Theorem ballotlem2 30550

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 3687	. . . . 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 30547	. . . . . 6 ⊢ 𝑂 ∈ V
6	5	elpw2 4828	. . . . 5 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 ↔ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂)
7	1, 6	mpbir 221	. . . 4 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
8		fveq2 6191	. . . . . 6 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → (#‘𝑥) = (#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}))
9	8	oveq1d 6665	. . . . 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 6678	. . . . 5 ⊢ ((#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)) ∈ V
12	9, 10, 11	fvmpt 6282	. . . 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 839	. . . . . . . . 9 ⊢ (((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ∧ (#‘𝑐) = 𝑀) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
15		2eluzge1 11734	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (ℤ≥‘1)
16		fzss1 12380	. . . . . . . . . . . . . . 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 4917	. . . . . . . . . . . . . 14 ⊢ ((2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁)) ↔ 𝒫 (2...(𝑀 + 𝑁)) ⊆ 𝒫 (1...(𝑀 + 𝑁)))
19	17, 18	mpbi 220	. . . . . . . . . . . . 13 ⊢ 𝒫 (2...(𝑀 + 𝑁)) ⊆ 𝒫 (1...(𝑀 + 𝑁))
20	19	sseli 3599	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
21		1lt2 11194	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 2
22		1re 10039	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
23		2re 11090	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
24	22, 23	ltnlei 10158	. . . . . . . . . . . . . . . . 17 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
25	21, 24	mpbi 220	. . . . . . . . . . . . . . . 16 ⊢ ¬ 2 ≤ 1
26		elfzle1 12344	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (2...(𝑀 + 𝑁)) → 2 ≤ 1)
27	25, 26	mto 188	. . . . . . . . . . . . . . 15 ⊢ ¬ 1 ∈ (2...(𝑀 + 𝑁))
28		elelpwi 4171	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) → 1 ∈ (2...(𝑀 + 𝑁)))
29	27, 28	mto 188	. . . . . . . . . . . . . 14 ⊢ ¬ (1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
30		ancom 466	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) ↔ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐))
31	29, 30	mtbi 312	. . . . . . . . . . . . 13 ⊢ ¬ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐)
32	31	imnani 439	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → ¬ 1 ∈ 𝑐)
33	20, 32	jca 554	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐))
34		ssin 3835	. . . . . . . . . . . . 13 ⊢ ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) ↔ 𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}))
35		1le2 11241	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ≤ 2
36		1p1e2 11134	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 + 1) = 2
37		nnge1 11046	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
38	2, 37	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ≤ 𝑀
39		nnge1 11046	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
40	3, 39	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ≤ 𝑁
41	2	nnrei 11029	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑀 ∈ ℝ
42	3	nnrei 11029	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑁 ∈ ℝ
43	22, 22, 41, 42	le2addi 10591	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ≤ 𝑀 ∧ 1 ≤ 𝑁) → (1 + 1) ≤ (𝑀 + 𝑁))
44	38, 40, 43	mp2an 708	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 + 1) ≤ (𝑀 + 𝑁)
45	36, 44	eqbrtrri 4676	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ≤ (𝑀 + 𝑁)
46	41, 42	readdcli 10053	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 + 𝑁) ∈ ℝ
47	22, 23, 46	letri 10166	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ≤ 2 ∧ 2 ≤ (𝑀 + 𝑁)) → 1 ≤ (𝑀 + 𝑁))
48	35, 45, 47	mp2an 708	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ≤ (𝑀 + 𝑁)
49		1z 11407	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℤ
50		nnaddcl 11042	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
51	2, 3, 50	mp2an 708	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 + 𝑁) ∈ ℕ
52	51	nnzi 11401	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 + 𝑁) ∈ ℤ
53		eluz 11701	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝑀 + 𝑁) ∈ (ℤ≥‘1) ↔ 1 ≤ (𝑀 + 𝑁)))
54	49, 52, 53	mp2an 708	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘1) ↔ 1 ≤ (𝑀 + 𝑁))
55	48, 54	mpbir 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 + 𝑁) ∈ (ℤ≥‘1)
56		elfzp12 12419	. . . . . . . . . . . . . . . . . . . 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 206	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) → (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁))))
59	58	orcanai 952	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁)))
60	36	oveq1i 6660	. . . . . . . . . . . . . . . . 17 ⊢ ((1 + 1)...(𝑀 + 𝑁)) = (2...(𝑀 + 𝑁))
61	59, 60	syl6eleq 2711	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ (2...(𝑀 + 𝑁)))
62	61	ss2abi 3674	. . . . . . . . . . . . . . 15 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} ⊆ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))}
63		inab 3895	. . . . . . . . . . . . . . . 16 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)}
64		abid2 2745	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} = (1...(𝑀 + 𝑁))
65	64	ineq1i 3810	. . . . . . . . . . . . . . . 16 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
66	63, 65	eqtr3i 2646	. . . . . . . . . . . . . . 15 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
67		abid2 2745	. . . . . . . . . . . . . . 15 ⊢ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))} = (2...(𝑀 + 𝑁))
68	62, 66, 67	3sstr3i 3643	. . . . . . . . . . . . . 14 ⊢ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))
69		sstr 3611	. . . . . . . . . . . . . 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 707	. . . . . . . . . . . . 13 ⊢ (𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
71	34, 70	sylbi 207	. . . . . . . . . . . 12 ⊢ ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
72		selpw 4165	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (1...(𝑀 + 𝑁)))
73		ssab 3672	. . . . . . . . . . . . . 14 ⊢ (𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1} ↔ ∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1))
74		df-ex 1705	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ¬ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
75	74	bicomi 214	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
76	75	con1bii 346	. . . . . . . . . . . . . . 15 ⊢ (¬ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
77		df-clel 2618	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ 𝑐 ↔ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
78	77	notbii 310	. . . . . . . . . . . . . . 15 ⊢ (¬ 1 ∈ 𝑐 ↔ ¬ ∃𝑖(𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
79		imnang 1769	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ ∀𝑖 ¬ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
80		ancom 466	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
81	80	notbii 310	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ¬ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
82	81	albii 1747	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐) ↔ ∀𝑖 ¬ (𝑖 ∈ 𝑐 ∧ 𝑖 = 1))
83	79, 82	bitr4i 267	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖 ∈ 𝑐))
84	76, 78, 83	3bitr4ri 293	. . . . . . . . . . . . . 14 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ ¬ 1 ∈ 𝑐)
85	73, 84	bitr2i 265	. . . . . . . . . . . . 13 ⊢ (¬ 1 ∈ 𝑐 ↔ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1})
86	72, 85	anbi12i 733	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ↔ (𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}))
87		selpw 4165	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (2...(𝑀 + 𝑁)))
88	71, 86, 87	3imtr4i 281	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) → 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
89	33, 88	impbii 199	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐))
90	89	anbi1i 731	. . . . . . . . 9 ⊢ ((𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ∧ (#‘𝑐) = 𝑀))
91	4	rabeq2i 3197	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝑂 ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀))
92	91	anbi1i 731	. . . . . . . . 9 ⊢ ((𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
93	14, 90, 92	3bitr4i 292	. . . . . . . 8 ⊢ ((𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀) ↔ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐))
94	93	abbii 2739	. . . . . . 7 ⊢ {𝑐 ∣ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀)} = {𝑐 ∣ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐)}
95		df-rab 2921	. . . . . . 7 ⊢ {𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀} = {𝑐 ∣ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (#‘𝑐) = 𝑀)}
96		df-rab 2921	. . . . . . 7 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} = {𝑐 ∣ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐)}
97	94, 95, 96	3eqtr4i 2654	. . . . . 6 ⊢ {𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀} = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
98	97	fveq2i 6194	. . . . 5 ⊢ (#‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}) = (#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐})
99		fzfi 12771	. . . . . . 7 ⊢ (2...(𝑀 + 𝑁)) ∈ Fin
100	2	nnzi 11401	. . . . . . 7 ⊢ 𝑀 ∈ ℤ
101		hashbc 13237	. . . . . . 7 ⊢ (((2...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑀 ∈ ℤ) → ((#‘(2...(𝑀 + 𝑁)))C𝑀) = (#‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}))
102	99, 100, 101	mp2an 708	. . . . . 6 ⊢ ((#‘(2...(𝑀 + 𝑁)))C𝑀) = (#‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀})
103		2z 11409	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
104	103	eluz1i 11695	. . . . . . . . . . 11 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘2) ↔ ((𝑀 + 𝑁) ∈ ℤ ∧ 2 ≤ (𝑀 + 𝑁)))
105	52, 45, 104	mpbir2an 955	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ (ℤ≥‘2)
106		hashfz 13214	. . . . . . . . . 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 11030	. . . . . . . . . . 11 ⊢ 𝑀 ∈ ℂ
109	3	nncni 11030	. . . . . . . . . . 11 ⊢ 𝑁 ∈ ℂ
110	108, 109	addcli 10044	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℂ
111		2cn 11091	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
112		ax-1cn 9994	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
113		subadd23 10293	. . . . . . . . . 10 ⊢ (((𝑀 + 𝑁) ∈ ℂ ∧ 2 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2)))
114	110, 111, 112, 113	mp3an 1424	. . . . . . . . 9 ⊢ (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2))
115	111, 112	negsubdi2i 10367	. . . . . . . . . . 11 ⊢ -(2 − 1) = (1 − 2)
116		2m1e1 11135	. . . . . . . . . . . 12 ⊢ (2 − 1) = 1
117	116	negeqi 10274	. . . . . . . . . . 11 ⊢ -(2 − 1) = -1
118	115, 117	eqtr3i 2646	. . . . . . . . . 10 ⊢ (1 − 2) = -1
119	118	oveq2i 6661	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) + (1 − 2)) = ((𝑀 + 𝑁) + -1)
120	107, 114, 119	3eqtri 2648	. . . . . . . 8 ⊢ (#‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) + -1)
121	110, 112	negsubi 10359	. . . . . . . 8 ⊢ ((𝑀 + 𝑁) + -1) = ((𝑀 + 𝑁) − 1)
122	120, 121	eqtri 2644	. . . . . . 7 ⊢ (#‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) − 1)
123	122	oveq1i 6660	. . . . . 6 ⊢ ((#‘(2...(𝑀 + 𝑁)))C𝑀) = (((𝑀 + 𝑁) − 1)C𝑀)
124	102, 123	eqtr3i 2646	. . . . 5 ⊢ (#‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}) = (((𝑀 + 𝑁) − 1)C𝑀)
125	98, 124	eqtr3i 2646	. . . 4 ⊢ (#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (((𝑀 + 𝑁) − 1)C𝑀)
126	2, 3, 4	ballotlem1 30548	. . . 4 ⊢ (#‘𝑂) = ((𝑀 + 𝑁)C𝑀)
127	125, 126	oveq12i 6662	. . 3 ⊢ ((#‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
128	13, 127	eqtri 2644	. 2 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
129		0le1 10551	. . . . 5 ⊢ 0 ≤ 1
130		0re 10040	. . . . . 6 ⊢ 0 ∈ ℝ
131	130, 22, 41	letri 10166	. . . . 5 ⊢ ((0 ≤ 1 ∧ 1 ≤ 𝑀) → 0 ≤ 𝑀)
132	129, 38, 131	mp2an 708	. . . 4 ⊢ 0 ≤ 𝑀
133	3	nngt0i 11054	. . . . . 6 ⊢ 0 < 𝑁
134	42, 133	elrpii 11835	. . . . 5 ⊢ 𝑁 ∈ ℝ+
135		ltaddrp 11867	. . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → 𝑀 < (𝑀 + 𝑁))
136	41, 134, 135	mp2an 708	. . . 4 ⊢ 𝑀 < (𝑀 + 𝑁)
137		0z 11388	. . . . 5 ⊢ 0 ∈ ℤ
138		elfzm11 12411	. . . . 5 ⊢ ((0 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < (𝑀 + 𝑁))))
139	137, 52, 138	mp2an 708	. . . 4 ⊢ (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < (𝑀 + 𝑁)))
140	100, 132, 136, 139	mpbir3an 1244	. . 3 ⊢ 𝑀 ∈ (0...((𝑀 + 𝑁) − 1))
141		bcm1n 29554	. . 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 708	. 2 ⊢ ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀)) = (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁))
143		pncan2 10288	. . . 4 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
144	108, 109, 143	mp2an 708	. . 3 ⊢ ((𝑀 + 𝑁) − 𝑀) = 𝑁
145	144	oveq1i 6660	. 2 ⊢ (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)) = (𝑁 / (𝑀 + 𝑁))
146	128, 142, 145	3eqtri 2648	1 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  {crab 2916   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075   − cmin 10266  -cneg 10267   / cdiv 10684  ℕcn 11020  2c2 11070  ℤcz 11377  ℤ_≥cuz 11687  ℝ⁺crp 11832  ...cfz 12326  Ccbc 13089  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-fac 13061  df-bc 13090  df-hash 13118

This theorem is referenced by:  ballotth  30599