ballotlem2 - Mathbox for Thierry Arnoux

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Theorem ballotlem2 34944

Description: The probability that the first vote picked in a count is a B. (Contributed by Thierry Arnoux, 23-Nov-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)))

**Assertion**
Ref	Expression
ballotlem2	⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))

Distinct variable groups: 𝑀,𝑐 𝑁,𝑐 𝑂,𝑐,𝑥 Allowed substitution hints: 𝑃(𝑥, 𝑐) 𝑀(𝑥) 𝑁(𝑥)

Proof of Theorem ballotlem2 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . . 6 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . . . . 6 ⊢ 𝑁 ∈ ℕ
3		ballotth.o	. . . . . 6 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
4	1, 2, 3	ballotlemoex 34941	. . . . 5 ⊢ 𝑂 ∈ V
5		ssrab2 4035	. . . . 5 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
6	4, 5	elpwi2 5308	. . . 4 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
7		fveq2 6885	. . . . . 6 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}))
8	7	oveq1d 7434	. . . . 5 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
9		ballotth.p	. . . . 5 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
10		ovex 7452	. . . . 5 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
11	8, 9, 10	fvmpt 6993	. . . 4 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 → (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
12	6, 11	ax-mp 5	. . 3 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
13		an32 659	. . . . . . . 8 ⊢ (((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ∧ (♯‘𝑐) = 𝑀) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
14		2eluzge1 12922	. . . . . . . . . . . . . 14 ⊢ 2 ∈ (ℤ_≥‘1)
15		fzss1 13608	. . . . . . . . . . . . . 14 ⊢ (2 ∈ (ℤ_≥‘1) → (2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁)))
16	14, 15	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁))
17	16	sspwi 4576	. . . . . . . . . . . 12 ⊢ 𝒫 (2...(𝑀 + 𝑁)) ⊆ 𝒫 (1...(𝑀 + 𝑁))
18	17	sseli 3934	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
19		1lt2 12428	. . . . . . . . . . . . . . . 16 ⊢ 1 < 2
20		1re 11223	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
21		2re 12330	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
22	20, 21	ltnlei 11346	. . . . . . . . . . . . . . . 16 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
23	19, 22	mpbi 233	. . . . . . . . . . . . . . 15 ⊢ ¬ 2 ≤ 1
24		elfzle1 13571	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (2...(𝑀 + 𝑁)) → 2 ≤ 1)
25	23, 24	mto 200	. . . . . . . . . . . . . 14 ⊢ ¬ 1 ∈ (2...(𝑀 + 𝑁))
26		elelpwi 4574	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) → 1 ∈ (2...(𝑀 + 𝑁)))
27	25, 26	mto 200	. . . . . . . . . . . . 13 ⊢ ¬ (1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
28		ancom 466	. . . . . . . . . . . . 13 ⊢ ((1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) ↔ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐))
29	27, 28	mtbi 325	. . . . . . . . . . . 12 ⊢ ¬ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐)
30	29	imnani 406	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → ¬ 1 ∈ 𝑐)
31	18, 30	jca 521	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐))
32		ssin 4191	. . . . . . . . . . . 12 ⊢ ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) ↔ 𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}))
33		1le2 12467	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ≤ 2
34		1p1e2 12379	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 + 1) = 2
35		nnge1 12279	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
36	1, 35	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ≤ 𝑀
37		nnge1 12279	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
38	2, 37	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ≤ 𝑁
39	1	nnrei 12257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑀 ∈ ℝ
40	2	nnrei 12257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑁 ∈ ℝ
41	20, 20, 39, 40	le2addi 11792	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ≤ 𝑀 ∧ 1 ≤ 𝑁) → (1 + 1) ≤ (𝑀 + 𝑁))
42	36, 38, 41	mp2an 705	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 + 1) ≤ (𝑀 + 𝑁)
43	34, 42	eqbrtrri 5136	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ≤ (𝑀 + 𝑁)
44	39, 40	readdcli 11239	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 + 𝑁) ∈ ℝ
45	20, 21, 44	letri 11354	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ≤ 2 ∧ 2 ≤ (𝑀 + 𝑁)) → 1 ≤ (𝑀 + 𝑁))
46	33, 43, 45	mp2an 705	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ≤ (𝑀 + 𝑁)
47		1z 12639	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℤ
48		nnaddcl 12271	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
49	1, 2, 48	mp2an 705	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 + 𝑁) ∈ ℕ
50	49	nnzi 12633	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 + 𝑁) ∈ ℤ
51		eluz 12892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) ↔ 1 ≤ (𝑀 + 𝑁)))
52	47, 50, 51	mp2an 705	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) ↔ 1 ≤ (𝑀 + 𝑁))
53	46, 52	mpbir 234	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘1)
54		elfzp12 13648	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↔ (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁)))))
55	53, 54	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↔ (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁))))
56	55	biimpi 219	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) → (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁))))
57	56	orcanai 1018	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁)))
58	34	oveq1i 7429	. . . . . . . . . . . . . . . 16 ⊢ ((1 + 1)...(𝑀 + 𝑁)) = (2...(𝑀 + 𝑁))
59	57, 58	eleqtrdi 2875	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ (2...(𝑀 + 𝑁)))
60	59	ss2abi 4021	. . . . . . . . . . . . . 14 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} ⊆ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))}
61		inab 4262	. . . . . . . . . . . . . . 15 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)}
62		abid2 2902	. . . . . . . . . . . . . . . 16 ⊢ {𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} = (1...(𝑀 + 𝑁))
63	62	ineq1i 4169	. . . . . . . . . . . . . . 15 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
64	61, 63	eqtr3i 2790	. . . . . . . . . . . . . 14 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
65		abid2 2902	. . . . . . . . . . . . . 14 ⊢ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))} = (2...(𝑀 + 𝑁))
66	60, 64, 65	3sstr3i 3988	. . . . . . . . . . . . 13 ⊢ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))
67		sstr 3946	. . . . . . . . . . . . 13 ⊢ ((𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ∧ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
68	66, 67	mpan2 704	. . . . . . . . . . . 12 ⊢ (𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
69	32, 68	sylbi 220	. . . . . . . . . . 11 ⊢ ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
70		velpw 4569	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (1...(𝑀 + 𝑁)))
71		ssab 4018	. . . . . . . . . . . . 13 ⊢ (𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1} ↔ ∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1))
72		df-ex 1813	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ¬ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
73	72	bicomi 227	. . . . . . . . . . . . . . 15 ⊢ (¬ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
74	73	con1bii 359	. . . . . . . . . . . . . 14 ⊢ (¬ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
75		dfclel 2841	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ 𝑐 ↔ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
76	75	notbii 323	. . . . . . . . . . . . . 14 ⊢ (¬ 1 ∈ 𝑐 ↔ ¬ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
77		imnang 1875	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ ∀𝑖 ¬ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
78		ancom 466	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
79	78	notbii 323	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ¬ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
80	79	albii 1852	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ∀𝑖 ¬ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
81	77, 80	bitr4i 281	. . . . . . . . . . . . . 14 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
82	74, 76, 81	3bitr4ri 307	. . . . . . . . . . . . 13 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ ¬ 1 ∈ 𝑐)
83	71, 82	bitr2i 279	. . . . . . . . . . . 12 ⊢ (¬ 1 ∈ 𝑐 ↔ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1})
84	70, 83	anbi12i 640	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ↔ (𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}))
85		velpw 4569	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (2...(𝑀 + 𝑁)))
86	69, 84, 85	3imtr4i 295	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) → 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
87	31, 86	impbii 212	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐))
88	87	anbi1i 636	. . . . . . . 8 ⊢ ((𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ∧ (♯‘𝑐) = 𝑀))
89	3	reqabi 3441	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝑂 ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀))
90	89	anbi1i 636	. . . . . . . 8 ⊢ ((𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
91	13, 88, 90	3bitr4i 306	. . . . . . 7 ⊢ ((𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ↔ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐))
92	91	rabbia2 3421	. . . . . 6 ⊢ {𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
93	92	fveq2i 6888	. . . . 5 ⊢ (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐})
94		fzfi 14026	. . . . . . 7 ⊢ (2...(𝑀 + 𝑁)) ∈ Fin
95	1	nnzi 12633	. . . . . . 7 ⊢ 𝑀 ∈ ℤ
96		hashbc 14508	. . . . . . 7 ⊢ (((2...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑀 ∈ ℤ) → ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}))
97	94, 95, 96	mp2an 705	. . . . . 6 ⊢ ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀})
98		2z 12641	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
99	98	eluz1i 12886	. . . . . . . . . . 11 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘2) ↔ ((𝑀 + 𝑁) ∈ ℤ ∧ 2 ≤ (𝑀 + 𝑁)))
100	50, 43, 99	mpbir2an 724	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘2)
101		hashfz 14482	. . . . . . . . . 10 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘2) → (♯‘(2...(𝑀 + 𝑁))) = (((𝑀 + 𝑁) − 2) + 1))
102	100, 101	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘(2...(𝑀 + 𝑁))) = (((𝑀 + 𝑁) − 2) + 1)
103	1	nncni 12258	. . . . . . . . . . 11 ⊢ 𝑀 ∈ ℂ
104	2	nncni 12258	. . . . . . . . . . 11 ⊢ 𝑁 ∈ ℂ
105	103, 104	addcli 11230	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℂ
106		2cn 12331	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
107		ax-1cn 11173	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
108		subadd23 11484	. . . . . . . . . 10 ⊢ (((𝑀 + 𝑁) ∈ ℂ ∧ 2 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2)))
109	105, 106, 107, 108	mp3an 1490	. . . . . . . . 9 ⊢ (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2))
110	106, 107	negsubdi2i 11559	. . . . . . . . . . 11 ⊢ -(2 − 1) = (1 − 2)
111		2m1e1 12380	. . . . . . . . . . . 12 ⊢ (2 − 1) = 1
112	111	negeqi 11465	. . . . . . . . . . 11 ⊢ -(2 − 1) = -1
113	110, 112	eqtr3i 2790	. . . . . . . . . 10 ⊢ (1 − 2) = -1
114	113	oveq2i 7430	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) + (1 − 2)) = ((𝑀 + 𝑁) + -1)
115	102, 109, 114	3eqtri 2792	. . . . . . . 8 ⊢ (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) + -1)
116	105, 107	negsubi 11551	. . . . . . . 8 ⊢ ((𝑀 + 𝑁) + -1) = ((𝑀 + 𝑁) − 1)
117	115, 116	eqtri 2788	. . . . . . 7 ⊢ (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) − 1)
118	117	oveq1i 7429	. . . . . 6 ⊢ ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (((𝑀 + 𝑁) − 1)C𝑀)
119	97, 118	eqtr3i 2790	. . . . 5 ⊢ (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) = (((𝑀 + 𝑁) − 1)C𝑀)
120	93, 119	eqtr3i 2790	. . . 4 ⊢ (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (((𝑀 + 𝑁) − 1)C𝑀)
121	1, 2, 3	ballotlem1 34942	. . . 4 ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
122	120, 121	oveq12i 7431	. . 3 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
123	12, 122	eqtri 2788	. 2 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
124		0le1 11752	. . . . 5 ⊢ 0 ≤ 1
125		0re 11225	. . . . . 6 ⊢ 0 ∈ ℝ
126	125, 20, 39	letri 11354	. . . . 5 ⊢ ((0 ≤ 1 ∧ 1 ≤ 𝑀) → 0 ≤ 𝑀)
127	124, 36, 126	mp2an 705	. . . 4 ⊢ 0 ≤ 𝑀
128	2	nngt0i 12290	. . . . . 6 ⊢ 0 < 𝑁
129	40, 128	elrpii 13035	. . . . 5 ⊢ 𝑁 ∈ ℝ⁺
130		ltaddrp 13071	. . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → 𝑀 < (𝑀 + 𝑁))
131	39, 129, 130	mp2an 705	. . . 4 ⊢ 𝑀 < (𝑀 + 𝑁)
132		0z 12617	. . . . 5 ⊢ 0 ∈ ℤ
133		elfzm11 13640	. . . . 5 ⊢ ((0 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < (𝑀 + 𝑁))))
134	132, 50, 133	mp2an 705	. . . 4 ⊢ (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < (𝑀 + 𝑁)))
135	95, 127, 131, 134	mpbir3an 1360	. . 3 ⊢ 𝑀 ∈ (0...((𝑀 + 𝑁) − 1))
136		bcm1n 33210	. . 3 ⊢ ((𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ∧ (𝑀 + 𝑁) ∈ ℕ) → ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀)) = (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)))
137	135, 49, 136	mp2an 705	. 2 ⊢ ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀)) = (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁))
138		pncan2 11479	. . . 4 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
139	103, 104, 138	mp2an 705	. . 3 ⊢ ((𝑀 + 𝑁) − 𝑀) = 𝑁
140	139	oveq1i 7429	. 2 ⊢ (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)) = (𝑁 / (𝑀 + 𝑁))
141	123, 137, 140	3eqtri 2792	1 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 ∀wal 1568 = wceq 1570 ∃wex 1812 ∈ wcel 2146 {cab 2743 {crab 3418 Vcvv 3457 ∩ cin 3905 ⊆ wss 3906 𝒫 cpw 4564 class class class wbr 5111 ↦ cmpt 5194 ‘cfv 6540 (class class class)co 7419 Fincfn 8949 ℂcc 11113 ℝcr 11114 0cc0 11115 1c1 11116 + caddc 11118 < clt 11258 ≤ cle 11259 − cmin 11456 -cneg 11457 / cdiv 11886 ℕcn 12248 2c2 12310 ℤcz 12606 ℤ_≥cuz 12878 ℝ⁺crp 13032 ...cfz 13551 Ccbc 14356 ♯chash 14384

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-oadd 8463 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-dju 9903 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-n0 12520 df-z 12607 df-uz 12879 df-rp 13033 df-fz 13552 df-seq 14056 df-fac 14328 df-bc 14357 df-hash 14385

This theorem is used by: ballotth 34993

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