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

Step	Hyp	Ref	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
7	5, 6	ltnlei 9500	. . . . . . . . . . . 12 |- ( 1 < 2 <-> -. 2 <_ 1 )
8	4, 7	mpbi 208	. . . . . . . . . . 11 |- -. 2 <_ 1
9		breq2 4301	. . . . . . . . . . 11 |- ( x = 1 -> ( 2 <_ x <-> 2 <_ 1 ) )
10	8, 9	mtbiri 303	. . . . . . . . . 10 |- ( x = 1 -> -. 2 <_ x )
11	10	necon2ai 2661	. . . . . . . . 9 |- ( 2 <_ x -> x =/= 1 )
12	2, 3, 11	3syl 20	. . . . . . . 8 |- ( x e. ( ( 2 ... ( N + 1 ) ) \ { M } ) -> x =/= 1 )
13		eldifsni 4006	. . . . . . . 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 2535	. . . . . 6 |- ( x e. K -> ( x =/= 1 /\ x =/= M ) )
17		neanior 2702	. . . . . 6 |- ( ( x =/= 1 /\ x =/= M ) <-> -. ( x = 1 \/ x = M ) )
18	16, 17	sylib 196	. . . . 5 |- ( x e. K -> -. ( x = 1 \/ x = M ) )
19		vex 2980	. . . . . 6 |- x e. _V
20	19	elpr 3900	. . . . 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 2791	. . 3 |- ( K i^i { 1 , M } ) = (/)
23	22	a1i 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 } )
27	25, 26	ax-mp 5	. . . . 5 |- ( 1 ... 1 ) = { 1 }
28	15	uneq1i 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 } )
30	28, 29	eqtr2i 2464	. . . . 5 |- ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( K u. { M } )
31	27, 30	uneq12i 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 } )
33	32	equncomi 3507	. . . . . 6 |- { 1 , M } = ( { M } u. { 1 } )
34	33	uneq2i 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 } ) )
36	34, 35	eqtr4i 2466	. . . 4 |- ( K u. { 1 , M } ) = ( ( K u. { M } ) u. { 1 } )
37	24, 31, 36	3eqtr4i 2473	. . 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 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 ) ) )
41	39, 40	sylib 196	. . . . . 6 |- ( ph -> ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( 2 ... ( N + 1 ) ) )
42		df-2 10385	. . . . . . 7 |- 2 = ( 1 + 1 )
43	42	oveq1i 6106	. . . . . 6 |- ( 2 ... ( N + 1 ) ) = ( ( 1 + 1 ) ... ( N + 1 ) )
44	41, 43	syl6eq 2491	. . . . 5 |- ( ph -> ( ( 2 ... ( N + 1 ) ) u. { M } ) = ( ( 1 + 1 ) ... ( N + 1 ) ) )
45	44	uneq2d 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 )
47	46	peano2nnd 10344	. . . . . 6 |- ( ph -> ( N + 1 ) e. NN )
48		nnuz 10901	. . . . . 6 |- NN = ( ZZ>= ` 1 )
49	47, 48	syl6eleq 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 ) ) ) )
52	49, 50, 51	3syl 20	. . . 4 |- ( ph -> ( 1 ... ( N + 1 ) ) = ( ( 1 ... 1 ) u. ( ( 1 + 1 ) ... ( N + 1 ) ) ) )
53	45, 52	eqtr4d 2478	. . 3 |- ( ph -> ( ( 1 ... 1 ) u. ( ( 2 ... ( N + 1 ) ) u. { M } ) ) = ( 1 ... ( N + 1 ) ) )
54	37, 53	syl5eqr 2489	. 2 |- ( ph -> ( K u. { 1 , M } ) = ( 1 ... ( N + 1 ) ) )
55	42	oveq2i 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 )
58	56, 57	ax-mp 5	. . . . . . . 8 |- ( ( 2 ... ( N + 1 ) ) \ { M } ) e. Fin
59	15, 58	eqeltri 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 } ) ) )
62	59, 60, 22, 61	mp3an 1314	. . . . . 6 |- ( # ` ( K u. { 1 , M } ) ) = ( ( # ` K ) + ( # ` { 1 , M } ) )
63	54	fveq2d 5700	. . . . . 6 |- ( ph -> ( # ` ( K u. { 1 , M } ) ) = ( # ` ( 1 ... ( N + 1 ) ) ) )
64		neeq1 2621	. . . . . . . . . . 11 |- ( x = M -> ( x =/= 1 <-> M =/= 1 ) )
65	3, 11	syl 16	. . . . . . . . . . 11 |- ( x e. ( 2 ... ( N + 1 ) ) -> x =/= 1 )
66	64, 65	vtoclga 3041	. . . . . . . . . 10 |- ( M e. ( 2 ... ( N + 1 ) ) -> M =/= 1 )
67	38, 66	syl 16	. . . . . . . . 9 |- ( ph -> M =/= 1 )
68	67	necomd 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 ) )
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 6112	. . . . . 6 |- ( ph -> ( ( # ` K ) + ( # ` { 1 , M } ) ) = ( ( # ` K ) + 2 ) )
75	62, 63, 74	3eqtr3a 2499	. . . . 5 |- ( ph -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( ( # ` K ) + 2 ) )
76	47	nnnn0d 10641	. . . . . 6 |- ( ph -> ( N + 1 ) e. NN0 )
77		hashfz1 12122	. . . . . 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 2477	. . . 4 |- ( ph -> ( ( # ` K ) + 2 ) = ( N + 1 ) )
80	47	nncnd 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 )
83	59, 82	ax-mp 5	. . . . . . 7 |- ( # ` K ) e. NN0
84	83	nn0cni 10596	. . . . . 6 |- ( # ` K ) e. CC
85	84	a1i 11	. . . . 5 |- ( ph -> ( # ` K ) e. CC )
86	80, 81, 85	subadd2d 9743	. . . 4 |- ( ph -> ( ( ( N + 1 ) - 2 ) = ( # ` K ) <-> ( ( # ` K ) + 2 ) = ( N + 1 ) ) )
87	79, 86	mpbird 232	. . 3 |- ( ph -> ( ( N + 1 ) - 2 ) = ( # ` K ) )
88	46	nncnd 10343	. . . 4 |- ( ph -> N e. CC )
89		ax-1cn 9345	. . . . 5 |- 1 e. CC
90	89	a1i 11	. . . 4 |- ( ph -> 1 e. CC )
91	88, 90, 90	pnpcan2d 9762	. . 3 |- ( ph -> ( ( N + 1 ) - ( 1 + 1 ) ) = ( N - 1 ) )
92	55, 87, 91	3eqtr3a 2499	. 2 |- ( ph -> ( # ` K ) = ( N - 1 ) )
93	23, 54, 92	3jca 1168	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 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