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Theorem ballotth 33525

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 4077	. . . . . 6 ⊢ {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} ⊆ 𝑂
3	1, 2	eqsstri 4016	. . . . 5 ⊢ 𝐸 ⊆ 𝑂
4		fzfi 13934	. . . . . . . . . . 11 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
5		pwfi 9175	. . . . . . . . . . 11 ⊢ ((1...(𝑀 + 𝑁)) ∈ Fin ↔ 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin)
6	4, 5	mpbi 229	. . . . . . . . . 10 ⊢ 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin
7		ballotth.o	. . . . . . . . . . 11 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
8		ssrab2 4077	. . . . . . . . . . 11 ⊢ {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} ⊆ 𝒫 (1...(𝑀 + 𝑁))
9	7, 8	eqsstri 4016	. . . . . . . . . 10 ⊢ 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))
10		ssfi 9170	. . . . . . . . . 10 ⊢ ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))) → 𝑂 ∈ Fin)
11	6, 9, 10	mp2an 691	. . . . . . . . 9 ⊢ 𝑂 ∈ Fin
12		ssfi 9170	. . . . . . . . 9 ⊢ ((𝑂 ∈ Fin ∧ 𝐸 ⊆ 𝑂) → 𝐸 ∈ Fin)
13	11, 3, 12	mp2an 691	. . . . . . . 8 ⊢ 𝐸 ∈ Fin
14	13	elexi 3494	. . . . . . 7 ⊢ 𝐸 ∈ V
15	14	elpw 4606	. . . . . 6 ⊢ (𝐸 ∈ 𝒫 𝑂 ↔ 𝐸 ⊆ 𝑂)
16		fveq2 6889	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸))
17	16	oveq1d 7421	. . . . . . 7 ⊢ (𝑥 = 𝐸 → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂)))
18		ballotth.p	. . . . . . 7 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
19		ovex 7439	. . . . . . 7 ⊢ ((♯‘𝐸) / (♯‘𝑂)) ∈ V
20	17, 18, 19	fvmpt 6996	. . . . . 6 ⊢ (𝐸 ∈ 𝒫 𝑂 → (𝑃‘𝐸) = ((♯‘𝐸) / (♯‘𝑂)))
21	15, 20	sylbir 234	. . . . 5 ⊢ (𝐸 ⊆ 𝑂 → (𝑃‘𝐸) = ((♯‘𝐸) / (♯‘𝑂)))
22	3, 21	ax-mp 5	. . . 4 ⊢ (𝑃‘𝐸) = ((♯‘𝐸) / (♯‘𝑂))
23		hashssdif 14369	. . . . . . . 8 ⊢ ((𝑂 ∈ Fin ∧ 𝐸 ⊆ 𝑂) → (♯‘(𝑂 ∖ 𝐸)) = ((♯‘𝑂) − (♯‘𝐸)))
24	11, 3, 23	mp2an 691	. . . . . . 7 ⊢ (♯‘(𝑂 ∖ 𝐸)) = ((♯‘𝑂) − (♯‘𝐸))
25	24	eqcomi 2742	. . . . . 6 ⊢ ((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂 ∖ 𝐸))
26		hashcl 14313	. . . . . . . . 9 ⊢ (𝑂 ∈ Fin → (♯‘𝑂) ∈ ℕ₀)
27	11, 26	ax-mp 5	. . . . . . . 8 ⊢ (♯‘𝑂) ∈ ℕ₀
28	27	nn0cni 12481	. . . . . . 7 ⊢ (♯‘𝑂) ∈ ℂ
29		hashcl 14313	. . . . . . . . 9 ⊢ (𝐸 ∈ Fin → (♯‘𝐸) ∈ ℕ₀)
30	13, 29	ax-mp 5	. . . . . . . 8 ⊢ (♯‘𝐸) ∈ ℕ₀
31	30	nn0cni 12481	. . . . . . 7 ⊢ (♯‘𝐸) ∈ ℂ
32		difss 4131	. . . . . . . . . 10 ⊢ (𝑂 ∖ 𝐸) ⊆ 𝑂
33		ssfi 9170	. . . . . . . . . 10 ⊢ ((𝑂 ∈ Fin ∧ (𝑂 ∖ 𝐸) ⊆ 𝑂) → (𝑂 ∖ 𝐸) ∈ Fin)
34	11, 32, 33	mp2an 691	. . . . . . . . 9 ⊢ (𝑂 ∖ 𝐸) ∈ Fin
35		hashcl 14313	. . . . . . . . 9 ⊢ ((𝑂 ∖ 𝐸) ∈ Fin → (♯‘(𝑂 ∖ 𝐸)) ∈ ℕ₀)
36	34, 35	ax-mp 5	. . . . . . . 8 ⊢ (♯‘(𝑂 ∖ 𝐸)) ∈ ℕ₀
37	36	nn0cni 12481	. . . . . . 7 ⊢ (♯‘(𝑂 ∖ 𝐸)) ∈ ℂ
38	28, 31, 37	subsub23i 11547	. . . . . 6 ⊢ (((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂 ∖ 𝐸)) ↔ ((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) = (♯‘𝐸))
39	25, 38	mpbi 229	. . . . 5 ⊢ ((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) = (♯‘𝐸)
40	39	oveq1i 7416	. . . 4 ⊢ (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂))
41	22, 40	eqtr4i 2764	. . 3 ⊢ (𝑃‘𝐸) = (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂))
42		ballotth.m	. . . . . . 7 ⊢ 𝑀 ∈ ℕ
43		ballotth.n	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
44	42, 43, 7	ballotlem1 33474	. . . . . 6 ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
45	42	nnnn0i 12477	. . . . . . . . 9 ⊢ 𝑀 ∈ ℕ₀
46		nnaddcl 12232	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
47	42, 43, 46	mp2an 691	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℕ
48	47	nnnn0i 12477	. . . . . . . . 9 ⊢ (𝑀 + 𝑁) ∈ ℕ₀
49	42	nnrei 12218	. . . . . . . . . 10 ⊢ 𝑀 ∈ ℝ
50	43	nnnn0i 12477	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℕ₀
51	49, 50	nn0addge1i 12517	. . . . . . . . 9 ⊢ 𝑀 ≤ (𝑀 + 𝑁)
52		elfz2nn0 13589	. . . . . . . . 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 14280	. . . . . . . 8 ⊢ (𝑀 ∈ (0...(𝑀 + 𝑁)) → ((𝑀 + 𝑁)C𝑀) ∈ ℕ)
55	53, 54	ax-mp 5	. . . . . . 7 ⊢ ((𝑀 + 𝑁)C𝑀) ∈ ℕ
56	55	nnne0i 12249	. . . . . 6 ⊢ ((𝑀 + 𝑁)C𝑀) ≠ 0
57	44, 56	eqnetri 3012	. . . . 5 ⊢ (♯‘𝑂) ≠ 0
58	28, 57	pm3.2i 472	. . . 4 ⊢ ((♯‘𝑂) ∈ ℂ ∧ (♯‘𝑂) ≠ 0)
59		divsubdir 11905	. . . 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 1462	. . 3 ⊢ (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂)) = (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
61	28, 57	dividi 11944	. . . 4 ⊢ ((♯‘𝑂) / (♯‘𝑂)) = 1
62	61	oveq1i 7416	. . 3 ⊢ (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂))) = (1 − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
63	41, 60, 62	3eqtri 2765	. 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 33524	. . . . . 6 ⊢ (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
70	69	oveq1i 7416	. . . . 5 ⊢ ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
71	70	oveq1i 7416	. . . 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 4386	. . . . . . 7 ⊢ (𝑂 ∖ 𝐸) = ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
73	72	fveq2i 6892	. . . . . 6 ⊢ (♯‘(𝑂 ∖ 𝐸)) = (♯‘({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
74		ssrab2 4077	. . . . . . . . . 10 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
75	74, 32	sstri 3991	. . . . . . . . 9 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝑂
76	75, 9	sstri 3991	. . . . . . . 8 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
77		ssfi 9170	. . . . . . . 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 691	. . . . . . 7 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∈ Fin
79		ssrab2 4077	. . . . . . . . . 10 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
80	79, 32	sstri 3991	. . . . . . . . 9 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
81	80, 9	sstri 3991	. . . . . . . 8 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
82		ssfi 9170	. . . . . . . 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 691	. . . . . . 7 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin
84		rabnc 4387	. . . . . . 7 ⊢ ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∩ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) = ∅
85		hashun 14339	. . . . . . 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 1462	. . . . . 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 2761	. . . . 5 ⊢ (♯‘(𝑂 ∖ 𝐸)) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
88	87	oveq1i 7416	. . . 4 ⊢ ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
89		ssrab2 4077	. . . . . . . . 9 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
90	11	elexi 3494	. . . . . . . . . 10 ⊢ 𝑂 ∈ V
91	90	elpw2 5345	. . . . . . . . 9 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 ↔ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂)
92	89, 91	mpbir 230	. . . . . . . 8 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
93		fveq2 6889	. . . . . . . . . 10 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}))
94	93	oveq1d 7421	. . . . . . . . 9 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
95		ovex 7439	. . . . . . . . 9 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
96	94, 18, 95	fvmpt 6996	. . . . . . . 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 33476	. . . . . . 7 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))
99		nfrab1 3452	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
100		nfrab1 3452	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
101	99, 100	dfss2f 3972	. . . . . . . . . . 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 33486	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝑂 → (¬ 1 ∈ 𝑐 → ¬ 𝑐 ∈ 𝐸))
103	102	imdistani 570	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐) → (𝑐 ∈ 𝑂 ∧ ¬ 𝑐 ∈ 𝐸))
104		rabid 3453	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐))
105		eldif 3958	. . . . . . . . . . . . 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 498	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ¬ 1 ∈ 𝑐)
108		rabid 3453	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐))
109	106, 107, 108	sylanbrc 584	. . . . . . . . . . 11 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
110	101, 109	mpgbir 1802	. . . . . . . . . 10 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
111		rabss2 4075	. . . . . . . . . . 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 3998	. . . . . . . . 9 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} = {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
114	113	fveq2i 6892	. . . . . . . 8 ⊢ (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
115	114	oveq1i 7416	. . . . . . 7 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
116	97, 98, 115	3eqtr3i 2769	. . . . . 6 ⊢ (𝑁 / (𝑀 + 𝑁)) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
117	116	oveq2i 7417	. . . . 5 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = (2 · ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
118		2cn 12284	. . . . . 6 ⊢ 2 ∈ ℂ
119		hashcl 14313	. . . . . . . 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 12481	. . . . . 6 ⊢ (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℂ
122	118, 121, 28, 57	divassi 11967	. . . . 5 ⊢ ((2 · (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (2 · ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
123	121	2timesi 12347	. . . . . 6 ⊢ (2 · (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
124	123	oveq1i 7416	. . . . 5 ⊢ ((2 · (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
125	117, 122, 124	3eqtr2i 2767	. . . 4 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
126	71, 88, 125	3eqtr4ri 2772	. . 3 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂))
127	126	oveq2i 7417	. 2 ⊢ (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = (1 − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
128	47	nncni 12219	. . . 4 ⊢ (𝑀 + 𝑁) ∈ ℂ
129	43	nncni 12219	. . . . 5 ⊢ 𝑁 ∈ ℂ
130	118, 129	mulcli 11218	. . . 4 ⊢ (2 · 𝑁) ∈ ℂ
131	47	nnne0i 12249	. . . . 5 ⊢ (𝑀 + 𝑁) ≠ 0
132	128, 131	pm3.2i 472	. . . 4 ⊢ ((𝑀 + 𝑁) ∈ ℂ ∧ (𝑀 + 𝑁) ≠ 0)
133		divsubdir 11905	. . . 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 1462	. . 3 ⊢ (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁)))
135	129	2timesi 12347	. . . . . 6 ⊢ (2 · 𝑁) = (𝑁 + 𝑁)
136	135	oveq2i 7417	. . . . 5 ⊢ ((𝑀 + 𝑁) − (2 · 𝑁)) = ((𝑀 + 𝑁) − (𝑁 + 𝑁))
137	42	nncni 12219	. . . . . . 7 ⊢ 𝑀 ∈ ℂ
138	137, 129, 129, 129	addsub4i 11553	. . . . . 6 ⊢ ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = ((𝑀 − 𝑁) + (𝑁 − 𝑁))
139	129	subidi 11528	. . . . . . 7 ⊢ (𝑁 − 𝑁) = 0
140	139	oveq2i 7417	. . . . . 6 ⊢ ((𝑀 − 𝑁) + (𝑁 − 𝑁)) = ((𝑀 − 𝑁) + 0)
141	137, 129	subcli 11533	. . . . . . 7 ⊢ (𝑀 − 𝑁) ∈ ℂ
142	141	addridi 11398	. . . . . 6 ⊢ ((𝑀 − 𝑁) + 0) = (𝑀 − 𝑁)
143	138, 140, 142	3eqtri 2765	. . . . 5 ⊢ ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = (𝑀 − 𝑁)
144	136, 143	eqtri 2761	. . . 4 ⊢ ((𝑀 + 𝑁) − (2 · 𝑁)) = (𝑀 − 𝑁)
145	144	oveq1i 7416	. . 3 ⊢ (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))
146	128, 131	dividi 11944	. . . 4 ⊢ ((𝑀 + 𝑁) / (𝑀 + 𝑁)) = 1
147	118, 129, 128, 131	divassi 11967	. . . 4 ⊢ ((2 · 𝑁) / (𝑀 + 𝑁)) = (2 · (𝑁 / (𝑀 + 𝑁)))
148	146, 147	oveq12i 7418	. . 3 ⊢ (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁))) = (1 − (2 · (𝑁 / (𝑀 + 𝑁))))
149	134, 145, 148	3eqtr3ri 2770	. 2 ⊢ (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))
150	63, 127, 149	3eqtr2i 2767	1 ⊢ (𝑃‘𝐸) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 {crab 3433 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ifcif 4528 𝒫 cpw 4602 class class class wbr 5148 ↦ cmpt 5231 “ cima 5679 ‘cfv 6541 (class class class)co 7406 Fincfn 8936 infcinf 9433 ℂcc 11105 ℝcr 11106 0cc0 11107 1c1 11108 + caddc 11110 · cmul 11112 < clt 11245 ≤ cle 11246 − cmin 11441 / cdiv 11868 ℕcn 12209 2c2 12264 ℕ₀cn0 12469 ℤcz 12555 ...cfz 13481 Ccbc 14259 ♯chash 14287

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-oadd 8467 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-dju 9893 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fz 13482 df-seq 13964 df-fac 14231 df-bc 14260 df-hash 14288

This theorem is referenced by: (None)

W3C validator