3eqtr3a 2508	. . . . 5 |- ( ph -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( ( # ` K ) + 2 ) )
76	47	nnnn0d 10859	. . . . . 6 |- ( ph -> ( N + 1 ) e. NN0 )
77		hashfz1 12401	. . . . . 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 2486	. . . 4 |- ( ph -> ( ( # ` K ) + 2 ) = ( N + 1 ) )
80	47	nncnd 10559	. . . . 5 |- ( ph -> ( N + 1 ) e. CC )
81		2cnd 10615	. . . . 5 |- ( ph -> 2 e. CC )
82		hashcl 12410	. . . . . . . 8 |- ( K e. Fin -> ( # ` K ) e. NN0 )
83	59, 82	ax-mp 5	. . . . . . 7 |- ( # ` K ) e. NN0
84	83	nn0cni 10814	. . . . . 6 |- ( # ` K ) e. CC
85	84	a1i 11	. . . . 5 |- ( ph -> ( # ` K ) e. CC )
86	80, 81, 85	subadd2d 9955	. . . 4 |- ( ph -> ( ( ( N + 1 ) - 2 ) = ( # ` K ) <-> ( ( # ` K ) + 2 ) = ( N + 1 ) ) )
87	79, 86	mpbird 232	. . 3 |- ( ph -> ( ( N + 1 ) - 2 ) = ( # ` K ) )
88	46	nncnd 10559	. . . 4 |- ( ph -> N e. CC )
89		1cnd 9615	. . . 4 |- ( ph -> 1 e. CC )
90	88, 89, 89	pnpcan2d 9974	. . 3 |- ( ph -> ( ( N + 1 ) - ( 1 + 1 ) ) = ( N - 1 ) )
91	55, 87, 90	3eqtr3a 2508	. 2 |- ( ph -> ( # ` K ) = ( N - 1 ) )
92	23, 54, 91	3jca 1177	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 974   = wceq 1383   e. wcel 1804   {cab 2428   =/= wne 2638   A.wral 2793   _Vcvv 3095   \ cdif 3458   u. cun 3459   i^i cin 3460   C_ wss 3461   (/)c0 3770   {csn 4014   {cpr 4016   class class class wbr 4437   |-> cmpt 4495   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281   Fincfn 7518   CCcc 9493   1c1 9496   + caddc 9498   < clt 9631   <_ cle 9632   - cmin 9810   NNcn 10543   2c2 10592   NN0cn0 10802   ZZcz 10871   ZZ>=cuz 11092   ...cfz 11683   #chash 12387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-hash 12388

This theorem is referenced by:  subfacp1lem2a  28602  subfacp1lem3  28604  subfacp1lem4  28605