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Theorem ballotth 29926

Description: Bertrand's ballot problem : the probability that A is ahead throughout the counting. 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 3650	. . . . . 6 ⊢ {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} ⊆ 𝑂
3	1, 2	eqsstri 3598	. . . . 5 ⊢ 𝐸 ⊆ 𝑂
4		fzfi 12633	. . . . . . . . . . 11 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
5		pwfi 8144	. . . . . . . . . . 11 ⊢ ((1...(𝑀 + 𝑁)) ∈ Fin ↔ 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin)
6	4, 5	mpbi 219	. . . . . . . . . 10 ⊢ 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin
7		ballotth.o	. . . . . . . . . . 11 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}
8		ssrab2 3650	. . . . . . . . . . 11 ⊢ {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀} ⊆ 𝒫 (1...(𝑀 + 𝑁))
9	7, 8	eqsstri 3598	. . . . . . . . . 10 ⊢ 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))
10		ssfi 8065	. . . . . . . . . 10 ⊢ ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))) → 𝑂 ∈ Fin)
11	6, 9, 10	mp2an 704	. . . . . . . . 9 ⊢ 𝑂 ∈ Fin
12		ssfi 8065	. . . . . . . . 9 ⊢ ((𝑂 ∈ Fin ∧ 𝐸 ⊆ 𝑂) → 𝐸 ∈ Fin)
13	11, 3, 12	mp2an 704	. . . . . . . 8 ⊢ 𝐸 ∈ Fin
14	13	elexi 3186	. . . . . . 7 ⊢ 𝐸 ∈ V
15	14	elpw 4114	. . . . . 6 ⊢ (𝐸 ∈ 𝒫 𝑂 ↔ 𝐸 ⊆ 𝑂)
16		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (#‘𝑥) = (#‘𝐸))
17	16	oveq1d 6564	. . . . . . 7 ⊢ (𝑥 = 𝐸 → ((#‘𝑥) / (#‘𝑂)) = ((#‘𝐸) / (#‘𝑂)))
18		ballotth.p	. . . . . . 7 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))
19		ovex 6577	. . . . . . 7 ⊢ ((#‘𝐸) / (#‘𝑂)) ∈ V
20	17, 18, 19	fvmpt 6191	. . . . . 6 ⊢ (𝐸 ∈ 𝒫 𝑂 → (𝑃‘𝐸) = ((#‘𝐸) / (#‘𝑂)))
21	15, 20	sylbir 224	. . . . 5 ⊢ (𝐸 ⊆ 𝑂 → (𝑃‘𝐸) = ((#‘𝐸) / (#‘𝑂)))
22	3, 21	ax-mp 5	. . . 4 ⊢ (𝑃‘𝐸) = ((#‘𝐸) / (#‘𝑂))
23		hashssdif 13061	. . . . . . . 8 ⊢ ((𝑂 ∈ Fin ∧ 𝐸 ⊆ 𝑂) → (#‘(𝑂 ∖ 𝐸)) = ((#‘𝑂) − (#‘𝐸)))
24	11, 3, 23	mp2an 704	. . . . . . 7 ⊢ (#‘(𝑂 ∖ 𝐸)) = ((#‘𝑂) − (#‘𝐸))
25	24	eqcomi 2619	. . . . . 6 ⊢ ((#‘𝑂) − (#‘𝐸)) = (#‘(𝑂 ∖ 𝐸))
26		hashcl 13009	. . . . . . . . 9 ⊢ (𝑂 ∈ Fin → (#‘𝑂) ∈ ℕ₀)
27	11, 26	ax-mp 5	. . . . . . . 8 ⊢ (#‘𝑂) ∈ ℕ₀
28	27	nn0cni 11181	. . . . . . 7 ⊢ (#‘𝑂) ∈ ℂ
29		hashcl 13009	. . . . . . . . 9 ⊢ (𝐸 ∈ Fin → (#‘𝐸) ∈ ℕ₀)
30	13, 29	ax-mp 5	. . . . . . . 8 ⊢ (#‘𝐸) ∈ ℕ₀
31	30	nn0cni 11181	. . . . . . 7 ⊢ (#‘𝐸) ∈ ℂ
32		difss 3699	. . . . . . . . . 10 ⊢ (𝑂 ∖ 𝐸) ⊆ 𝑂
33		ssfi 8065	. . . . . . . . . 10 ⊢ ((𝑂 ∈ Fin ∧ (𝑂 ∖ 𝐸) ⊆ 𝑂) → (𝑂 ∖ 𝐸) ∈ Fin)
34	11, 32, 33	mp2an 704	. . . . . . . . 9 ⊢ (𝑂 ∖ 𝐸) ∈ Fin
35		hashcl 13009	. . . . . . . . 9 ⊢ ((𝑂 ∖ 𝐸) ∈ Fin → (#‘(𝑂 ∖ 𝐸)) ∈ ℕ₀)
36	34, 35	ax-mp 5	. . . . . . . 8 ⊢ (#‘(𝑂 ∖ 𝐸)) ∈ ℕ₀
37	36	nn0cni 11181	. . . . . . 7 ⊢ (#‘(𝑂 ∖ 𝐸)) ∈ ℂ
38	28, 31, 37	subsub23i 10250	. . . . . 6 ⊢ (((#‘𝑂) − (#‘𝐸)) = (#‘(𝑂 ∖ 𝐸)) ↔ ((#‘𝑂) − (#‘(𝑂 ∖ 𝐸))) = (#‘𝐸))
39	25, 38	mpbi 219	. . . . 5 ⊢ ((#‘𝑂) − (#‘(𝑂 ∖ 𝐸))) = (#‘𝐸)
40	39	oveq1i 6559	. . . 4 ⊢ (((#‘𝑂) − (#‘(𝑂 ∖ 𝐸))) / (#‘𝑂)) = ((#‘𝐸) / (#‘𝑂))
41	22, 40	eqtr4i 2635	. . 3 ⊢ (𝑃‘𝐸) = (((#‘𝑂) − (#‘(𝑂 ∖ 𝐸))) / (#‘𝑂))
42		ballotth.m	. . . . . . 7 ⊢ 𝑀 ∈ ℕ
43		ballotth.n	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
44	42, 43, 7	ballotlem1 29875	. . . . . 6 ⊢ (#‘𝑂) = ((𝑀 + 𝑁)C𝑀)
45	42	nnnn0i 11177	. . . . . . . . 9 ⊢ 𝑀 ∈ ℕ₀
46		nnaddcl 10919	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
47	42, 43, 46	mp2an 704	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℕ
48	47	nnnn0i 11177	. . . . . . . . 9 ⊢ (𝑀 + 𝑁) ∈ ℕ₀
49	42	nnrei 10906	. . . . . . . . . 10 ⊢ 𝑀 ∈ ℝ
50	43	nnnn0i 11177	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℕ₀
51	49, 50	nn0addge1i 11218	. . . . . . . . 9 ⊢ 𝑀 ≤ (𝑀 + 𝑁)
52		elfz2nn0 12300	. . . . . . . . 9 ⊢ (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ (𝑀 ∈ ℕ₀ ∧ (𝑀 + 𝑁) ∈ ℕ₀ ∧ 𝑀 ≤ (𝑀 + 𝑁)))
53	45, 48, 51, 52	mpbir3an 1237	. . . . . . . 8 ⊢ 𝑀 ∈ (0...(𝑀 + 𝑁))
54		bccl2 12972	. . . . . . . 8 ⊢ (𝑀 ∈ (0...(𝑀 + 𝑁)) → ((𝑀 + 𝑁)C𝑀) ∈ ℕ)
55	53, 54	ax-mp 5	. . . . . . 7 ⊢ ((𝑀 + 𝑁)C𝑀) ∈ ℕ
56	55	nnne0i 10932	. . . . . 6 ⊢ ((𝑀 + 𝑁)C𝑀) ≠ 0
57	44, 56	eqnetri 2852	. . . . 5 ⊢ (#‘𝑂) ≠ 0
58	28, 57	pm3.2i 470	. . . 4 ⊢ ((#‘𝑂) ∈ ℂ ∧ (#‘𝑂) ≠ 0)
59		divsubdir 10600	. . . 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 1416	. . 3 ⊢ (((#‘𝑂) − (#‘(𝑂 ∖ 𝐸))) / (#‘𝑂)) = (((#‘𝑂) / (#‘𝑂)) − ((#‘(𝑂 ∖ 𝐸)) / (#‘𝑂)))
61	28, 57	dividi 10637	. . . 4 ⊢ ((#‘𝑂) / (#‘𝑂)) = 1
62	61	oveq1i 6559	. . 3 ⊢ (((#‘𝑂) / (#‘𝑂)) − ((#‘(𝑂 ∖ 𝐸)) / (#‘𝑂))) = (1 − ((#‘(𝑂 ∖ 𝐸)) / (#‘𝑂)))
63	41, 60, 62	3eqtri 2636	. 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 29925	. . . . . 6 ⊢ (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) = (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
70	69	oveq1i 6559	. . . . 5 ⊢ ((#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
71	70	oveq1i 6559	. . . 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 3915	. . . . . . 7 ⊢ (𝑂 ∖ 𝐸) = ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
73	72	fveq2i 6106	. . . . . 6 ⊢ (#‘(𝑂 ∖ 𝐸)) = (#‘({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
74		ssrab2 3650	. . . . . . . . . 10 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
75	74, 32	sstri 3577	. . . . . . . . 9 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝑂
76	75, 9	sstri 3577	. . . . . . . 8 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
77		ssfi 8065	. . . . . . . 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 704	. . . . . . 7 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∈ Fin
79		ssrab2 3650	. . . . . . . . . 10 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
80	79, 32	sstri 3577	. . . . . . . . 9 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
81	80, 9	sstri 3577	. . . . . . . 8 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
82		ssfi 8065	. . . . . . . 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 704	. . . . . . 7 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin
84		rabnc 3916	. . . . . . 7 ⊢ ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∩ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) = ∅
85		hashun 13032	. . . . . . 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 1416	. . . . . 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 2632	. . . . 5 ⊢ (#‘(𝑂 ∖ 𝐸)) = ((#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
88	87	oveq1i 6559	. . . 4 ⊢ ((#‘(𝑂 ∖ 𝐸)) / (#‘𝑂)) = (((#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (#‘𝑂))
89		ssrab2 3650	. . . . . . . . 9 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
90	11	elexi 3186	. . . . . . . . . 10 ⊢ 𝑂 ∈ V
91	90	elpw2 4755	. . . . . . . . 9 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 ↔ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂)
92	89, 91	mpbir 220	. . . . . . . 8 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
93		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → (#‘𝑥) = (#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}))
94	93	oveq1d 6564	. . . . . . . . 9 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ((#‘𝑥) / (#‘𝑂)) = ((#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)))
95		ovex 6577	. . . . . . . . 9 ⊢ ((#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)) ∈ V
96	94, 18, 95	fvmpt 6191	. . . . . . . 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 29877	. . . . . . 7 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))
99		nfrab1 3099	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
100		nfrab1 3099	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
101	99, 100	dfss2f 3559	. . . . . . . . . . 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 29887	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝑂 → (¬ 1 ∈ 𝑐 → ¬ 𝑐 ∈ 𝐸))
103	102	imdistani 722	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐) → (𝑐 ∈ 𝑂 ∧ ¬ 𝑐 ∈ 𝐸))
104		rabid 3095	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐))
105		eldif 3550	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↔ (𝑐 ∈ 𝑂 ∧ ¬ 𝑐 ∈ 𝐸))
106	103, 104, 105	3imtr4i 280	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ (𝑂 ∖ 𝐸))
107	104	simprbi 479	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ¬ 1 ∈ 𝑐)
108		rabid 3095	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐))
109	106, 107, 108	sylanbrc 695	. . . . . . . . . . 11 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
110	101, 109	mpgbir 1717	. . . . . . . . . 10 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
111		rabss2 3648	. . . . . . . . . . 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 3584	. . . . . . . . 9 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} = {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
114	113	fveq2i 6106	. . . . . . . 8 ⊢ (#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
115	114	oveq1i 6559	. . . . . . 7 ⊢ ((#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)) = ((#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂))
116	97, 98, 115	3eqtr3i 2640	. . . . . 6 ⊢ (𝑁 / (𝑀 + 𝑁)) = ((#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂))
117	116	oveq2i 6560	. . . . 5 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = (2 · ((#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)))
118		2cn 10968	. . . . . 6 ⊢ 2 ∈ ℂ
119		hashcl 13009	. . . . . . . 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 11181	. . . . . 6 ⊢ (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℂ
122	118, 121, 28, 57	divassi 10660	. . . . 5 ⊢ ((2 · (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (#‘𝑂)) = (2 · ((#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)))
123	121	2timesi 11024	. . . . . 6 ⊢ (2 · (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
124	123	oveq1i 6559	. . . . 5 ⊢ ((2 · (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (#‘𝑂)) = (((#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (#‘𝑂))
125	117, 122, 124	3eqtr2i 2638	. . . 4 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = (((#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (#‘𝑂))
126	71, 88, 125	3eqtr4ri 2643	. . 3 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = ((#‘(𝑂 ∖ 𝐸)) / (#‘𝑂))
127	126	oveq2i 6560	. 2 ⊢ (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = (1 − ((#‘(𝑂 ∖ 𝐸)) / (#‘𝑂)))
128	47	nncni 10907	. . . 4 ⊢ (𝑀 + 𝑁) ∈ ℂ
129	43	nncni 10907	. . . . 5 ⊢ 𝑁 ∈ ℂ
130	118, 129	mulcli 9924	. . . 4 ⊢ (2 · 𝑁) ∈ ℂ
131	47	nnne0i 10932	. . . . 5 ⊢ (𝑀 + 𝑁) ≠ 0
132	128, 131	pm3.2i 470	. . . 4 ⊢ ((𝑀 + 𝑁) ∈ ℂ ∧ (𝑀 + 𝑁) ≠ 0)
133		divsubdir 10600	. . . 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 1416	. . 3 ⊢ (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁)))
135	129	2timesi 11024	. . . . . 6 ⊢ (2 · 𝑁) = (𝑁 + 𝑁)
136	135	oveq2i 6560	. . . . 5 ⊢ ((𝑀 + 𝑁) − (2 · 𝑁)) = ((𝑀 + 𝑁) − (𝑁 + 𝑁))
137	42	nncni 10907	. . . . . . 7 ⊢ 𝑀 ∈ ℂ
138	137, 129, 129, 129	addsub4i 10256	. . . . . 6 ⊢ ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = ((𝑀 − 𝑁) + (𝑁 − 𝑁))
139	129	subidi 10231	. . . . . . 7 ⊢ (𝑁 − 𝑁) = 0
140	139	oveq2i 6560	. . . . . 6 ⊢ ((𝑀 − 𝑁) + (𝑁 − 𝑁)) = ((𝑀 − 𝑁) + 0)
141	137, 129	subcli 10236	. . . . . . 7 ⊢ (𝑀 − 𝑁) ∈ ℂ
142	141	addid1i 10102	. . . . . 6 ⊢ ((𝑀 − 𝑁) + 0) = (𝑀 − 𝑁)
143	138, 140, 142	3eqtri 2636	. . . . 5 ⊢ ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = (𝑀 − 𝑁)
144	136, 143	eqtri 2632	. . . 4 ⊢ ((𝑀 + 𝑁) − (2 · 𝑁)) = (𝑀 − 𝑁)
145	144	oveq1i 6559	. . 3 ⊢ (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))
146	128, 131	dividi 10637	. . . 4 ⊢ ((𝑀 + 𝑁) / (𝑀 + 𝑁)) = 1
147	118, 129, 128, 131	divassi 10660	. . . 4 ⊢ ((2 · 𝑁) / (𝑀 + 𝑁)) = (2 · (𝑁 / (𝑀 + 𝑁)))
148	146, 147	oveq12i 6561	. . 3 ⊢ (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁))) = (1 − (2 · (𝑁 / (𝑀 + 𝑁))))
149	134, 145, 148	3eqtr3ri 2641	. 2 ⊢ (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))
150	63, 127, 149	3eqtr2i 2638	1 ⊢ (𝑃‘𝐸) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ifcif 4036 𝒫 cpw 4108 class class class wbr 4583 ↦ cmpt 4643 “ cima 5041 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 infcinf 8230 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ...cfz 12197 Ccbc 12951 #chash 12979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-seq 12664 df-fac 12923 df-bc 12952 df-hash 12980

This theorem is referenced by: (None)

W3C validator