Theorem ballotfilemth 13225

Description: Lemma for ballotfi 13226. 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 3328	. . . . . 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 13181	. . . . . . . 8 |- E e. Fin
9	8	elexi 2828	. . . . . . 7 |- E e. _V
10	9	elpw 3680	. . . . . 6 |- ( E e. ~P O <-> E C_ O )
11	2, 10	mpbir 146	. . . . 5 |- E e. ~P O
12		elin 3406	. . . . . 6 |- ( E e. ( ~P O i^i Fin ) <-> ( E e. ~P O /\ E e. Fin ) )
13		fveq2 5675	. . . . . . . 8 |- ( x = E -> (♯ ` x ) = (♯ ` E ) )
14	13	oveq1d 6073	. . . . . . 7 |- ( x = E -> ( (♯ ` x ) / (♯ ` O ) ) = ( (♯ ` E ) / (♯ ` O ) ) )
15		hashcl 11169	. . . . . . . . . 10 |- ( E e. Fin -> (♯ ` E ) e. NN0 )
16	8, 15	ax-mp 5	. . . . . . . . 9 |- (♯ ` E ) e. NN0
17	3, 4, 5	ballotfilemonn 13165	. . . . . . . . 9 |- (♯ ` O ) e. NN
18		nn0nndivcl 9579	. . . . . . . . 9 |- ( ( (♯ ` E ) e. NN0 /\ (♯ ` O ) e. NN ) -> ( (♯ ` E ) / (♯ ` O ) ) e. RR )
19	16, 17, 18	mp2an 426	. . . . . . . 8 |- ( (♯ ` E ) / (♯ ` O ) ) e. RR
20	19	elexi 2828	. . . . . . 7 |- ( (♯ ` E ) / (♯ ` O ) ) e. _V
21	14, 6, 20	fvmpt 5759	. . . . . 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 426	. . . 4 |- ( P ` E ) = ( (♯ ` E ) / (♯ ` O ) )
24	3, 4, 5	ballotfilemofi 13163	. . . . . . . 8 |- O e. Fin
25		fihashssdif 11208	. . . . . . . 8 |- ( ( O e. Fin /\ E e. Fin /\ E C_ O ) -> (♯ ` ( O \ E ) ) = ( (♯ ` O ) - (♯ ` E ) ) )
26	24, 8, 2, 25	mp3an 1374	. . . . . . 7 |- (♯ ` ( O \ E ) ) = ( (♯ ` O ) - (♯ ` E ) )
27	26	eqcomi 2238	. . . . . 6 |- ( (♯ ` O ) - (♯ ` E ) ) = (♯ ` ( O \ E ) )
28		hashcl 11169	. . . . . . . . 9 |- ( O e. Fin -> (♯ ` O ) e. NN0 )
29	24, 28	ax-mp 5	. . . . . . . 8 |- (♯ ` O ) e. NN0
30	29	nn0cni 9525	. . . . . . 7 |- (♯ ` O ) e. CC
31	16	nn0cni 9525	. . . . . . 7 |- (♯ ` E ) e. CC
32		diffifi 7164	. . . . . . . . . 10 |- ( ( O e. Fin /\ E e. Fin /\ E C_ O ) -> ( O \ E ) e. Fin )
33	24, 8, 2, 32	mp3an 1374	. . . . . . . . 9 |- ( O \ E ) e. Fin
34		hashcl 11169	. . . . . . . . 9 |- ( ( O \ E ) e. Fin -> (♯ ` ( O \ E ) ) e. NN0 )
35	33, 34	ax-mp 5	. . . . . . . 8 |- (♯ ` ( O \ E ) ) e. NN0
36	35	nn0cni 9525	. . . . . . 7 |- (♯ ` ( O \ E ) ) e. CC
37	30, 31, 36	subsub23i 8579	. . . . . 6 |- ( ( (♯ ` O ) - (♯ ` E ) ) = (♯ ` ( O \ E ) ) <-> ( (♯ ` O ) - (♯ ` ( O \ E ) ) ) = (♯ ` E ) )
38	27, 37	mpbi 145	. . . . 5 |- ( (♯ ` O ) - (♯ ` ( O \ E ) ) ) = (♯ ` E )
39	38	oveq1i 6068	. . . 4 |- ( ( (♯ ` O ) - (♯ ` ( O \ E ) ) ) / (♯ ` O ) ) = ( (♯ ` E ) / (♯ ` O ) )
40	23, 39	eqtr4i 2258	. . 3 |- ( P ` E ) = ( ( (♯ ` O ) - (♯ ` ( O \ E ) ) ) / (♯ ` O ) )
41	3, 4, 5	ballotfilem1 13164	. . . . . 6 |- (♯ ` O ) = ( ( M + N ) _C M )
42	3	nnnn0i 9521	. . . . . . . . 9 |- M e. NN0
43		nnaddcl 9274	. . . . . . . . . . 11 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
44	3, 4, 43	mp2an 426	. . . . . . . . . 10 |- ( M + N ) e. NN
45	44	nnnn0i 9521	. . . . . . . . 9 |- ( M + N ) e. NN0
46	3	nnrei 9263	. . . . . . . . . 10 |- M e. RR
47	4	nnnn0i 9521	. . . . . . . . . 10 |- N e. NN0
48	46, 47	nn0addge1i 9561	. . . . . . . . 9 |- M <_ ( M + N )
49		elfz2nn0 10468	. . . . . . . . 9 |- ( M e. ( 0 ... ( M + N ) ) <-> ( M e. NN0 /\ ( M + N ) e. NN0 /\ M <_ ( M + N ) ) )
50	42, 45, 48, 49	mpbir3an 1206	. . . . . . . 8 |- M e. ( 0 ... ( M + N ) )
51		bccl2 11155	. . . . . . . 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 9285	. . . . . 6 |- ( ( M + N ) _C M ) # 0
54	41, 53	eqbrtri 4135	. . . . 5 |- (♯ ` O ) # 0
55	30, 54	pm3.2i 272	. . . 4 |- ( (♯ ` O ) e. CC /\ (♯ ` O ) # 0 )
56		divsubdirap 8999	. . . 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 1374	. . 3 |- ( ( (♯ ` O ) - (♯ ` ( O \ E ) ) ) / (♯ ` O ) ) = ( ( (♯ ` O ) / (♯ ` O ) ) - ( (♯ ` ( O \ E ) ) / (♯ ` O ) ) )
58	30, 54	dividapi 9036	. . . 4 |- ( (♯ ` O ) / (♯ ` O ) ) = 1
59	58	oveq1i 6068	. . 3 |- ( ( (♯ ` O ) / (♯ ` O ) ) - ( (♯ ` ( O \ E ) ) / (♯ ` O ) ) ) = ( 1 - ( (♯ ` ( O \ E ) ) / (♯ ` O ) ) )
60	40, 57, 59	3eqtri 2259	. 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 13224	. . . . . 6 |- (♯ ` { c e. ( O \ E ) | 1 e. c } ) = (♯ ` { c e. ( O \ E ) | -. 1 e. c } )
66	65	oveq1i 6068	. . . . 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 6068	. . . 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 3544	. . . . . . . 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 3345	. . . . . . . . 9 |- ( c e. ( O \ E ) -> c e. O )
70		1zzd 9621	. . . . . . . . 9 |- ( c e. ( O \ E ) -> 1 e. ZZ )
71	3, 4, 5, 69, 70	ballotfilemcdc 13167	. . . . . . . 8 |- ( c e. ( O \ E ) -> DECID 1 e. c )
72	68, 71	mprg 2601	. . . . . . 7 |- ( O \ E ) = ( { c e. ( O \ E ) | 1 e. c } u. { c e. ( O \ E ) | -. 1 e. c } )
73	72	fveq2i 5678	. . . . . 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 13182	. . . . . . 7 |- { c e. ( O \ E ) | 1 e. c } e. Fin
75	3, 4, 5, 6, 7, 1	ballotfilembfi 13183	. . . . . . 7 |- { c e. ( O \ E ) | -. 1 e. c } e. Fin
76		rabnc 3545	. . . . . . 7 |- ( { c e. ( O \ E ) | 1 e. c } i^i { c e. ( O \ E ) | -. 1 e. c } ) = (/)
77		hashun 11194	. . . . . . 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 1374	. . . . . 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 2255	. . . . 5 |- (♯ ` ( O \ E ) ) = ( (♯ ` { c e. ( O \ E ) | 1 e. c } ) + (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) )
80	79	oveq1i 6068	. . . 4 |- ( (♯ ` ( O \ E ) ) / (♯ ` O ) ) = ( ( (♯ ` { c e. ( O \ E ) | 1 e. c } ) + (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) ) / (♯ ` O ) )
81		ssrab2 3327	. . . . . . . . . 10 |- { c e. O | -. 1 e. c } C_ O
82	24, 81	elpwi2 4275	. . . . . . . . 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 13185	. . . . . . . . . . . . . . 15 |- ( c e. O -> ( -. 1 e. c -> -. c e. E ) )
85	84	imdistani 445	. . . . . . . . . . . . . 14 |- ( ( c e. O /\ -. 1 e. c ) -> ( c e. O /\ -. c e. E ) )
86		eldif 3223	. . . . . . . . . . . . . 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 2800	. . . . . . . . . 10 |- { c e. ( O \ E ) | -. 1 e. c } = { c e. O | -. 1 e. c }
92	91, 75	eqeltrri 2308	. . . . . . . . 9 |- { c e. O | -. 1 e. c } e. Fin
93	82, 92	elini 3407	. . . . . . . 8 |- { c e. O | -. 1 e. c } e. ( ~P O i^i Fin )
94		fveq2 5675	. . . . . . . . . 10 |- ( x = { c e. O | -. 1 e. c } -> (♯ ` x ) = (♯ ` { c e. O | -. 1 e. c } ) )
95	94	oveq1d 6073	. . . . . . . . 9 |- ( x = { c e. O | -. 1 e. c } -> ( (♯ ` x ) / (♯ ` O ) ) = ( (♯ ` { c e. O | -. 1 e. c } ) / (♯ ` O ) ) )
96		hashcl 11169	. . . . . . . . . . . 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 9579	. . . . . . . . . . 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 426	. . . . . . . . . 10 |- ( (♯ ` { c e. O | -. 1 e. c } ) / (♯ ` O ) ) e. RR
100	99	elexi 2828	. . . . . . . . 9 |- ( (♯ ` { c e. O | -. 1 e. c } ) / (♯ ` O ) ) e. _V
101	95, 6, 100	fvmpt 5759	. . . . . . . 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 13172	. . . . . . 7 |- ( P ` { c e. O | -. 1 e. c } ) = ( N / ( M + N ) )
104		nfrab1 2726	. . . . . . . . . . . 12 |- F/_ c { c e. O | -. 1 e. c }
105		nfrab1 2726	. . . . . . . . . . . 12 |- F/_ c { c e. ( O \ E ) | -. 1 e. c }
106	104, 105	dfssf 3232	. . . . . . . . . . 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 2721	. . . . . . . . . . . . 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 2721	. . . . . . . . . . . 12 |- ( c e. { c e. ( O \ E ) | -. 1 e. c } <-> ( c e. ( O \ E ) /\ -. 1 e. c ) )
111	108, 109, 110	sylanbrc 417	. . . . . . . . . . 11 |- ( c e. { c e. O | -. 1 e. c } -> c e. { c e. ( O \ E ) | -. 1 e. c } )
112	106, 111	mpgbir 1502	. . . . . . . . . 10 |- { c e. O | -. 1 e. c } C_ { c e. ( O \ E ) | -. 1 e. c }
113		difss 3349	. . . . . . . . . . 11 |- ( O \ E ) C_ O
114		rabss2 3325	. . . . . . . . . . 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 3258	. . . . . . . . 9 |- { c e. O | -. 1 e. c } = { c e. ( O \ E ) | -. 1 e. c }
117	116	fveq2i 5678	. . . . . . . 8 |- (♯ ` { c e. O | -. 1 e. c } ) = (♯ ` { c e. ( O \ E ) | -. 1 e. c } )
118	117	oveq1i 6068	. . . . . . 7 |- ( (♯ ` { c e. O | -. 1 e. c } ) / (♯ ` O ) ) = ( (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) / (♯ ` O ) )
119	102, 103, 118	3eqtr3i 2263	. . . . . 6 |- ( N / ( M + N ) ) = ( (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) / (♯ ` O ) )
120	119	oveq2i 6069	. . . . 5 |- ( 2 x. ( N / ( M + N ) ) ) = ( 2 x. ( (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) / (♯ ` O ) ) )
121		2cn 9325	. . . . . 6 |- 2 e. CC
122		hashcl 11169	. . . . . . . 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 9525	. . . . . 6 |- (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) e. CC
125	121, 124, 30, 54	divassapi 9059	. . . . 5 |- ( ( 2 x. (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) ) / (♯ ` O ) ) = ( 2 x. ( (♯ ` { c e. ( O \ E ) | -. 1 e. c } ) / (♯ ` O ) ) )
126	124	2timesi 9384	. . . . . 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 6068	. . . . 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 2261	. . . 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 2266	. . 3 |- ( 2 x. ( N / ( M + N ) ) ) = ( (♯ ` ( O \ E ) ) / (♯ ` O ) )
130	129	oveq2i 6069	. 2 |- ( 1 - ( 2 x. ( N / ( M + N ) ) ) ) = ( 1 - ( (♯ ` ( O \ E ) ) / (♯ ` O ) ) )
131	44	nncni 9264	. . . 4 |- ( M + N ) e. CC
132	4	nncni 9264	. . . . 5 |- N e. CC
133	121, 132	mulcli 8295	. . . 4 |- ( 2 x. N ) e. CC
134	44	nnap0i 9285	. . . . 5 |- ( M + N ) # 0
135	131, 134	pm3.2i 272	. . . 4 |- ( ( M + N ) e. CC /\ ( M + N ) # 0 )
136		divsubdirap 8999	. . . 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 1374	. . 3 |- ( ( ( M + N ) - ( 2 x. N ) ) / ( M + N ) ) = ( ( ( M + N ) / ( M + N ) ) - ( ( 2 x. N ) / ( M + N ) ) )
138	132	2timesi 9384	. . . . . 6 |- ( 2 x. N ) = ( N + N )
139	138	oveq2i 6069	. . . . 5 |- ( ( M + N ) - ( 2 x. N ) ) = ( ( M + N ) - ( N + N ) )
140	3	nncni 9264	. . . . . . 7 |- M e. CC
141	140, 132, 132, 132	addsub4i 8585	. . . . . 6 |- ( ( M + N ) - ( N + N ) ) = ( ( M - N ) + ( N - N ) )
142	132	subidi 8560	. . . . . . 7 |- ( N - N ) = 0
143	142	oveq2i 6069	. . . . . 6 |- ( ( M - N ) + ( N - N ) ) = ( ( M - N ) + 0 )
144	140, 132	subcli 8565	. . . . . . 7 |- ( M - N ) e. CC
145	144	addridi 8431	. . . . . 6 |- ( ( M - N ) + 0 ) = ( M - N )
146	141, 143, 145	3eqtri 2259	. . . . 5 |- ( ( M + N ) - ( N + N ) ) = ( M - N )
147	139, 146	eqtri 2255	. . . 4 |- ( ( M + N ) - ( 2 x. N ) ) = ( M - N )
148	147	oveq1i 6068	. . 3 |- ( ( ( M + N ) - ( 2 x. N ) ) / ( M + N ) ) = ( ( M - N ) / ( M + N ) )
149	131, 134	dividapi 9036	. . . 4 |- ( ( M + N ) / ( M + N ) ) = 1
150	121, 132, 131, 134	divassapi 9059	. . . 4 |- ( ( 2 x. N ) / ( M + N ) ) = ( 2 x. ( N / ( M + N ) ) )
151	149, 150	oveq12i 6070	. . 3 |- ( ( ( M + N ) / ( M + N ) ) - ( ( 2 x. N ) / ( M + N ) ) ) = ( 1 - ( 2 x. ( N / ( M + N ) ) ) )
152	137, 148, 151	3eqtr3ri 2264	. 2 |- ( 1 - ( 2 x. ( N / ( M + N ) ) ) ) = ( ( M - N ) / ( M + N ) )
153	60, 130, 152	3eqtr2i 2261	1 |- ( P ` E ) = ( ( M - N ) / ( M + N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 842   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   \ cdif 3211   u. cun 3212   i^i cin 3213   C_ wss 3214   (/)c0 3512   ifcif 3624   ~Pcpw 3674   class class class wbr 4114   |-> cmpt 4176   "cima 4757   ` cfv 5357  (class class class)co 6058   Fincfn 6988  infcinf 7287   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   <_ cle 8325   - cmin 8460   # cap 8872   / cdiv 8963   NNcn 9254   2c2 9305   NN0cn0 9513   ZZcz 9594   ...cfz 10361   _C cbc 11134  ♯chash 11163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-fac 11113  df-bc 11135  df-ihash 11164

This theorem is referenced by:  ballotfi  13226