Theorem ballotth 35170

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 4028	. . . . . 6 ⊢ {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} ⊆ 𝑂
3	1, 2	eqsstri 3977	. . . . 5 ⊢ 𝐸 ⊆ 𝑂
4		fzfi 14115	. . . . . . . . . . 11 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
5		pwfi 9310	. . . . . . . . . . 11 ⊢ ((1...(𝑀 + 𝑁)) ∈ Fin ↔ 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin)
6	4, 5	mpbi 233	. . . . . . . . . 10 ⊢ 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin
7		ballotth.o	. . . . . . . . . . 11 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
8		ssrab2 4028	. . . . . . . . . . 11 ⊢ {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} ⊆ 𝒫 (1...(𝑀 + 𝑁))
9	7, 8	eqsstri 3977	. . . . . . . . . 10 ⊢ 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))
10		ssfi 9188	. . . . . . . . . 10 ⊢ ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))) → 𝑂 ∈ Fin)
11	6, 9, 10	mp2an 705	. . . . . . . . 9 ⊢ 𝑂 ∈ Fin
12		ssfi 9188	. . . . . . . . 9 ⊢ ((𝑂 ∈ Fin ∧ 𝐸 ⊆ 𝑂) → 𝐸 ∈ Fin)
13	11, 3, 12	mp2an 705	. . . . . . . 8 ⊢ 𝐸 ∈ Fin
14	13	elexi 3473	. . . . . . 7 ⊢ 𝐸 ∈ V
15	14	elpw 4561	. . . . . 6 ⊢ (𝐸 ∈ 𝒫 𝑂 ↔ 𝐸 ⊆ 𝑂)
16		fveq2 6885	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸))
17	16	oveq1d 7435	. . . . . . 7 ⊢ (𝑥 = 𝐸 → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂)))
18		ballotth.p	. . . . . . 7 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
19		ovex 7453	. . . . . . 7 ⊢ ((♯‘𝐸) / (♯‘𝑂)) ∈ V
20	17, 18, 19	fvmpt 6993	. . . . . 6 ⊢ (𝐸 ∈ 𝒫 𝑂 → (𝑃‘𝐸) = ((♯‘𝐸) / (♯‘𝑂)))
21	15, 20	sylbir 238	. . . . 5 ⊢ (𝐸 ⊆ 𝑂 → (𝑃‘𝐸) = ((♯‘𝐸) / (♯‘𝑂)))
22	3, 21	ax-mp 5	. . . 4 ⊢ (𝑃‘𝐸) = ((♯‘𝐸) / (♯‘𝑂))
23		hashssdif 14557	. . . . . . . 8 ⊢ ((𝑂 ∈ Fin ∧ 𝐸 ⊆ 𝑂) → (♯‘(𝑂 ∖ 𝐸)) = ((♯‘𝑂) − (♯‘𝐸)))
24	11, 3, 23	mp2an 705	. . . . . . 7 ⊢ (♯‘(𝑂 ∖ 𝐸)) = ((♯‘𝑂) − (♯‘𝐸))
25	24	eqcomi 2770	. . . . . 6 ⊢ ((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂 ∖ 𝐸))
26		hashcl 14500	. . . . . . . . 9 ⊢ (𝑂 ∈ Fin → (♯‘𝑂) ∈ ℕ0)
27	11, 26	ax-mp 5	. . . . . . . 8 ⊢ (♯‘𝑂) ∈ ℕ0
28	27	nn0cni 12618	. . . . . . 7 ⊢ (♯‘𝑂) ∈ ℂ
29		hashcl 14500	. . . . . . . . 9 ⊢ (𝐸 ∈ Fin → (♯‘𝐸) ∈ ℕ0)
30	13, 29	ax-mp 5	. . . . . . . 8 ⊢ (♯‘𝐸) ∈ ℕ0
31	30	nn0cni 12618	. . . . . . 7 ⊢ (♯‘𝐸) ∈ ℂ
32		difss 4083	. . . . . . . . . 10 ⊢ (𝑂 ∖ 𝐸) ⊆ 𝑂
33		ssfi 9188	. . . . . . . . . 10 ⊢ ((𝑂 ∈ Fin ∧ (𝑂 ∖ 𝐸) ⊆ 𝑂) → (𝑂 ∖ 𝐸) ∈ Fin)
34	11, 32, 33	mp2an 705	. . . . . . . . 9 ⊢ (𝑂 ∖ 𝐸) ∈ Fin
35		hashcl 14500	. . . . . . . . 9 ⊢ ((𝑂 ∖ 𝐸) ∈ Fin → (♯‘(𝑂 ∖ 𝐸)) ∈ ℕ0)
36	34, 35	ax-mp 5	. . . . . . . 8 ⊢ (♯‘(𝑂 ∖ 𝐸)) ∈ ℕ0
37	36	nn0cni 12618	. . . . . . 7 ⊢ (♯‘(𝑂 ∖ 𝐸)) ∈ ℂ
38	28, 31, 37	subsub23i 11648	. . . . . 6 ⊢ (((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂 ∖ 𝐸)) ↔ ((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) = (♯‘𝐸))
39	25, 38	mpbi 233	. . . . 5 ⊢ ((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) = (♯‘𝐸)
40	39	oveq1i 7430	. . . 4 ⊢ (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂))
41	22, 40	eqtr4i 2787	. . 3 ⊢ (𝑃‘𝐸) = (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂))
42		ballotth.m	. . . . . . 7 ⊢ 𝑀 ∈ ℕ
43		ballotth.n	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
44	42, 43, 7	ballotlem1 35119	. . . . . 6 ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
45	42	nnnn0i 12614	. . . . . . . . 9 ⊢ 𝑀 ∈ ℕ0
46		nnaddcl 12358	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
47	42, 43, 46	mp2an 705	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℕ
48	47	nnnn0i 12614	. . . . . . . . 9 ⊢ (𝑀 + 𝑁) ∈ ℕ0
49	42	nnrei 12344	. . . . . . . . . 10 ⊢ 𝑀 ∈ ℝ
50	43	nnnn0i 12614	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℕ0
51	49, 50	nn0addge1i 12654	. . . . . . . . 9 ⊢ 𝑀 ≤ (𝑀 + 𝑁)
52		elfz2nn0 13752	. . . . . . . . 9 ⊢ (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ (𝑀 ∈ ℕ0 ∧ (𝑀 + 𝑁) ∈ ℕ0 ∧ 𝑀 ≤ (𝑀 + 𝑁)))
53	45, 48, 51, 52	mpbir3an 1360	. . . . . . . 8 ⊢ 𝑀 ∈ (0...(𝑀 + 𝑁))
54		bccl2 14467	. . . . . . . 8 ⊢ (𝑀 ∈ (0...(𝑀 + 𝑁)) → ((𝑀 + 𝑁)C𝑀) ∈ ℕ)
55	53, 54	ax-mp 5	. . . . . . 7 ⊢ ((𝑀 + 𝑁)C𝑀) ∈ ℕ
56	55	nnne0i 12378	. . . . . 6 ⊢ ((𝑀 + 𝑁)C𝑀) ≠ 0
57	44, 56	eqnetri 3026	. . . . 5 ⊢ (♯‘𝑂) ≠ 0
58	28, 57	pm3.2i 476	. . . 4 ⊢ ((♯‘𝑂) ∈ ℂ ∧ (♯‘𝑂) ≠ 0)
59		divsubdir 12010	. . . 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 1490	. . 3 ⊢ (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂)) = (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
61	28, 57	dividi 12050	. . . 4 ⊢ ((♯‘𝑂) / (♯‘𝑂)) = 1
62	61	oveq1i 7430	. . 3 ⊢ (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂))) = (1 − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
63	41, 60, 62	3eqtri 2788	. 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 35169	. . . . . 6 ⊢ (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
70	69	oveq1i 7430	. . . . 5 ⊢ ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
71	70	oveq1i 7430	. . . 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 4340	. . . . . . 7 ⊢ (𝑂 ∖ 𝐸) = ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
73	72	fveq2i 6888	. . . . . 6 ⊢ (♯‘(𝑂 ∖ 𝐸)) = (♯‘({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
74		ssrab2 4028	. . . . . . . . . 10 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
75	74, 32	sstri 3940	. . . . . . . . 9 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝑂
76	75, 9	sstri 3940	. . . . . . . 8 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
77		ssfi 9188	. . . . . . . 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 705	. . . . . . 7 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∈ Fin
79		ssrab2 4028	. . . . . . . . . 10 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
80	79, 32	sstri 3940	. . . . . . . . 9 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
81	80, 9	sstri 3940	. . . . . . . 8 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
82		ssfi 9188	. . . . . . . 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 705	. . . . . . 7 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin
84		rabnc 4341	. . . . . . 7 ⊢ ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∩ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) = ∅
85		hashun 14526	. . . . . . 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 1490	. . . . . 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 2784	. . . . 5 ⊢ (♯‘(𝑂 ∖ 𝐸)) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
88	87	oveq1i 7430	. . . 4 ⊢ ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
89		ssrab2 4028	. . . . . . . . 9 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
90	11	elexi 3473	. . . . . . . . . 10 ⊢ 𝑂 ∈ V
91	90	elpw2 5296	. . . . . . . . 9 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 ↔ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂)
92	89, 91	mpbir 234	. . . . . . . 8 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
93		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}))
94	93	oveq1d 7435	. . . . . . . . 9 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
95		ovex 7453	. . . . . . . . 9 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
96	94, 18, 95	fvmpt 6993	. . . . . . . 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 35121	. . . . . . 7 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))
99		nfrab1 3432	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
100		nfrab1 3432	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
101	99, 100	dfssf 3922	. . . . . . . . . . 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 35131	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝑂 → (¬ 1 ∈ 𝑐 → ¬ 𝑐 ∈ 𝐸))
103	102	imdistani 579	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐) → (𝑐 ∈ 𝑂 ∧ ¬ 𝑐 ∈ 𝐸))
104		rabid 3433	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐))
105		eldif 3909	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↔ (𝑐 ∈ 𝑂 ∧ ¬ 𝑐 ∈ 𝐸))
106	103, 104, 105	3imtr4i 295	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ (𝑂 ∖ 𝐸))
107	104	simprbi 503	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ¬ 1 ∈ 𝑐)
108		rabid 3433	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐))
109	106, 107, 108	sylanbrc 595	. . . . . . . . . . 11 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
110	101, 109	mpgbir 1832	. . . . . . . . . 10 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
111		rabss2 4025	. . . . . . . . . . 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 3947	. . . . . . . . 9 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} = {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
114	113	fveq2i 6888	. . . . . . . 8 ⊢ (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
115	114	oveq1i 7430	. . . . . . 7 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
116	97, 98, 115	3eqtr3i 2792	. . . . . 6 ⊢ (𝑁 / (𝑀 + 𝑁)) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
117	116	oveq2i 7431	. . . . 5 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = (2 · ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
118		2cn 12418	. . . . . 6 ⊢ 2 ∈ ℂ
119		hashcl 14500	. . . . . . . 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 12618	. . . . . 6 ⊢ (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℂ
122	118, 121, 28, 57	divassi 12073	. . . . 5 ⊢ ((2 · (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (2 · ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
123	121	2timesi 12480	. . . . . 6 ⊢ (2 · (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
124	123	oveq1i 7430	. . . . 5 ⊢ ((2 · (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
125	117, 122, 124	3eqtr2i 2790	. . . 4 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
126	71, 88, 125	3eqtr4ri 2795	. . 3 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂))
127	126	oveq2i 7431	. 2 ⊢ (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = (1 − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
128	47	nncni 12345	. . . 4 ⊢ (𝑀 + 𝑁) ∈ ℂ
129	43	nncni 12345	. . . . 5 ⊢ 𝑁 ∈ ℂ
130	118, 129	mulcli 11316	. . . 4 ⊢ (2 · 𝑁) ∈ ℂ
131	47	nnne0i 12378	. . . . 5 ⊢ (𝑀 + 𝑁) ≠ 0
132	128, 131	pm3.2i 476	. . . 4 ⊢ ((𝑀 + 𝑁) ∈ ℂ ∧ (𝑀 + 𝑁) ≠ 0)
133		divsubdir 12010	. . . 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 1490	. . 3 ⊢ (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁)))
135	129	2timesi 12480	. . . . . 6 ⊢ (2 · 𝑁) = (𝑁 + 𝑁)
136	135	oveq2i 7431	. . . . 5 ⊢ ((𝑀 + 𝑁) − (2 · 𝑁)) = ((𝑀 + 𝑁) − (𝑁 + 𝑁))
137	42	nncni 12345	. . . . . . 7 ⊢ 𝑀 ∈ ℂ
138	137, 129, 129, 129	addsub4i 11654	. . . . . 6 ⊢ ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = ((𝑀 − 𝑁) + (𝑁 − 𝑁))
139	129	subidi 11629	. . . . . . 7 ⊢ (𝑁 − 𝑁) = 0
140	139	oveq2i 7431	. . . . . 6 ⊢ ((𝑀 − 𝑁) + (𝑁 − 𝑁)) = ((𝑀 − 𝑁) + 0)
141	137, 129	subcli 11634	. . . . . . 7 ⊢ (𝑀 − 𝑁) ∈ ℂ
142	141	addridi 11497	. . . . . 6 ⊢ ((𝑀 − 𝑁) + 0) = (𝑀 − 𝑁)
143	138, 140, 142	3eqtri 2788	. . . . 5 ⊢ ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = (𝑀 − 𝑁)
144	136, 143	eqtri 2784	. . . 4 ⊢ ((𝑀 + 𝑁) − (2 · 𝑁)) = (𝑀 − 𝑁)
145	144	oveq1i 7430	. . 3 ⊢ (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))
146	128, 131	dividi 12050	. . . 4 ⊢ ((𝑀 + 𝑁) / (𝑀 + 𝑁)) = 1
147	118, 129, 128, 131	divassi 12073	. . . 4 ⊢ ((2 · 𝑁) / (𝑀 + 𝑁)) = (2 · (𝑁 / (𝑀 + 𝑁)))
148	146, 147	oveq12i 7432	. . 3 ⊢ (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁))) = (1 − (2 · (𝑁 / (𝑀 + 𝑁))))
149	134, 145, 148	3eqtr3ri 2793	. 2 ⊢ (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))
150	63, 127, 149	3eqtr2i 2790	1 ⊢ (𝑃‘𝐸) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  {crab 3413   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ifcif 4482  𝒫 cpw 4557   class class class wbr 5103   ↦ cmpt 5186   “ cima 5654  ‘cfv 6538  (class class class)co 7420  Fincfn 8973  infcinf 9433  ℂcc 11198  ℝcr 11199  0cc0 11200  1c1 11201   + caddc 11203   · cmul 11205   < clt 11343   ≤ cle 11344   − cmin 11541   / cdiv 11973  ℕcn 12335  2c2 12397  ℕ₀cn0 12606  ℤcz 12693  ...cfz 13639  Ccbc 14446  ♯chash 14474

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-oadd 8480  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-inf 9435  df-dju 9982  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-n0 12607  df-z 12694  df-uz 12966  df-rp 13121  df-fz 13640  df-seq 14145  df-fac 14418  df-bc 14447  df-hash 14475

This theorem is used by: (None)