Theorem subfacp1lem1 29687

Description: Lemma for subfacp1 29694. 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 3830	. . . 4 |- ( ( K i^i { 1 , M } ) = (/) <-> A. x e. K -. x e. { 1 , M } )
2		eldifi 3584	. . . . . . . . 9 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> x e. ( 2 ... ( N + 1 ) ) )
3		elfzle1 11789	. . . . . . . . 9 |- ( x e. ( 2 ... ( N + 1 ) ) -> 2 <_ x )
4		1lt2 10765	. . . . . . . . . . . 12 |- 1 < 2
5		1re 9631	. . . . . . . . . . . . 13 |- 1 e. RR
6		2re 10668	. . . . . . . . . . . . 13 |- 2 e. RR
7	5, 6	ltnlei 9744	. . . . . . . . . . . 12 |- ( 1 < 2 <-> -. 2 <_ 1 )
8	4, 7	mpbi 211	. . . . . . . . . . 11 |- -. 2 <_ 1
9		breq2 4421	. . . . . . . . . . 11 |- ( x = 1 -> ( 2 <_ x <-> 2 <_ 1 ) )
10	8, 9	mtbiri 304	. . . . . . . . . 10 |- ( x = 1 -> -. 2 <_ x )
11	10	necon2ai 2657	. . . . . . . . 9 |- ( 2 <_ x -> x =/= 1 )
12	2, 3, 11	3syl 18	. . . . . . . 8 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> x =/= 1 )
13		eldifsni 4120	. . . . . . . 8 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> x =/= M )
14	12, 13	jca 534	. . . . . . 7 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> ( x =/= 1 /\ x =/= M ) )
15		subfacp1lem1.k	. . . . . . 7 |- K = ( ( 2 ... ( N + 1 ) ) \ { M } )
16	14, 15	eleq2s 2528	. . . . . 6 |- ( x e. K -> ( x =/= 1 /\ x =/= M ) )
17		neanior 2747	. . . . . 6 |- ( ( x =/= 1 /\ x =/= M ) <-> -. ( x = 1 \/ x = M ) )
18	16, 17	sylib 199	. . . . 5 |- ( x e. K -> -. ( x = 1 \/ x = M ) )
19		vex 3081	. . . . . 6 |- x e. _V
20	19	elpr 4011	. . . . 5 |- ( x e. { 1 , M } <-> ( x = 1 \/ x = M ) )
21	18, 20	sylnibr 306	. . . 4 |- ( x e. K -> -. x e. { 1 , M } )
22	1, 21	mprgbir 2787	. . 3 |- ( K i^i { 1 , M } ) = (/)
23	22	a1i 11	. 2 |- ( ph -> ( K i^i { 1 , M } ) = (/) )
24		uncom 3607	. . . 4 |- ( { 1 } u. ( K u. { M } ) ) = ( ( K u. { M } ) u. { 1 } )
25		1z 10956	. . . . . 6 |- 1 e. ZZ
26		fzsn 11827	. . . . . 6 |- ( 1 e. ZZ -> ( 1 ... 1 ) = { 1 } )
27	25, 26	ax-mp 5	. . . . 5 |- ( 1 ... 1 ) = { 1 }
28	15	uneq1i 3613	. . . . . 6 |- ( K u. { M } ) = ( ( ( 2 ... ( N + 1 ) ) \ { M } ) u. { M } )
29		undif1 3867	. . . . . 6 |- ( ( ( 2 ... ( N + 1 ) ) \ { M } ) u. { M } ) = ( ( 2 ... ( N + 1 ) ) u. { M } )
30	28, 29	eqtr2i 2450	. . . . 5 |- ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( K u. { M } )
31	27, 30	uneq12i 3615	. . . 4 |- ( ( 1 ... 1 ) u. ( ( 2 ... ( N + 1 ) ) u. { M } ) ) = ( { 1 } u. ( K u. { M } ) )
32		df-pr 3996	. . . . . . 7 |- { 1 , M } = ( { 1 } u. { M } )
33	32	equncomi 3609	. . . . . 6 |- { 1 , M } = ( { M } u. { 1 } )
34	33	uneq2i 3614	. . . . 5 |- ( K u. { 1 , M } ) = ( K u. ( { M } u. { 1 } ) )
35		unass 3620	. . . . 5 |- ( ( K u. { M } ) u. { 1 } ) = ( K u. ( { M } u. { 1 } ) )
36	34, 35	eqtr4i 2452	. . . 4 |- ( K u. { 1 , M } ) = ( ( K u. { M } ) u. { 1 } )
37	24, 31, 36	3eqtr4i 2459	. . 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 4139	. . . . . . 7 |- ( ph -> { M } C_ ( 2 ... ( N + 1 ) ) )
40		ssequn2 3636	. . . . . . 7 |- ( { M } C_ ( 2 ... ( N + 1 ) ) <-> ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( 2 ... ( N + 1 ) ) )
41	39, 40	sylib 199	. . . . . 6 |- ( ph -> ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( 2 ... ( N + 1 ) ) )
42		df-2 10657	. . . . . . 7 |- 2 = ( 1 + 1 )
43	42	oveq1i 6306	. . . . . 6 |- ( 2 ... ( N + 1 ) ) = ( ( 1 + 1 ) ... ( N + 1 ) )
44	41, 43	syl6eq 2477	. . . . 5 |- ( ph -> ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( ( 1 + 1 ) ... ( N + 1 ) ) )
45	44	uneq2d 3617	. . . 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 10615	. . . . . 6 |- ( ph -> ( N + 1 ) e. NN )
48		nnuz 11183	. . . . . 6 |- NN = ( ZZ>= ` 1 )
49	47, 48	syl6eleq 2518	. . . . 5 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` 1 ) )
50		eluzfz1 11793	. . . . 5 |- ( ( N + 1 ) e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... ( N + 1 ) ) )
51		fzsplit 11812	. . . . 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 2464	. . 3 |- ( ph -> ( ( 1 ... 1 ) u. ( ( 2 ... ( N + 1 ) ) u. { M } ) ) = ( 1 ... ( N + 1 ) ) )
54	37, 53	syl5eqr 2475	. 2 |- ( ph -> ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) )
55	42	oveq2i 6307	. . 3 |- ( ( N + 1 ) - 2 ) = ( ( N + 1 ) - ( 1 + 1 ) )
56		fzfi 12171	. . . . . . . . 9 |- ( 2 ... ( N + 1 ) ) e. Fin
57		diffi 7800	. . . . . . . . 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 2504	. . . . . . 7 |- K e. Fin
60		prfi 7843	. . . . . . 7 |- { 1 , M } e. Fin
61		hashun 12547	. . . . . . 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 1360	. . . . . 6 |- ( # ` ( K u. { 1 , M } ) ) = ( ( # ` K ) + ( # ` { 1 , M } ) )
63	54	fveq2d 5876	. . . . . 6 |- ( ph -> ( # ` ( K u. { 1 , M } ) ) = ( # ` ( 1 ... ( N + 1 ) ) ) )
64		neeq1 2703	. . . . . . . . . . 11 |- ( x = M -> ( x =/= 1 <-> M =/= 1 ) )
65	3, 11	syl 17	. . . . . . . . . . 11 |- ( x e. ( 2 ... ( N + 1 ) ) -> x =/= 1 )
66	64, 65	vtoclga 3142	. . . . . . . . . 10 |- ( M e. ( 2 ... ( N + 1 ) ) -> M =/= 1 )
67	38, 66	syl 17	. . . . . . . . 9 |- ( ph -> M =/= 1 )
68	67	necomd 2693	. . . . . . . 8 |- ( ph -> 1 =/= M )
69		1ex 9627	. . . . . . . . 9 |- 1 e. _V
70		subfacp1lem1.x	. . . . . . . . 9 |- M e. _V
71		hashprg 12558	. . . . . . . . 9 |- ( ( 1 e. _V /\ M e. _V ) -> ( 1 =/= M <-> ( # ` { 1 , M } ) = 2 ) )
72	69, 70, 71	mp2an 676	. . . . . . . 8 |- ( 1 =/= M <-> ( # ` { 1 , M } ) = 2 )
73	68, 72	sylib 199	. . . . . . 7 |- ( ph -> ( # ` { 1 , M } ) = 2 )
74	73	oveq2d 6312	. . . . . 6 |- ( ph -> ( ( # ` K ) + ( # ` { 1 , M } ) ) = ( ( # ` K ) + 2 ) )
75	62, 63, 74	3eqtr3a 2485	. . . . 5 |- ( ph -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( ( # ` K ) + 2 ) )
76	47	nnnn0d 10914	. . . . . 6 |- ( ph -> ( N + 1 ) e. NN0 )
77		hashfz1 12515	. . . . . 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 2463	. . . 4 |- ( ph -> ( ( # ` K ) + 2 ) = ( N + 1 ) )
80	47	nncnd 10614	. . . . 5 |- ( ph -> ( N + 1 ) e. CC )
81		2cnd 10671	. . . . 5 |- ( ph -> 2 e. CC )
82		hashcl 12524	. . . . . . . 8 |- ( K e. Fin -> ( # ` K ) e. NN0 )
83	59, 82	ax-mp 5	. . . . . . 7 |- ( # ` K ) e. NN0
84	83	nn0cni 10870	. . . . . 6 |- ( # ` K ) e. CC
85	84	a1i 11	. . . . 5 |- ( ph -> ( # ` K ) e. CC )
86	80, 81, 85	subadd2d 9994	. . . 4 |- ( ph -> ( ( ( N + 1 ) - 2 ) = ( # ` K ) <-> ( ( # ` K ) + 2 ) = ( N + 1 ) ) )
87	79, 86	mpbird 235	. . 3 |- ( ph -> ( ( N + 1 ) - 2 ) = ( # ` K ) )
88	46	nncnd 10614	. . . 4 |- ( ph -> N e. CC )
89		1cnd 9648	. . . 4 |- ( ph -> 1 e. CC )
90	88, 89, 89	pnpcan2d 10013	. . 3 |- ( ph -> ( ( N + 1 ) - ( 1 + 1 ) ) = ( N - 1 ) )
91	55, 87, 90	3eqtr3a 2485	. 2 |- ( ph -> ( # ` K ) = ( N - 1 ) )
92	23, 54, 91	3jca 1185	1 |- ( ph -> ( ( K i^i { 1 , M } ) = (/) /\ ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) /\ ( # ` K ) = ( N - 1 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   \/ wo 369   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1867   {cab 2405   =/= wne 2616   A.wral 2773   _Vcvv 3078   \ cdif 3430   u. cun 3431   i^i cin 3432   C_ wss 3433   (/)c0 3758   {csn 3993   {cpr 3995   class class class wbr 4417   |-> cmpt 4475   -1-1-onto->wf1o 5591   ` cfv 5592  (class class class)co 6296   Fincfn 7568   CCcc 9526   1c1 9529   + caddc 9531   < clt 9664   <_ cle 9665   - cmin 9849   NNcn 10598   2c2 10648   NN0cn0 10858   ZZcz 10926   ZZ>=cuz 11148   ...cfz 11771   #chash 12501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  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-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-hash 12502

This theorem is referenced by:  subfacp1lem2a  29688  subfacp1lem3  29690  subfacp1lem4  29691