Theorem ballotth 34540

Description: Bertrand's ballot problem : the probability that A is ahead throughout the counting. The proof formalized here is a proof "by reflection", as opposed to other known proofs "by induction" or "by permutation". This is Metamath 100 proof #30. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Hypotheses

Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotth.o	⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
ballotth.p	⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
ballotth.f	⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e	⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
ballotth.mgtn	⊢ 𝑁 < 𝑀
ballotth.i	⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s	⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r	⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))

Assertion

Ref	Expression
ballotth	⊢ (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 ballotth

Step	Hyp	Ref	Expression
1		ballotth.e	. . . . . 6 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
2		ssrab2 4080	. . . . . 6 ⊢ {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} ⊆ 𝑂
3	1, 2	eqsstri 4030	. . . . 5 ⊢ 𝐸 ⊆ 𝑂
4		fzfi 14013	. . . . . . . . . . 11 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
5		pwfi 9357	. . . . . . . . . . 11 ⊢ ((1...(𝑀 + 𝑁)) ∈ Fin ↔ 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin)
6	4, 5	mpbi 230	. . . . . . . . . 10 ⊢ 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin
7		ballotth.o	. . . . . . . . . . 11 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
8		ssrab2 4080	. . . . . . . . . . 11 ⊢ {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} ⊆ 𝒫 (1...(𝑀 + 𝑁))
9	7, 8	eqsstri 4030	. . . . . . . . . 10 ⊢ 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))
10		ssfi 9213	. . . . . . . . . 10 ⊢ ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))) → 𝑂 ∈ Fin)
11	6, 9, 10	mp2an 692	. . . . . . . . 9 ⊢ 𝑂 ∈ Fin
12		ssfi 9213	. . . . . . . . 9 ⊢ ((𝑂 ∈ Fin ∧ 𝐸 ⊆ 𝑂) → 𝐸 ∈ Fin)
13	11, 3, 12	mp2an 692	. . . . . . . 8 ⊢ 𝐸 ∈ Fin
14	13	elexi 3503	. . . . . . 7 ⊢ 𝐸 ∈ V
15	14	elpw 4604	. . . . . 6 ⊢ (𝐸 ∈ 𝒫 𝑂 ↔ 𝐸 ⊆ 𝑂)
16		fveq2 6906	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸))
17	16	oveq1d 7446	. . . . . . 7 ⊢ (𝑥 = 𝐸 → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂)))
18		ballotth.p	. . . . . . 7 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
19		ovex 7464	. . . . . . 7 ⊢ ((♯‘𝐸) / (♯‘𝑂)) ∈ V
20	17, 18, 19	fvmpt 7016	. . . . . 6 ⊢ (𝐸 ∈ 𝒫 𝑂 → (𝑃‘𝐸) = ((♯‘𝐸) / (♯‘𝑂)))
21	15, 20	sylbir 235	. . . . 5 ⊢ (𝐸 ⊆ 𝑂 → (𝑃‘𝐸) = ((♯‘𝐸) / (♯‘𝑂)))
22	3, 21	ax-mp 5	. . . 4 ⊢ (𝑃‘𝐸) = ((♯‘𝐸) / (♯‘𝑂))
23		hashssdif 14451	. . . . . . . 8 ⊢ ((𝑂 ∈ Fin ∧ 𝐸 ⊆ 𝑂) → (♯‘(𝑂 ∖ 𝐸)) = ((♯‘𝑂) − (♯‘𝐸)))
24	11, 3, 23	mp2an 692	. . . . . . 7 ⊢ (♯‘(𝑂 ∖ 𝐸)) = ((♯‘𝑂) − (♯‘𝐸))
25	24	eqcomi 2746	. . . . . 6 ⊢ ((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂 ∖ 𝐸))
26		hashcl 14395	. . . . . . . . 9 ⊢ (𝑂 ∈ Fin → (♯‘𝑂) ∈ ℕ0)
27	11, 26	ax-mp 5	. . . . . . . 8 ⊢ (♯‘𝑂) ∈ ℕ0
28	27	nn0cni 12538	. . . . . . 7 ⊢ (♯‘𝑂) ∈ ℂ
29		hashcl 14395	. . . . . . . . 9 ⊢ (𝐸 ∈ Fin → (♯‘𝐸) ∈ ℕ0)
30	13, 29	ax-mp 5	. . . . . . . 8 ⊢ (♯‘𝐸) ∈ ℕ0
31	30	nn0cni 12538	. . . . . . 7 ⊢ (♯‘𝐸) ∈ ℂ
32		difss 4136	. . . . . . . . . 10 ⊢ (𝑂 ∖ 𝐸) ⊆ 𝑂
33		ssfi 9213	. . . . . . . . . 10 ⊢ ((𝑂 ∈ Fin ∧ (𝑂 ∖ 𝐸) ⊆ 𝑂) → (𝑂 ∖ 𝐸) ∈ Fin)
34	11, 32, 33	mp2an 692	. . . . . . . . 9 ⊢ (𝑂 ∖ 𝐸) ∈ Fin
35		hashcl 14395	. . . . . . . . 9 ⊢ ((𝑂 ∖ 𝐸) ∈ Fin → (♯‘(𝑂 ∖ 𝐸)) ∈ ℕ0)
36	34, 35	ax-mp 5	. . . . . . . 8 ⊢ (♯‘(𝑂 ∖ 𝐸)) ∈ ℕ0
37	36	nn0cni 12538	. . . . . . 7 ⊢ (♯‘(𝑂 ∖ 𝐸)) ∈ ℂ
38	28, 31, 37	subsub23i 11599	. . . . . 6 ⊢ (((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂 ∖ 𝐸)) ↔ ((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) = (♯‘𝐸))
39	25, 38	mpbi 230	. . . . 5 ⊢ ((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) = (♯‘𝐸)
40	39	oveq1i 7441	. . . 4 ⊢ (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂))
41	22, 40	eqtr4i 2768	. . 3 ⊢ (𝑃‘𝐸) = (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂))
42		ballotth.m	. . . . . . 7 ⊢ 𝑀 ∈ ℕ
43		ballotth.n	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
44	42, 43, 7	ballotlem1 34489	. . . . . 6 ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
45	42	nnnn0i 12534	. . . . . . . . 9 ⊢ 𝑀 ∈ ℕ0
46		nnaddcl 12289	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
47	42, 43, 46	mp2an 692	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℕ
48	47	nnnn0i 12534	. . . . . . . . 9 ⊢ (𝑀 + 𝑁) ∈ ℕ0
49	42	nnrei 12275	. . . . . . . . . 10 ⊢ 𝑀 ∈ ℝ
50	43	nnnn0i 12534	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℕ0
51	49, 50	nn0addge1i 12574	. . . . . . . . 9 ⊢ 𝑀 ≤ (𝑀 + 𝑁)
52		elfz2nn0 13658	. . . . . . . . 9 ⊢ (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ (𝑀 ∈ ℕ0 ∧ (𝑀 + 𝑁) ∈ ℕ0 ∧ 𝑀 ≤ (𝑀 + 𝑁)))
53	45, 48, 51, 52	mpbir3an 1342	. . . . . . . 8 ⊢ 𝑀 ∈ (0...(𝑀 + 𝑁))
54		bccl2 14362	. . . . . . . 8 ⊢ (𝑀 ∈ (0...(𝑀 + 𝑁)) → ((𝑀 + 𝑁)C𝑀) ∈ ℕ)
55	53, 54	ax-mp 5	. . . . . . 7 ⊢ ((𝑀 + 𝑁)C𝑀) ∈ ℕ
56	55	nnne0i 12306	. . . . . 6 ⊢ ((𝑀 + 𝑁)C𝑀) ≠ 0
57	44, 56	eqnetri 3011	. . . . 5 ⊢ (♯‘𝑂) ≠ 0
58	28, 57	pm3.2i 470	. . . 4 ⊢ ((♯‘𝑂) ∈ ℂ ∧ (♯‘𝑂) ≠ 0)
59		divsubdir 11961	. . . 4 ⊢ (((♯‘𝑂) ∈ ℂ ∧ (♯‘(𝑂 ∖ 𝐸)) ∈ ℂ ∧ ((♯‘𝑂) ∈ ℂ ∧ (♯‘𝑂) ≠ 0)) → (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂)) = (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂))))
60	28, 37, 58, 59	mp3an 1463	. . 3 ⊢ (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂)) = (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
61	28, 57	dividi 12000	. . . 4 ⊢ ((♯‘𝑂) / (♯‘𝑂)) = 1
62	61	oveq1i 7441	. . 3 ⊢ (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂))) = (1 − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
63	41, 60, 62	3eqtri 2769	. 2 ⊢ (𝑃‘𝐸) = (1 − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
64		ballotth.f	. . . . . . 7 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
65		ballotth.mgtn	. . . . . . 7 ⊢ 𝑁 < 𝑀
66		ballotth.i	. . . . . . 7 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
67		ballotth.s	. . . . . . 7 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
68		ballotth.r	. . . . . . 7 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
69	42, 43, 7, 18, 64, 1, 65, 66, 67, 68	ballotlem8 34539	. . . . . 6 ⊢ (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
70	69	oveq1i 7441	. . . . 5 ⊢ ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
71	70	oveq1i 7441	. . . 4 ⊢ (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
72		rabxm 4390	. . . . . . 7 ⊢ (𝑂 ∖ 𝐸) = ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
73	72	fveq2i 6909	. . . . . 6 ⊢ (♯‘(𝑂 ∖ 𝐸)) = (♯‘({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
74		ssrab2 4080	. . . . . . . . . 10 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
75	74, 32	sstri 3993	. . . . . . . . 9 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝑂
76	75, 9	sstri 3993	. . . . . . . 8 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
77		ssfi 9213	. . . . . . . 8 ⊢ ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))) → {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∈ Fin)
78	6, 76, 77	mp2an 692	. . . . . . 7 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∈ Fin
79		ssrab2 4080	. . . . . . . . . 10 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
80	79, 32	sstri 3993	. . . . . . . . 9 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
81	80, 9	sstri 3993	. . . . . . . 8 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
82		ssfi 9213	. . . . . . . 8 ⊢ ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))) → {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin)
83	6, 81, 82	mp2an 692	. . . . . . 7 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin
84		rabnc 4391	. . . . . . 7 ⊢ ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∩ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) = ∅
85		hashun 14421	. . . . . . 7 ⊢ (({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∈ Fin ∧ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin ∧ ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∩ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) = ∅) → (♯‘({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})))
86	78, 83, 84, 85	mp3an 1463	. . . . . 6 ⊢ (♯‘({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
87	73, 86	eqtri 2765	. . . . 5 ⊢ (♯‘(𝑂 ∖ 𝐸)) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
88	87	oveq1i 7441	. . . 4 ⊢ ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
89		ssrab2 4080	. . . . . . . . 9 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
90	11	elexi 3503	. . . . . . . . . 10 ⊢ 𝑂 ∈ V
91	90	elpw2 5334	. . . . . . . . 9 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 ↔ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂)
92	89, 91	mpbir 231	. . . . . . . 8 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
93		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}))
94	93	oveq1d 7446	. . . . . . . . 9 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
95		ovex 7464	. . . . . . . . 9 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
96	94, 18, 95	fvmpt 7016	. . . . . . . 8 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 → (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
97	92, 96	ax-mp 5	. . . . . . 7 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
98	42, 43, 7, 18	ballotlem2 34491	. . . . . . 7 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))
99		nfrab1 3457	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
100		nfrab1 3457	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
101	99, 100	dfssf 3974	. . . . . . . . . . 11 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ ∀𝑐(𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
102	42, 43, 7, 18, 64, 1	ballotlem4 34501	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝑂 → (¬ 1 ∈ 𝑐 → ¬ 𝑐 ∈ 𝐸))
103	102	imdistani 568	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐) → (𝑐 ∈ 𝑂 ∧ ¬ 𝑐 ∈ 𝐸))
104		rabid 3458	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐))
105		eldif 3961	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↔ (𝑐 ∈ 𝑂 ∧ ¬ 𝑐 ∈ 𝐸))
106	103, 104, 105	3imtr4i 292	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ (𝑂 ∖ 𝐸))
107	104	simprbi 496	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ¬ 1 ∈ 𝑐)
108		rabid 3458	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐))
109	106, 107, 108	sylanbrc 583	. . . . . . . . . . 11 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
110	101, 109	mpgbir 1799	. . . . . . . . . 10 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
111		rabss2 4078	. . . . . . . . . . 11 ⊢ ((𝑂 ∖ 𝐸) ⊆ 𝑂 → {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐})
112	32, 111	ax-mp 5	. . . . . . . . . 10 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
113	110, 112	eqssi 4000	. . . . . . . . 9 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} = {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
114	113	fveq2i 6909	. . . . . . . 8 ⊢ (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
115	114	oveq1i 7441	. . . . . . 7 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
116	97, 98, 115	3eqtr3i 2773	. . . . . 6 ⊢ (𝑁 / (𝑀 + 𝑁)) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
117	116	oveq2i 7442	. . . . 5 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = (2 · ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
118		2cn 12341	. . . . . 6 ⊢ 2 ∈ ℂ
119		hashcl 14395	. . . . . . . 8 ⊢ ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin → (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ0)
120	83, 119	ax-mp 5	. . . . . . 7 ⊢ (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ0
121	120	nn0cni 12538	. . . . . 6 ⊢ (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℂ
122	118, 121, 28, 57	divassi 12023	. . . . 5 ⊢ ((2 · (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (2 · ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
123	121	2timesi 12404	. . . . . 6 ⊢ (2 · (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
124	123	oveq1i 7441	. . . . 5 ⊢ ((2 · (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
125	117, 122, 124	3eqtr2i 2771	. . . 4 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
126	71, 88, 125	3eqtr4ri 2776	. . 3 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂))
127	126	oveq2i 7442	. 2 ⊢ (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = (1 − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
128	47	nncni 12276	. . . 4 ⊢ (𝑀 + 𝑁) ∈ ℂ
129	43	nncni 12276	. . . . 5 ⊢ 𝑁 ∈ ℂ
130	118, 129	mulcli 11268	. . . 4 ⊢ (2 · 𝑁) ∈ ℂ
131	47	nnne0i 12306	. . . . 5 ⊢ (𝑀 + 𝑁) ≠ 0
132	128, 131	pm3.2i 470	. . . 4 ⊢ ((𝑀 + 𝑁) ∈ ℂ ∧ (𝑀 + 𝑁) ≠ 0)
133		divsubdir 11961	. . . 4 ⊢ (((𝑀 + 𝑁) ∈ ℂ ∧ (2 · 𝑁) ∈ ℂ ∧ ((𝑀 + 𝑁) ∈ ℂ ∧ (𝑀 + 𝑁) ≠ 0)) → (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁))))
134	128, 130, 132, 133	mp3an 1463	. . 3 ⊢ (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁)))
135	129	2timesi 12404	. . . . . 6 ⊢ (2 · 𝑁) = (𝑁 + 𝑁)
136	135	oveq2i 7442	. . . . 5 ⊢ ((𝑀 + 𝑁) − (2 · 𝑁)) = ((𝑀 + 𝑁) − (𝑁 + 𝑁))
137	42	nncni 12276	. . . . . . 7 ⊢ 𝑀 ∈ ℂ
138	137, 129, 129, 129	addsub4i 11605	. . . . . 6 ⊢ ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = ((𝑀 − 𝑁) + (𝑁 − 𝑁))
139	129	subidi 11580	. . . . . . 7 ⊢ (𝑁 − 𝑁) = 0
140	139	oveq2i 7442	. . . . . 6 ⊢ ((𝑀 − 𝑁) + (𝑁 − 𝑁)) = ((𝑀 − 𝑁) + 0)
141	137, 129	subcli 11585	. . . . . . 7 ⊢ (𝑀 − 𝑁) ∈ ℂ
142	141	addridi 11448	. . . . . 6 ⊢ ((𝑀 − 𝑁) + 0) = (𝑀 − 𝑁)
143	138, 140, 142	3eqtri 2769	. . . . 5 ⊢ ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = (𝑀 − 𝑁)
144	136, 143	eqtri 2765	. . . 4 ⊢ ((𝑀 + 𝑁) − (2 · 𝑁)) = (𝑀 − 𝑁)
145	144	oveq1i 7441	. . 3 ⊢ (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))
146	128, 131	dividi 12000	. . . 4 ⊢ ((𝑀 + 𝑁) / (𝑀 + 𝑁)) = 1
147	118, 129, 128, 131	divassi 12023	. . . 4 ⊢ ((2 · 𝑁) / (𝑀 + 𝑁)) = (2 · (𝑁 / (𝑀 + 𝑁)))
148	146, 147	oveq12i 7443	. . 3 ⊢ (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁))) = (1 − (2 · (𝑁 / (𝑀 + 𝑁))))
149	134, 145, 148	3eqtr3ri 2774	. 2 ⊢ (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))
150	63, 127, 149	3eqtr2i 2771	1 ⊢ (𝑃‘𝐸) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  {crab 3436   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  ifcif 4525  𝒫 cpw 4600   class class class wbr 5143   ↦ cmpt 5225   “ cima 5688  ‘cfv 6561  (class class class)co 7431  Fincfn 8985  infcinf 9481  ℂcc 11153  ℝcr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295   ≤ cle 11296   − cmin 11492   / cdiv 11920  ℕcn 12266  2c2 12321  ℕ₀cn0 12526  ℤcz 12613  ...cfz 13547  Ccbc 14341  ♯chash 14369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-seq 14043  df-fac 14313  df-bc 14342  df-hash 14370

This theorem is referenced by: (None)