ballotlem2 - Mathbox for Thierry Arnoux

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Theorem ballotlem2 29877

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		ssrab2 3650	. . . . 5 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
2		ballotth.m	. . . . . . 7 ⊢ 𝑀 ∈ ℕ
3		ballotth.n	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
4		ballotth.o	. . . . . . 7 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}
5	2, 3, 4	ballotlemoex 29874	. . . . . 6 ⊢ 𝑂 ∈ V
6	5	elpw2 4755	. . . . 5 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 ↔ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂)
7	1, 6	mpbir 220	. . . 4 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
8		fveq2 6103	. . . . . 6 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → (#‘𝑥) = (#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}))
9	8	oveq1d 6564	. . . . 5 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ((#‘𝑥) / (#‘𝑂)) = ((#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)))
10		ballotth.p	. . . . 5 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))
11		ovex 6577	. . . . 5 ⊢ ((#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)) ∈ V
12	9, 10, 11	fvmpt 6191	. . . 4 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 → (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)))
13	7, 12	ax-mp 5	. . 3 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂))
14		an32 835	. . . . . . . . 9 ⊢ (((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ∧ (#‘𝑐) = 𝑀) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
15		2eluzge1 11610	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (ℤ_≥‘1)
16		fzss1 12251	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ (ℤ_≥‘1) → (2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁)))
17	15, 16	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁))
18		sspwb 4844	. . . . . . . . . . . . . 14 ⊢ ((2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁)) ↔ 𝒫 (2...(𝑀 + 𝑁)) ⊆ 𝒫 (1...(𝑀 + 𝑁)))
19	17, 18	mpbi 219	. . . . . . . . . . . . 13 ⊢ 𝒫 (2...(𝑀 + 𝑁)) ⊆ 𝒫 (1...(𝑀 + 𝑁))
20	19	sseli 3564	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
21		1lt2 11071	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 2
22		1re 9918	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
23		2re 10967	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
24	22, 23	ltnlei 10037	. . . . . . . . . . . . . . . . 17 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
25	21, 24	mpbi 219	. . . . . . . . . . . . . . . 16 ⊢ ¬ 2 ≤ 1
26		elfzle1 12215	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (2...(𝑀 + 𝑁)) → 2 ≤ 1)
27	25, 26	mto 187	. . . . . . . . . . . . . . 15 ⊢ ¬ 1 ∈ (2...(𝑀 + 𝑁))
28		elelpwi 4119	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) → 1 ∈ (2...(𝑀 + 𝑁)))
29	27, 28	mto 187	. . . . . . . . . . . . . 14 ⊢ ¬ (1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
30		ancom 465	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) ↔ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐))
31	29, 30	mtbi 311	. . . . . . . . . . . . 13 ⊢ ¬ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐)
32	31	imnani 438	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → ¬ 1 ∈ 𝑐)
33	20, 32	jca 553	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐))
34		ssin 3797	. . . . . . . . . . . . 13 ⊢ ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) ↔ 𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}))
35		1le2 11118	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ≤ 2
36		1p1e2 11011	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 + 1) = 2
37		nnge1 10923	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
38	2, 37	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ≤ 𝑀
39		nnge1 10923	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
40	3, 39	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ≤ 𝑁
41	2	nnrei 10906	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑀 ∈ ℝ
42	3	nnrei 10906	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑁 ∈ ℝ
43	22, 22, 41, 42	le2addi 10470	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ≤ 𝑀 ∧ 1 ≤ 𝑁) → (1 + 1) ≤ (𝑀 + 𝑁))
44	38, 40, 43	mp2an 704	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 + 1) ≤ (𝑀 + 𝑁)
45	36, 44	eqbrtrri 4606	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ≤ (𝑀 + 𝑁)
46	41, 42	readdcli 9932	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 + 𝑁) ∈ ℝ
47	22, 23, 46	letri 10045	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ≤ 2 ∧ 2 ≤ (𝑀 + 𝑁)) → 1 ≤ (𝑀 + 𝑁))
48	35, 45, 47	mp2an 704	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ≤ (𝑀 + 𝑁)
49		1z 11284	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℤ
50		nnaddcl 10919	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
51	2, 3, 50	mp2an 704	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 + 𝑁) ∈ ℕ
52	51	nnzi 11278	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 + 𝑁) ∈ ℤ
53		eluz 11577	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) ↔ 1 ≤ (𝑀 + 𝑁)))
54	49, 52, 53	mp2an 704	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) ↔ 1 ≤ (𝑀 + 𝑁))
55	48, 54	mpbir 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘1)
56		elfzp12 12288	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↔ (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁)))))
57	55, 56	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↔ (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁))))
58	57	biimpi 205	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) → (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁))))
59	58	orcanai 950	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁)))
60	36	oveq1i 6559	. . . . . . . . . . . . . . . . 17 ⊢ ((1 + 1)...(𝑀 + 𝑁)) = (2...(𝑀 + 𝑁))
61	59, 60	syl6eleq 2698	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ (2...(𝑀 + 𝑁)))
62	61	ss2abi 3637	. . . . . . . . . . . . . . 15 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} ⊆ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))}
63		inab 3854	. . . . . . . . . . . . . . . 16 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)}
64		abid2 2732	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} = (1...(𝑀 + 𝑁))
65	64	ineq1i 3772	. . . . . . . . . . . . . . . 16 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
66	63, 65	eqtr3i 2634	. . . . . . . . . . . . . . 15 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
67		abid2 2732	. . . . . . . . . . . . . . 15 ⊢ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))} = (2...(𝑀 + 𝑁))
68	62, 66, 67	3sstr3i 3606	. . . . . . . . . . . . . 14 ⊢ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))
69		sstr 3576	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ∧ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
70	68, 69	mpan2 703	. . . . . . . . . . . . 13 ⊢ (𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
71	34, 70	sylbi 206	. . . . . . . . . . . 12 ⊢ ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
72		selpw 4115	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (1...(𝑀 + 𝑁)))
73		ssab 3635	. . . . . . . . . . . . . 14 ⊢ (𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1} ↔ ∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1))
74		df-ex 1696	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ¬ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
75	74	bicomi 213	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
76	75	con1bii 345	. . . . . . . . . . . . . . 15 ⊢ (¬ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
77		df-clel 2606	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ 𝑐 ↔ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
78	77	notbii 309	. . . . . . . . . . . . . . 15 ⊢ (¬ 1 ∈ 𝑐 ↔ ¬ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
79		imnang 1758	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ ∀𝑖 ¬ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
80		ancom 465	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
81	80	notbii 309	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ¬ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
82	81	albii 1737	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ∀𝑖 ¬ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
83	79, 82	bitr4i 266	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
84	76, 78, 83	3bitr4ri 292	. . . . . . . . . . . . . 14 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ ¬ 1 ∈ 𝑐)
85	73, 84	bitr2i 264	. . . . . . . . . . . . 13 ⊢ (¬ 1 ∈ 𝑐 ↔ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1})
86	72, 85	anbi12i 729	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ↔ (𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}))
87		selpw 4115	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (2...(𝑀 + 𝑁)))
88	71, 86, 87	3imtr4i 280	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) → 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
89	33, 88	impbii 198	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐))
90	89	anbi1i 727	. . . . . . . . 9 ⊢ ((𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ∧ (#‘𝑐) = 𝑀))
91	4	rabeq2i 3170	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝑂 ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀))
92	91	anbi1i 727	. . . . . . . . 9 ⊢ ((𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
93	14, 90, 92	3bitr4i 291	. . . . . . . 8 ⊢ ((𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀) ↔ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐))
94	93	abbii 2726	. . . . . . 7 ⊢ {𝑐 ∣ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀)} = {𝑐 ∣ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐)}
95		df-rab 2905	. . . . . . 7 ⊢ {𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀} = {𝑐 ∣ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀)}
96		df-rab 2905	. . . . . . 7 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} = {𝑐 ∣ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐)}
97	94, 95, 96	3eqtr4i 2642	. . . . . 6 ⊢ {𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀} = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
98	97	fveq2i 6106	. . . . 5 ⊢ (#‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}) = (#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐})
99		fzfi 12633	. . . . . . 7 ⊢ (2...(𝑀 + 𝑁)) ∈ Fin
100	2	nnzi 11278	. . . . . . 7 ⊢ 𝑀 ∈ ℤ
101		hashbc 13094	. . . . . . 7 ⊢ (((2...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑀 ∈ ℤ) → ((#‘(2...(𝑀 + 𝑁)))C𝑀) = (#‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}))
102	99, 100, 101	mp2an 704	. . . . . 6 ⊢ ((#‘(2...(𝑀 + 𝑁)))C𝑀) = (#‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀})
103		2z 11286	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
104	103	eluz1i 11571	. . . . . . . . . . 11 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘2) ↔ ((𝑀 + 𝑁) ∈ ℤ ∧ 2 ≤ (𝑀 + 𝑁)))
105	52, 45, 104	mpbir2an 957	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘2)
106		hashfz 13074	. . . . . . . . . 10 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘2) → (#‘(2...(𝑀 + 𝑁))) = (((𝑀 + 𝑁) − 2) + 1))
107	105, 106	ax-mp 5	. . . . . . . . 9 ⊢ (#‘(2...(𝑀 + 𝑁))) = (((𝑀 + 𝑁) − 2) + 1)
108	2	nncni 10907	. . . . . . . . . . 11 ⊢ 𝑀 ∈ ℂ
109	3	nncni 10907	. . . . . . . . . . 11 ⊢ 𝑁 ∈ ℂ
110	108, 109	addcli 9923	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℂ
111		2cn 10968	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
112		ax-1cn 9873	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
113		subadd23 10172	. . . . . . . . . 10 ⊢ (((𝑀 + 𝑁) ∈ ℂ ∧ 2 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2)))
114	110, 111, 112, 113	mp3an 1416	. . . . . . . . 9 ⊢ (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2))
115	111, 112	negsubdi2i 10246	. . . . . . . . . . 11 ⊢ -(2 − 1) = (1 − 2)
116		2m1e1 11012	. . . . . . . . . . . 12 ⊢ (2 − 1) = 1
117	116	negeqi 10153	. . . . . . . . . . 11 ⊢ -(2 − 1) = -1
118	115, 117	eqtr3i 2634	. . . . . . . . . 10 ⊢ (1 − 2) = -1
119	118	oveq2i 6560	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) + (1 − 2)) = ((𝑀 + 𝑁) + -1)
120	107, 114, 119	3eqtri 2636	. . . . . . . 8 ⊢ (#‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) + -1)
121	110, 112	negsubi 10238	. . . . . . . 8 ⊢ ((𝑀 + 𝑁) + -1) = ((𝑀 + 𝑁) − 1)
122	120, 121	eqtri 2632	. . . . . . 7 ⊢ (#‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) − 1)
123	122	oveq1i 6559	. . . . . 6 ⊢ ((#‘(2...(𝑀 + 𝑁)))C𝑀) = (((𝑀 + 𝑁) − 1)C𝑀)
124	102, 123	eqtr3i 2634	. . . . 5 ⊢ (#‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}) = (((𝑀 + 𝑁) − 1)C𝑀)
125	98, 124	eqtr3i 2634	. . . 4 ⊢ (#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (((𝑀 + 𝑁) − 1)C𝑀)
126	2, 3, 4	ballotlem1 29875	. . . 4 ⊢ (#‘𝑂) = ((𝑀 + 𝑁)C𝑀)
127	125, 126	oveq12i 6561	. . 3 ⊢ ((#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
128	13, 127	eqtri 2632	. 2 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
129		0le1 10430	. . . . 5 ⊢ 0 ≤ 1
130		0re 9919	. . . . . 6 ⊢ 0 ∈ ℝ
131	130, 22, 41	letri 10045	. . . . 5 ⊢ ((0 ≤ 1 ∧ 1 ≤ 𝑀) → 0 ≤ 𝑀)
132	129, 38, 131	mp2an 704	. . . 4 ⊢ 0 ≤ 𝑀
133	3	nngt0i 10931	. . . . . 6 ⊢ 0 < 𝑁
134	42, 133	elrpii 11711	. . . . 5 ⊢ 𝑁 ∈ ℝ⁺
135		ltaddrp 11743	. . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → 𝑀 < (𝑀 + 𝑁))
136	41, 134, 135	mp2an 704	. . . 4 ⊢ 𝑀 < (𝑀 + 𝑁)
137		0z 11265	. . . . 5 ⊢ 0 ∈ ℤ
138		elfzm11 12280	. . . . 5 ⊢ ((0 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < (𝑀 + 𝑁))))
139	137, 52, 138	mp2an 704	. . . 4 ⊢ (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < (𝑀 + 𝑁)))
140	100, 132, 136, 139	mpbir3an 1237	. . 3 ⊢ 𝑀 ∈ (0...((𝑀 + 𝑁) − 1))
141		bcm1n 28941	. . 3 ⊢ ((𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ∧ (𝑀 + 𝑁) ∈ ℕ) → ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀)) = (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)))
142	140, 51, 141	mp2an 704	. 2 ⊢ ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀)) = (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁))
143		pncan2 10167	. . . 4 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
144	108, 109, 143	mp2an 704	. . 3 ⊢ ((𝑀 + 𝑁) − 𝑀) = 𝑁
145	144	oveq1i 6559	. 2 ⊢ (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)) = (𝑁 / (𝑀 + 𝑁))
146	128, 142, 145	3eqtri 2636	1 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 {crab 2900 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 ℤcz 11254 ℤ_≥cuz 11563 ℝ⁺crp 11708 ...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-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: ballotth 29926

W3C validator