Theorem ballotfilemth 13283

Description: Lemma for ballotfi 13284. The result, with several additional hypotheses which are for use during the proof. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Hypotheses

Ref	Expression
ballotth.m	|- M e. NN
ballotth.n	|- N e. NN
ballotfilem.o	|- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
ballotfilem.p	|- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
ballotth.f	|- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
ballotth.e	|- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
ballotth.mgtn	|- N < M
ballotth.i	|- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
ballotth.s	|- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
ballotth.r	|- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )

Assertion

Ref	Expression
ballotfilemth	|- ( P ` E ) = ( ( M - N ) / ( M + N ) )

Distinct variable groups:   M, c   N, c   O, c   i, M   i, N   i, O   k, M   k, N   k, O   i, c, F, k   i, E, k   k, I, c   E, c   i, I, c   S, k, i, c   R, i, k   x, c, F   x, M   x, N, k, i   x, E   x, O

Allowed substitution hints:   P( x, i, k, c)   R( x, c)   S( x)   I( x)

Proof of Theorem ballotfilemth

Step	Hyp	Ref	Expression
1		ballotth.e	. . . . . . 7 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
2	1	ssrab3 3334	. . . . . 6 |- E C_ O
3		ballotth.m	. . . . . . . . 9 |- M e. NN
4		ballotth.n	. . . . . . . . 9 |- N e. NN
5		ballotfilem.o	. . . . . . . . 9 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
6		ballotfilem.p	. . . . . . . . 9 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
7		ballotth.f	. . . . . . . . 9 |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
8	3, 4, 5, 6, 7, 1	ballotfilemefi 13239	. . . . . . . 8 |- E e. Fin
9	8	elexi 2834	. . . . . . 7 |- E e. _V
10	9	elpw 3694	. . . . . 6 |- ( E e. ~P O <-> E C_ O )
11	2, 10	mpbir 146	. . . . 5 |- E e. ~P O
12		elin 3412	. . . . . 6 |- ( E e. ( ~P O i^i Fin ) <-> ( E e. ~P O /\ E e. Fin ) )
13		fveq2 5695	. . . . . . . 8 |- ( x = E -> (♯ ` x ) = (♯ ` E ) )
14	13	oveq1d 6100	. . . . . . 7 |- ( x = E -> ( (♯ ` x ) / (♯ ` O ) ) = ( (♯ ` E ) / (♯ ` O ) ) )
15		hashcl 11222	. . . . . . . . . 10 |- ( E e. Fin -> (♯ ` E ) e. NN0 )
16	8, 15	ax-mp 5	. . . . . . . . 9 |- (♯ ` E ) e. NN0
17	3, 4, 5	ballotfilemonn 13223	. . . . . . . . 9 |- (♯ ` O ) e. NN
18		nn0nndivcl 9631	. . . . . . . . 9 |- ( ( (♯ ` E ) e. NN0 /\ (♯ ` O ) e. NN ) -> ( (♯ ` E ) / (♯ ` O ) ) e. RR )
19	16, 17, 18	mp2an 430	. . . . . . . 8 |- ( (♯ ` E ) / (♯ ` O ) ) e. RR
20	19	elexi 2834	. . . . . . 7 |- ( (♯ ` E ) / (♯ ` O ) ) e. _V
21	14, 6, 20	fvmpt 5782	. . . . . 6 |- ( E e. ( ~P O i^i Fin ) -> ( P ` E ) = ( (♯ ` E ) / (♯ ` O ) ) )
22	12, 21	sylbir 135	. . . . 5 |- ( ( E e. ~P O /\ E e. Fin ) -> ( P ` E ) = ( (♯ ` E ) / (♯ ` O ) ) )
23	11, 8, 22	mp2an 430	. . . 4 |- ( P ` E ) = ( (♯ ` E ) / (♯ ` O ) )
24	3, 4, 5	ballotfilemofi 13221	. . . . . . . 8 |- O e. Fin
25		fihashssdif 11261	. . . . . . . 8 |- ( ( O e. Fin /\ E e. Fin /\ E C_ O ) -> (♯ ` ( O \ E ) ) = ( (♯ ` O ) - (♯ ` E ) ) )
26	24, 8, 2, 25	mp3an 1378	. . . . . . 7 |- (♯ ` ( O \ E ) ) = ( (♯ ` O ) - (♯ ` E ) )
27	26	eqcomi 2242	. . . . . 6 |- ( (♯ ` O ) - (♯ ` E ) ) = (♯ ` ( O \ E ) )
28		hashcl 11222	. . . . . . . . 9 |- ( O e. Fin -> (♯ ` O ) e. NN0 )
29	24, 28	ax-mp 5	. . . . . . . 8 |- (♯ ` O ) e. NN0
30	29	nn0cni 9577	. . . . . . 7 |- (♯ ` O ) e. CC
31	16	nn0cni 9577	. . . . . . 7 |- (♯ ` E ) e. CC
32		diffifi 7198	. . . . . . . . . 10 |- ( ( O e. Fin /\ E e. Fin /\ E C_ O ) -> ( O \ E ) e. Fin )
33	24, 8, 2, 32	mp3an 1378	. . . . . . . . 9 |- ( O \ E ) e. Fin
34		hashcl 11222	. . . . . . . . 9 |- ( ( O \ E ) e. Fin -> (♯ ` ( O \ E ) ) e. NN0 )
35	33, 34	ax-mp 5	. . . . . . . 8 |- (♯ ` ( O \ E ) ) e. NN0
36	35	nn0cni 9577	. . . . . . 7 |- (♯ ` ( O \ E ) ) e. CC
37	30, 31, 36	subsub23i 8616	. . . . . 6 |- ( ( (♯ ` O ) - (♯ ` E ) ) = (♯ ` ( O \ E ) ) <-> ( (♯ ` O ) - (♯ ` ( O \ E ) ) ) = (♯ ` E ) )
38	27, 37	mpbi 145	. . . . 5 |- ( (♯ ` O ) - (♯ ` ( O \ E ) ) ) = (♯ ` E )
39	38	oveq1i 6095	. . . 4 |- ( ( (♯ ` O ) - (♯ ` ( O \ E ) ) ) / (♯ ` O ) ) = ( (♯ ` E ) / (♯ ` O ) )
40	23, 39	eqtr4i 2262	. . 3 |- ( P ` E ) = ( ( (♯ ` O ) - (♯ ` ( O \ E ) ) ) / (♯ ` O ) )
41	3, 4, 5	ballotfilem1 13222	. . . . . 6 |- (♯ ` O ) = ( ( M + N ) _C M )
42	3	nnnn0i 9573	. . . . . . . . 9 |- M e. NN0
43		nnaddcl 9325	. . . . . . . . . . 11 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
44	3, 4, 43	mp2an 430	. . . . . . . . . 10 |- ( M + N ) e. NN
45	44	nnnn0i 9573	. . . . . . . . 9 |- ( M + N ) e. NN0
46	3	nnrei 9314	. . . . . . . . . 10 |- M e. RR
47	4	nnnn0i 9573	. . . . . . . . . 10 |- N e. NN0
48	46, 47	nn0addge1i 9613	. . . . . . . . 9 |- M <_ ( M + N )
49		elfz2nn0 10521	. . . . . . . . 9 |- ( M e. ( 0 ... ( M + N ) ) <-> ( M e. NN0 /\ ( M + N ) e. NN0 /\ M <_ ( M + N ) ) )
50	42, 45, 48, 49	mpbir3an 1210	. . . . . . . 8 |- M e. ( 0 ... ( M + N ) )
51		bccl2 11208	. . . . . . . 8 |- ( M e. ( 0 ... ( M + N ) ) -> ( ( M + N ) _C M ) e. NN )
52	50, 51	ax-mp 5	. . . . . . 7 |- ( ( M + N ) _C M ) e. NN
53	52	nnap0i 9336	. . . . . 6 |- ( ( M + N ) _C M ) # 0
54	41, 53	eqbrtri 4151	. . . . 5 |- (♯ ` O ) # 0
55	30, 54	pm3.2i 272	. . . 4 |- ( (♯ ` O ) e. CC /\ (♯ ` O ) # 0 )
56		divsubdirap 9039	. . . 4 |- ( ( (♯ ` O ) e. CC /\ (♯ ` ( O \ E ) ) e. CC /\ ( (♯ ` O ) e. CC /\ (♯ ` O ) # 0 ) ) -> ( ( (♯ ` O ) - (♯ ` ( O \ E ) ) ) / (♯ ` O ) ) = ( ( (♯ ` O ) / (♯ ` O ) ) - ( (♯ ` ( O \ E ) ) / (♯ ` O ) ) ) )
57	30, 36, 55, 56	mp3an 1378	. . 3 |- ( ( (♯ ` O ) - (♯ ` ( O \ E ) ) ) / (♯ ` O ) ) = ( ( (♯ ` O ) / (♯ ` O ) ) - ( (♯ ` ( O \ E ) ) / (♯ ` O ) ) )
58	30, 54	dividapi 9076	. . . 4 |- ( (♯ ` O ) / (♯ ` O ) ) = 1
59	58	oveq1i 6095	. . 3 |- ( ( (♯ ` O ) / (♯ ` O ) ) - ( (♯ ` ( O \ E ) ) / (♯ ` O ) ) ) = ( 1 - ( (♯ ` ( O \ E ) ) / (♯ ` O ) ) )
60	40, 57, 59	3eqtri 2263	. 2 |- ( P ` E ) = ( 1 - ( (♯ ` ( O \ E ) ) / (♯ ` O ) ) )
61		ballotth.mgtn	. . . . . . 7 |- N < M
62		ballotth.i	. . . . . . 7 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
63		ballotth.s	. . . . . . 7 |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
64		ballotth.r	. . . . . . 7 |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
65	3, 4, 5, 6, 7, 1, 61, 62, 63, 64	ballotfilem8 13282	. . . . . 6 |- (♯ ` { c e. ( O \ E ) | 1 e. c } ) = (♯ ` { c e. ( O \ E ) | -. 1 e. c } )
66	65	oveq1i 6095	. . . . 5 |- ( (♯ ` { c e. ( O \ E ) | 1 e. c } ) + (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) ) = ( (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) + (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) )
67	66	oveq1i 6095	. . . 4 |- ( ( (♯ ` { c e. ( O \ E ) | 1 e. c } ) + (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) ) / (♯ ` O ) ) = ( ( (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) + (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) ) / (♯ ` O ) )
68		rabxmdc 3554	. . . . . . . 8 |- ( A. c e. ( O \ E )DECID 1 e. c -> ( O \ E ) = ( { c e. ( O \ E ) | 1 e. c } u. { c e. ( O \ E ) | -. 1 e. c } ) )
69		eldifi 3351	. . . . . . . . 9 |- ( c e. ( O \ E ) -> c e. O )
70		1zzd 9673	. . . . . . . . 9 |- ( c e. ( O \ E ) -> 1 e. ZZ )
71	3, 4, 5, 69, 70	ballotfilemcdc 13225	. . . . . . . 8 |- ( c e. ( O \ E ) -> DECID 1 e. c )
72	68, 71	mprg 2607	. . . . . . 7 |- ( O \ E ) = ( { c e. ( O \ E ) | 1 e. c } u. { c e. ( O \ E ) | -. 1 e. c } )
73	72	fveq2i 5698	. . . . . 6 |- (♯ ` ( O \ E ) ) = (♯ ` ( { c e. ( O \ E ) | 1 e. c } u. { c e. ( O \ E ) | -. 1 e. c } ) )
74	3, 4, 5, 6, 7, 1	ballotfilemafi 13240	. . . . . . 7 |- { c e. ( O \ E ) | 1 e. c } e. Fin
75	3, 4, 5, 6, 7, 1	ballotfilembfi 13241	. . . . . . 7 |- { c e. ( O \ E ) | -. 1 e. c } e. Fin
76		rabnc 3555	. . . . . . 7 |- ( { c e. ( O \ E ) | 1 e. c } i^i { c e. ( O \ E ) | -. 1 e. c } ) = (/)
77		hashun 11247	. . . . . . 7 |- ( ( { c e. ( O \ E ) | 1 e. c } e. Fin /\ { c e. ( O \ E ) | -. 1 e. c } e. Fin /\ ( { c e. ( O \ E ) | 1 e. c } i^i { c e. ( O \ E ) | -. 1 e. c } ) = (/) ) -> (♯ ` ( { c e. ( O \ E ) | 1 e. c } u. { c e. ( O \ E ) | -. 1 e. c } ) ) = ( (♯ ` { c e. ( O \ E ) | 1 e. c } ) + (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) ) )
78	74, 75, 76, 77	mp3an 1378	. . . . . 6 |- (♯ ` ( { c e. ( O \ E ) | 1 e. c } u. { c e. ( O \ E ) | -. 1 e. c } ) ) = ( (♯ ` { c e. ( O \ E ) | 1 e. c } ) + (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) )
79	73, 78	eqtri 2259	. . . . 5 |- (♯ ` ( O \ E ) ) = ( (♯ ` { c e. ( O \ E ) | 1 e. c } ) + (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) )
80	79	oveq1i 6095	. . . 4 |- ( (♯ ` ( O \ E ) ) / (♯ ` O ) ) = ( ( (♯ ` { c e. ( O \ E ) | 1 e. c } ) + (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) ) / (♯ ` O ) )
81		ssrab2 3333	. . . . . . . . . 10 |- { c e. O | -. 1 e. c } C_ O
82	24, 81	elpwi2 4294	. . . . . . . . 9 |- { c e. O | -. 1 e. c } e. ~P O
83	69	anim1i 340	. . . . . . . . . . . 12 |- ( ( c e. ( O \ E ) /\ -. 1 e. c ) -> ( c e. O /\ -. 1 e. c ) )
84	3, 4, 5, 6, 7, 1	ballotfilem4 13243	. . . . . . . . . . . . . . 15 |- ( c e. O -> ( -. 1 e. c -> -. c e. E ) )
85	84	imdistani 449	. . . . . . . . . . . . . 14 |- ( ( c e. O /\ -. 1 e. c ) -> ( c e. O /\ -. c e. E ) )
86		eldif 3229	. . . . . . . . . . . . . 14 |- ( c e. ( O \ E ) <-> ( c e. O /\ -. c e. E ) )
87	85, 86	sylibr 134	. . . . . . . . . . . . 13 |- ( ( c e. O /\ -. 1 e. c ) -> c e. ( O \ E ) )
88		simpr 110	. . . . . . . . . . . . 13 |- ( ( c e. O /\ -. 1 e. c ) -> -. 1 e. c )
89	87, 88	jca 306	. . . . . . . . . . . 12 |- ( ( c e. O /\ -. 1 e. c ) -> ( c e. ( O \ E ) /\ -. 1 e. c ) )
90	83, 89	impbii 126	. . . . . . . . . . 11 |- ( ( c e. ( O \ E ) /\ -. 1 e. c ) <-> ( c e. O /\ -. 1 e. c ) )
91	90	rabbia2 2806	. . . . . . . . . 10 |- { c e. ( O \ E ) | -. 1 e. c } = { c e. O | -. 1 e. c }
92	91, 75	eqeltrri 2312	. . . . . . . . 9 |- { c e. O | -. 1 e. c } e. Fin
93	82, 92	elini 3413	. . . . . . . 8 |- { c e. O | -. 1 e. c } e. ( ~P O i^i Fin )
94		fveq2 5695	. . . . . . . . . 10 |- ( x = { c e. O | -. 1 e. c } -> (♯ ` x ) = (♯ ` { c e. O | -. 1 e. c } ) )
95	94	oveq1d 6100	. . . . . . . . 9 |- ( x = { c e. O | -. 1 e. c } -> ( (♯ ` x ) / (♯ ` O ) ) = ( (♯ ` { c e. O | -. 1 e. c } ) / (♯ ` O ) ) )
96		hashcl 11222	. . . . . . . . . . . 12 |- ( { c e. O | -. 1 e. c } e. Fin -> (♯ ` { c e. O | -. 1 e. c } ) e. NN0 )
97	92, 96	ax-mp 5	. . . . . . . . . . 11 |- (♯ ` { c e. O | -. 1 e. c } ) e. NN0
98		nn0nndivcl 9631	. . . . . . . . . . 11 |- ( ( (♯ ` { c e. O | -. 1 e. c } ) e. NN0 /\ (♯ ` O ) e. NN ) -> ( (♯ ` { c e. O | -. 1 e. c } ) / (♯ ` O ) ) e. RR )
99	97, 17, 98	mp2an 430	. . . . . . . . . 10 |- ( (♯ ` { c e. O | -. 1 e. c } ) / (♯ ` O ) ) e. RR
100	99	elexi 2834	. . . . . . . . 9 |- ( (♯ ` { c e. O | -. 1 e. c } ) / (♯ ` O ) ) e. _V
101	95, 6, 100	fvmpt 5782	. . . . . . . 8 |- ( { c e. O | -. 1 e. c } e. ( ~P O i^i Fin ) -> ( P ` { c e. O | -. 1 e. c } ) = ( (♯ ` { c e. O | -. 1 e. c } ) / (♯ ` O ) ) )
102	93, 101	ax-mp 5	. . . . . . 7 |- ( P ` { c e. O | -. 1 e. c } ) = ( (♯ ` { c e. O | -. 1 e. c } ) / (♯ ` O ) )
103	3, 4, 5, 6	ballotfilem2 13230	. . . . . . 7 |- ( P ` { c e. O | -. 1 e. c } ) = ( N / ( M + N ) )
104		nfrab1 2732	. . . . . . . . . . . 12 |- F/_ c { c e. O | -. 1 e. c }
105		nfrab1 2732	. . . . . . . . . . . 12 |- F/_ c { c e. ( O \ E ) | -. 1 e. c }
106	104, 105	dfssf 3238	. . . . . . . . . . 11 |- ( { c e. O | -. 1 e. c } C_ { c e. ( O \ E ) | -. 1 e. c } <-> A. c ( c e. { c e. O | -. 1 e. c } -> c e. { c e. ( O \ E ) | -. 1 e. c } ) )
107		rabid 2727	. . . . . . . . . . . . 13 |- ( c e. { c e. O | -. 1 e. c } <-> ( c e. O /\ -. 1 e. c ) )
108	85, 107, 86	3imtr4i 201	. . . . . . . . . . . 12 |- ( c e. { c e. O | -. 1 e. c } -> c e. ( O \ E ) )
109	107	simprbi 275	. . . . . . . . . . . 12 |- ( c e. { c e. O | -. 1 e. c } -> -. 1 e. c )
110		rabid 2727	. . . . . . . . . . . 12 |- ( c e. { c e. ( O \ E ) | -. 1 e. c } <-> ( c e. ( O \ E ) /\ -. 1 e. c ) )
111	108, 109, 110	sylanbrc 421	. . . . . . . . . . 11 |- ( c e. { c e. O | -. 1 e. c } -> c e. { c e. ( O \ E ) | -. 1 e. c } )
112	106, 111	mpgbir 1506	. . . . . . . . . 10 |- { c e. O | -. 1 e. c } C_ { c e. ( O \ E ) | -. 1 e. c }
113		difss 3355	. . . . . . . . . . 11 |- ( O \ E ) C_ O
114		rabss2 3331	. . . . . . . . . . 11 |- ( ( O \ E ) C_ O -> { c e. ( O \ E ) | -. 1 e. c } C_ { c e. O | -. 1 e. c } )
115	113, 114	ax-mp 5	. . . . . . . . . 10 |- { c e. ( O \ E ) | -. 1 e. c } C_ { c e. O | -. 1 e. c }
116	112, 115	eqssi 3264	. . . . . . . . 9 |- { c e. O | -. 1 e. c } = { c e. ( O \ E ) | -. 1 e. c }
117	116	fveq2i 5698	. . . . . . . 8 |- (♯ ` { c e. O | -. 1 e. c } ) = (♯ ` { c e. ( O \ E ) | -. 1 e. c } )
118	117	oveq1i 6095	. . . . . . 7 |- ( (♯ ` { c e. O | -. 1 e. c } ) / (♯ ` O ) ) = ( (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) / (♯ ` O ) )
119	102, 103, 118	3eqtr3i 2267	. . . . . 6 |- ( N / ( M + N ) ) = ( (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) / (♯ ` O ) )
120	119	oveq2i 6096	. . . . 5 |- ( 2 x. ( N / ( M + N ) ) ) = ( 2 x. ( (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) / (♯ ` O ) ) )
121		2cn 9376	. . . . . 6 |- 2 e. CC
122		hashcl 11222	. . . . . . . 8 |- ( { c e. ( O \ E ) | -. 1 e. c } e. Fin -> (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) e. NN0 )
123	75, 122	ax-mp 5	. . . . . . 7 |- (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) e. NN0
124	123	nn0cni 9577	. . . . . 6 |- (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) e. CC
125	121, 124, 30, 54	divassapi 9099	. . . . 5 |- ( ( 2 x. (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) ) / (♯ ` O ) ) = ( 2 x. ( (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) / (♯ ` O ) ) )
126	124	2timesi 9435	. . . . . 6 |- ( 2 x. (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) ) = ( (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) + (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) )
127	126	oveq1i 6095	. . . . 5 |- ( ( 2 x. (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) ) / (♯ ` O ) ) = ( ( (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) + (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) ) / (♯ ` O ) )
128	120, 125, 127	3eqtr2i 2265	. . . 4 |- ( 2 x. ( N / ( M + N ) ) ) = ( ( (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) + (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) ) / (♯ ` O ) )
129	67, 80, 128	3eqtr4ri 2270	. . 3 |- ( 2 x. ( N / ( M + N ) ) ) = ( (♯ ` ( O \ E ) ) / (♯ ` O ) )
130	129	oveq2i 6096	. 2 |- ( 1 - ( 2 x. ( N / ( M + N ) ) ) ) = ( 1 - ( (♯ ` ( O \ E ) ) / (♯ ` O ) ) )
131	44	nncni 9315	. . . 4 |- ( M + N ) e. CC
132	4	nncni 9315	. . . . 5 |- N e. CC
133	121, 132	mulcli 8331	. . . 4 |- ( 2 x. N ) e. CC
134	44	nnap0i 9336	. . . . 5 |- ( M + N ) # 0
135	131, 134	pm3.2i 272	. . . 4 |- ( ( M + N ) e. CC /\ ( M + N ) # 0 )
136		divsubdirap 9039	. . . 4 |- ( ( ( M + N ) e. CC /\ ( 2 x. N ) e. CC /\ ( ( M + N ) e. CC /\ ( M + N ) # 0 ) ) -> ( ( ( M + N ) - ( 2 x. N ) ) / ( M + N ) ) = ( ( ( M + N ) / ( M + N ) ) - ( ( 2 x. N ) / ( M + N ) ) ) )
137	131, 133, 135, 136	mp3an 1378	. . 3 |- ( ( ( M + N ) - ( 2 x. N ) ) / ( M + N ) ) = ( ( ( M + N ) / ( M + N ) ) - ( ( 2 x. N ) / ( M + N ) ) )
138	132	2timesi 9435	. . . . . 6 |- ( 2 x. N ) = ( N + N )
139	138	oveq2i 6096	. . . . 5 |- ( ( M + N ) - ( 2 x. N ) ) = ( ( M + N ) - ( N + N ) )
140	3	nncni 9315	. . . . . . 7 |- M e. CC
141	140, 132, 132, 132	addsub4i 8622	. . . . . 6 |- ( ( M + N ) - ( N + N ) ) = ( ( M - N ) + ( N - N ) )
142	132	subidi 8597	. . . . . . 7 |- ( N - N ) = 0
143	142	oveq2i 6096	. . . . . 6 |- ( ( M - N ) + ( N - N ) ) = ( ( M - N ) + 0 )
144	140, 132	subcli 8602	. . . . . . 7 |- ( M - N ) e. CC
145	144	addridi 8468	. . . . . 6 |- ( ( M - N ) + 0 ) = ( M - N )
146	141, 143, 145	3eqtri 2263	. . . . 5 |- ( ( M + N ) - ( N + N ) ) = ( M - N )
147	139, 146	eqtri 2259	. . . 4 |- ( ( M + N ) - ( 2 x. N ) ) = ( M - N )
148	147	oveq1i 6095	. . 3 |- ( ( ( M + N ) - ( 2 x. N ) ) / ( M + N ) ) = ( ( M - N ) / ( M + N ) )
149	131, 134	dividapi 9076	. . . 4 |- ( ( M + N ) / ( M + N ) ) = 1
150	121, 132, 131, 134	divassapi 9099	. . . 4 |- ( ( 2 x. N ) / ( M + N ) ) = ( 2 x. ( N / ( M + N ) ) )
151	149, 150	oveq12i 6097	. . 3 |- ( ( ( M + N ) / ( M + N ) ) - ( ( 2 x. N ) / ( M + N ) ) ) = ( 1 - ( 2 x. ( N / ( M + N ) ) ) )
152	137, 148, 151	3eqtr3ri 2268	. 2 |- ( 1 - ( 2 x. ( N / ( M + N ) ) ) ) = ( ( M - N ) / ( M + N ) )
153	60, 130, 152	3eqtr2i 2265	1 |- ( P ` E ) = ( ( M - N ) / ( M + N ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   \ cdif 3217   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   ifcif 3638   ~Pcpw 3688   class class class wbr 4130   |-> cmpt 4192   "cima 4777   ` cfv 5377  (class class class)co 6085   Fincfn 7022  infcinf 7323   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   <_ cle 8361   - cmin 8497   # cap 8910   / cdiv 9003   NNcn 9305   2c2 9356   NN0cn0 9565   ZZcz 9646   ...cfz 10413   _C cbc 11187  ♯chash 11216

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-fz 10414  df-fzo 10552  df-seqfrec 10887  df-fac 11166  df-bc 11188  df-ihash 11217

This theorem is used by:  ballotfi  13284