ballotlem2 - Mathbox for Thierry Arnoux

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Theorem ballotlem2 31748

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 31745	. . . . 5 ⊢ 𝑂 ∈ V
5		ssrab2 4058	. . . . 5 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
6	4, 5	elpwi2 5251	. . . 4 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
7		fveq2 6672	. . . . . 6 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}))
8	7	oveq1d 7173	. . . . 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 7191	. . . . 5 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
11	8, 9, 10	fvmpt 6770	. . . 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 644	. . . . . . . 8 ⊢ (((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ∧ (♯‘𝑐) = 𝑀) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
14		2eluzge1 12297	. . . . . . . . . . . . . 14 ⊢ 2 ∈ (ℤ_≥‘1)
15		fzss1 12949	. . . . . . . . . . . . . 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 4555	. . . . . . . . . . . 12 ⊢ 𝒫 (2...(𝑀 + 𝑁)) ⊆ 𝒫 (1...(𝑀 + 𝑁))
18	17	sseli 3965	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
19		1lt2 11811	. . . . . . . . . . . . . . . 16 ⊢ 1 < 2
20		1re 10643	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
21		2re 11714	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
22	20, 21	ltnlei 10763	. . . . . . . . . . . . . . . 16 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
23	19, 22	mpbi 232	. . . . . . . . . . . . . . 15 ⊢ ¬ 2 ≤ 1
24		elfzle1 12913	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (2...(𝑀 + 𝑁)) → 2 ≤ 1)
25	23, 24	mto 199	. . . . . . . . . . . . . 14 ⊢ ¬ 1 ∈ (2...(𝑀 + 𝑁))
26		elelpwi 4553	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) → 1 ∈ (2...(𝑀 + 𝑁)))
27	25, 26	mto 199	. . . . . . . . . . . . 13 ⊢ ¬ (1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
28		ancom 463	. . . . . . . . . . . . 13 ⊢ ((1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) ↔ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐))
29	27, 28	mtbi 324	. . . . . . . . . . . 12 ⊢ ¬ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐)
30	29	imnani 403	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → ¬ 1 ∈ 𝑐)
31	18, 30	jca 514	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐))
32		ssin 4209	. . . . . . . . . . . 12 ⊢ ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) ↔ 𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}))
33		1le2 11849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ≤ 2
34		1p1e2 11765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 + 1) = 2
35		nnge1 11668	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
36	1, 35	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ≤ 𝑀
37		nnge1 11668	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
38	2, 37	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ≤ 𝑁
39	1	nnrei 11649	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑀 ∈ ℝ
40	2	nnrei 11649	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑁 ∈ ℝ
41	20, 20, 39, 40	le2addi 11205	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ≤ 𝑀 ∧ 1 ≤ 𝑁) → (1 + 1) ≤ (𝑀 + 𝑁))
42	36, 38, 41	mp2an 690	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 + 1) ≤ (𝑀 + 𝑁)
43	34, 42	eqbrtrri 5091	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ≤ (𝑀 + 𝑁)
44	39, 40	readdcli 10658	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 + 𝑁) ∈ ℝ
45	20, 21, 44	letri 10771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ≤ 2 ∧ 2 ≤ (𝑀 + 𝑁)) → 1 ≤ (𝑀 + 𝑁))
46	33, 43, 45	mp2an 690	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ≤ (𝑀 + 𝑁)
47		1z 12015	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℤ
48		nnaddcl 11663	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
49	1, 2, 48	mp2an 690	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 + 𝑁) ∈ ℕ
50	49	nnzi 12009	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 + 𝑁) ∈ ℤ
51		eluz 12260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) ↔ 1 ≤ (𝑀 + 𝑁)))
52	47, 50, 51	mp2an 690	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) ↔ 1 ≤ (𝑀 + 𝑁))
53	46, 52	mpbir 233	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘1)
54		elfzp12 12989	. . . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) → (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁))))
57	56	orcanai 999	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁)))
58	34	oveq1i 7168	. . . . . . . . . . . . . . . 16 ⊢ ((1 + 1)...(𝑀 + 𝑁)) = (2...(𝑀 + 𝑁))
59	57, 58	eleqtrdi 2925	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ (2...(𝑀 + 𝑁)))
60	59	ss2abi 4045	. . . . . . . . . . . . . 14 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} ⊆ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))}
61		inab 4273	. . . . . . . . . . . . . . 15 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)}
62		abid2 2959	. . . . . . . . . . . . . . . 16 ⊢ {𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} = (1...(𝑀 + 𝑁))
63	62	ineq1i 4187	. . . . . . . . . . . . . . 15 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
64	61, 63	eqtr3i 2848	. . . . . . . . . . . . . 14 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
65		abid2 2959	. . . . . . . . . . . . . 14 ⊢ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))} = (2...(𝑀 + 𝑁))
66	60, 64, 65	3sstr3i 4011	. . . . . . . . . . . . 13 ⊢ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))
67		sstr 3977	. . . . . . . . . . . . 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 689	. . . . . . . . . . . 12 ⊢ (𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
69	32, 68	sylbi 219	. . . . . . . . . . 11 ⊢ ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
70		velpw 4546	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (1...(𝑀 + 𝑁)))
71		ssab 4043	. . . . . . . . . . . . 13 ⊢ (𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1} ↔ ∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1))
72		df-ex 1781	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ¬ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
73	72	bicomi 226	. . . . . . . . . . . . . . 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 2896	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ 𝑐 ↔ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
76	75	notbii 322	. . . . . . . . . . . . . 14 ⊢ (¬ 1 ∈ 𝑐 ↔ ¬ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
77		imnang 1842	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ ∀𝑖 ¬ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
78		ancom 463	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
79	78	notbii 322	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ¬ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
80	79	albii 1820	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ∀𝑖 ¬ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
81	77, 80	bitr4i 280	. . . . . . . . . . . . . 14 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
82	74, 76, 81	3bitr4ri 306	. . . . . . . . . . . . 13 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ ¬ 1 ∈ 𝑐)
83	71, 82	bitr2i 278	. . . . . . . . . . . 12 ⊢ (¬ 1 ∈ 𝑐 ↔ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1})
84	70, 83	anbi12i 628	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ↔ (𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}))
85		velpw 4546	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (2...(𝑀 + 𝑁)))
86	69, 84, 85	3imtr4i 294	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) → 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
87	31, 86	impbii 211	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐))
88	87	anbi1i 625	. . . . . . . 8 ⊢ ((𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ∧ (♯‘𝑐) = 𝑀))
89	3	rabeq2i 3489	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝑂 ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀))
90	89	anbi1i 625	. . . . . . . 8 ⊢ ((𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
91	13, 88, 90	3bitr4i 305	. . . . . . 7 ⊢ ((𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ↔ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐))
92	91	rabbia2 3479	. . . . . 6 ⊢ {𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
93	92	fveq2i 6675	. . . . 5 ⊢ (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐})
94		fzfi 13343	. . . . . . 7 ⊢ (2...(𝑀 + 𝑁)) ∈ Fin
95	1	nnzi 12009	. . . . . . 7 ⊢ 𝑀 ∈ ℤ
96		hashbc 13814	. . . . . . 7 ⊢ (((2...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑀 ∈ ℤ) → ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}))
97	94, 95, 96	mp2an 690	. . . . . 6 ⊢ ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀})
98		2z 12017	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
99	98	eluz1i 12254	. . . . . . . . . . 11 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘2) ↔ ((𝑀 + 𝑁) ∈ ℤ ∧ 2 ≤ (𝑀 + 𝑁)))
100	50, 43, 99	mpbir2an 709	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘2)
101		hashfz 13791	. . . . . . . . . 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 11650	. . . . . . . . . . 11 ⊢ 𝑀 ∈ ℂ
104	2	nncni 11650	. . . . . . . . . . 11 ⊢ 𝑁 ∈ ℂ
105	103, 104	addcli 10649	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℂ
106		2cn 11715	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
107		ax-1cn 10597	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
108		subadd23 10900	. . . . . . . . . 10 ⊢ (((𝑀 + 𝑁) ∈ ℂ ∧ 2 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2)))
109	105, 106, 107, 108	mp3an 1457	. . . . . . . . 9 ⊢ (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2))
110	106, 107	negsubdi2i 10974	. . . . . . . . . . 11 ⊢ -(2 − 1) = (1 − 2)
111		2m1e1 11766	. . . . . . . . . . . 12 ⊢ (2 − 1) = 1
112	111	negeqi 10881	. . . . . . . . . . 11 ⊢ -(2 − 1) = -1
113	110, 112	eqtr3i 2848	. . . . . . . . . 10 ⊢ (1 − 2) = -1
114	113	oveq2i 7169	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) + (1 − 2)) = ((𝑀 + 𝑁) + -1)
115	102, 109, 114	3eqtri 2850	. . . . . . . 8 ⊢ (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) + -1)
116	105, 107	negsubi 10966	. . . . . . . 8 ⊢ ((𝑀 + 𝑁) + -1) = ((𝑀 + 𝑁) − 1)
117	115, 116	eqtri 2846	. . . . . . 7 ⊢ (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) − 1)
118	117	oveq1i 7168	. . . . . 6 ⊢ ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (((𝑀 + 𝑁) − 1)C𝑀)
119	97, 118	eqtr3i 2848	. . . . 5 ⊢ (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) = (((𝑀 + 𝑁) − 1)C𝑀)
120	93, 119	eqtr3i 2848	. . . 4 ⊢ (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (((𝑀 + 𝑁) − 1)C𝑀)
121	1, 2, 3	ballotlem1 31746	. . . 4 ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
122	120, 121	oveq12i 7170	. . 3 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
123	12, 122	eqtri 2846	. 2 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
124		0le1 11165	. . . . 5 ⊢ 0 ≤ 1
125		0re 10645	. . . . . 6 ⊢ 0 ∈ ℝ
126	125, 20, 39	letri 10771	. . . . 5 ⊢ ((0 ≤ 1 ∧ 1 ≤ 𝑀) → 0 ≤ 𝑀)
127	124, 36, 126	mp2an 690	. . . 4 ⊢ 0 ≤ 𝑀
128	2	nngt0i 11679	. . . . . 6 ⊢ 0 < 𝑁
129	40, 128	elrpii 12395	. . . . 5 ⊢ 𝑁 ∈ ℝ⁺
130		ltaddrp 12429	. . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → 𝑀 < (𝑀 + 𝑁))
131	39, 129, 130	mp2an 690	. . . 4 ⊢ 𝑀 < (𝑀 + 𝑁)
132		0z 11995	. . . . 5 ⊢ 0 ∈ ℤ
133		elfzm11 12981	. . . . 5 ⊢ ((0 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < (𝑀 + 𝑁))))
134	132, 50, 133	mp2an 690	. . . 4 ⊢ (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < (𝑀 + 𝑁)))
135	95, 127, 131, 134	mpbir3an 1337	. . 3 ⊢ 𝑀 ∈ (0...((𝑀 + 𝑁) − 1))
136		bcm1n 30520	. . 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 690	. 2 ⊢ ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀)) = (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁))
138		pncan2 10895	. . . 4 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
139	103, 104, 138	mp2an 690	. . 3 ⊢ ((𝑀 + 𝑁) − 𝑀) = 𝑁
140	139	oveq1i 7168	. 2 ⊢ (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)) = (𝑁 / (𝑀 + 𝑁))
141	123, 137, 140	3eqtri 2850	1 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2801 {crab 3144 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 𝒫 cpw 4541 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 < clt 10677 ≤ cle 10678 − cmin 10872 -cneg 10873 / cdiv 11299 ℕcn 11640 2c2 11695 ℤcz 11984 ℤ_≥cuz 12246 ℝ⁺crp 12392 ...cfz 12895 Ccbc 13665 ♯chash 13693

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-seq 13373 df-fac 13637 df-bc 13666 df-hash 13694

This theorem is referenced by: ballotth 31797

W3C validator