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Theorem ballotfilem2 13206

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
ballotfilem.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
ballotfilem.p	⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))

**Assertion**
Ref	Expression
ballotfilem2	⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))

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

Proof of Theorem ballotfilem2 Dummy variables 𝑖 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . . . 7 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
3		ballotfilem.o	. . . . . . 7 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
4	1, 2, 3	ballotfilemofi 13197	. . . . . 6 ⊢ 𝑂 ∈ Fin
5		ssrab2 3333	. . . . . 6 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
6	4, 5	elpwi2 4289	. . . . 5 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
7	4	a1i 9	. . . . . . 7 ⊢ (⊤ → 𝑂 ∈ Fin)
8		1z 9649	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℤ
9		nnaddcl 9303	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
10	1, 2, 9	mp2an 430	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 + 𝑁) ∈ ℕ
11	10	nnzi 9644	. . . . . . . . . . . . . . 15 ⊢ (𝑀 + 𝑁) ∈ ℤ
12		fzfig 10845	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (1...(𝑀 + 𝑁)) ∈ Fin)
13	8, 11, 12	mp2an 430	. . . . . . . . . . . . . 14 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
14		fidceq 7161	. . . . . . . . . . . . . 14 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑥 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → DECID 𝑥 = 𝑦)
15	13, 14	mp3an1 1365	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → DECID 𝑥 = 𝑦)
16	15	rgen2 2636	. . . . . . . . . . . 12 ⊢ ∀𝑥 ∈ (1...(𝑀 + 𝑁))∀𝑦 ∈ (1...(𝑀 + 𝑁))DECID 𝑥 = 𝑦
17	16	a1i 9	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝑂 → ∀𝑥 ∈ (1...(𝑀 + 𝑁))∀𝑦 ∈ (1...(𝑀 + 𝑁))DECID 𝑥 = 𝑦)
18		nnuz 9937	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
19	10, 18	eleqtri 2313	. . . . . . . . . . . 12 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘1)
20		eluzfz1 10414	. . . . . . . . . . . 12 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) → 1 ∈ (1...(𝑀 + 𝑁)))
21	19, 20	mp1i 10	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝑂 → 1 ∈ (1...(𝑀 + 𝑁)))
22	3	reqabi 2728	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝑂 ↔ (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ (♯‘𝑐) = 𝑀))
23	22	simplbi 274	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝑂 → 𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin))
24		elin 3412	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ 𝑐 ∈ Fin))
25	23, 24	sylib 122	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝑂 → (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ 𝑐 ∈ Fin))
26	25	simpld 112	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝑂 → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
27	26	elpwid 3696	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝑂 → 𝑐 ⊆ (1...(𝑀 + 𝑁)))
28	25	simprd 114	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝑂 → 𝑐 ∈ Fin)
29	17, 21, 27, 28	elssdc 7199	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝑂 → DECID 1 ∈ 𝑐)
30		dcn 854	. . . . . . . . . 10 ⊢ (DECID 1 ∈ 𝑐 → DECID ¬ 1 ∈ 𝑐)
31	29, 30	syl 14	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝑂 → DECID ¬ 1 ∈ 𝑐)
32	31	rgen 2603	. . . . . . . 8 ⊢ ∀𝑐 ∈ 𝑂 DECID ¬ 1 ∈ 𝑐
33	32	a1i 9	. . . . . . 7 ⊢ (⊤ → ∀𝑐 ∈ 𝑂 DECID ¬ 1 ∈ 𝑐)
34	7, 33	ssfirab 7234	. . . . . 6 ⊢ (⊤ → {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ Fin)
35	34	mptru 1411	. . . . 5 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ Fin
36	6, 35	elini 3413	. . . 4 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ (𝒫 𝑂 ∩ Fin)
37		fveq2 5690	. . . . . 6 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}))
38	37	oveq1d 6090	. . . . 5 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
39		ballotfilem.p	. . . . 5 ⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
40		hashcl 11198	. . . . . . . 8 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ Fin → (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ₀)
41	35, 40	ax-mp 5	. . . . . . 7 ⊢ (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ₀
42	1, 2, 3	ballotfilemonn 13199	. . . . . . 7 ⊢ (♯‘𝑂) ∈ ℕ
43		nn0nndivcl 9608	. . . . . . 7 ⊢ (((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ₀ ∧ (♯‘𝑂) ∈ ℕ) → ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ ℝ)
44	41, 42, 43	mp2an 430	. . . . . 6 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ ℝ
45	44	elexi 2834	. . . . 5 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
46	38, 39, 45	fvmpt 5776	. . . 4 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ (𝒫 𝑂 ∩ Fin) → (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
47	36, 46	ax-mp 5	. . 3 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
48		an32 568	. . . . . . . 8 ⊢ (((𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ ¬ 1 ∈ 𝑐) ∧ (♯‘𝑐) = 𝑀) ↔ ((𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ (♯‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
49		2eluzge1 9955	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (ℤ_≥‘1)
50		fzss1 10447	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ (ℤ_≥‘1) → (2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁)))
51	49, 50	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁))
52	51	sspwi 3699	. . . . . . . . . . . . 13 ⊢ 𝒫 (2...(𝑀 + 𝑁)) ⊆ 𝒫 (1...(𝑀 + 𝑁))
53		elinel1 3415	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
54	52, 53	sselid 3246	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
55		elinel2 3416	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ Fin)
56	54, 55	elind 3414	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin))
57		1lt2 9453	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 2
58		2z 9651	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℤ
59		zltnle 9669	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℤ ∧ 2 ∈ ℤ) → (1 < 2 ↔ ¬ 2 ≤ 1))
60	8, 58, 59	mp2an 430	. . . . . . . . . . . . . . . . 17 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
61	57, 60	mpbi 145	. . . . . . . . . . . . . . . 16 ⊢ ¬ 2 ≤ 1
62		elfzle1 10410	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (2...(𝑀 + 𝑁)) → 2 ≤ 1)
63	61, 62	mto 672	. . . . . . . . . . . . . . 15 ⊢ ¬ 1 ∈ (2...(𝑀 + 𝑁))
64		elelpwi 3697	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) → 1 ∈ (2...(𝑀 + 𝑁)))
65	63, 64	mto 672	. . . . . . . . . . . . . 14 ⊢ ¬ (1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
66		ancom 266	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) ↔ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐))
67	65, 66	mtbi 681	. . . . . . . . . . . . 13 ⊢ ¬ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐)
68	67	imnani 702	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → ¬ 1 ∈ 𝑐)
69	53, 68	syl 14	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → ¬ 1 ∈ 𝑐)
70	56, 69	jca 306	. . . . . . . . . 10 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ ¬ 1 ∈ 𝑐))
71		elinel1 3415	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
72		velpw 3692	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (1...(𝑀 + 𝑁)))
73		ssab 3318	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1} ↔ ∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1))
74		eqid 2238	. . . . . . . . . . . . . . . . 17 ⊢ 1 = 1
75		1ex 8311	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ V
76		eleq1 2301	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 1 → (𝑖 ∈ 𝑐 ↔ 1 ∈ 𝑐))
77		eqeq1 2245	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 1 → (𝑖 = 1 ↔ 1 = 1))
78	77	notbid 677	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 1 → (¬ 𝑖 = 1 ↔ ¬ 1 = 1))
79	76, 78	imbi12d 234	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 1 → ((𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ (1 ∈ 𝑐 → ¬ 1 = 1)))
80	75, 79	spcv 2919	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) → (1 ∈ 𝑐 → ¬ 1 = 1))
81	74, 80	mt2i 653	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) → ¬ 1 ∈ 𝑐)
82		simpr 110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((¬ 1 ∈ 𝑐 ∧ 𝑖 ∈ 𝑐) ∧ 𝑖 = 1) → 𝑖 = 1)
83		simplr 533	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((¬ 1 ∈ 𝑐 ∧ 𝑖 ∈ 𝑐) ∧ 𝑖 = 1) → 𝑖 ∈ 𝑐)
84	82, 83	eqeltrrd 2316	. . . . . . . . . . . . . . . . . . 19 ⊢ (((¬ 1 ∈ 𝑐 ∧ 𝑖 ∈ 𝑐) ∧ 𝑖 = 1) → 1 ∈ 𝑐)
85		simpll 531	. . . . . . . . . . . . . . . . . . 19 ⊢ (((¬ 1 ∈ 𝑐 ∧ 𝑖 ∈ 𝑐) ∧ 𝑖 = 1) → ¬ 1 ∈ 𝑐)
86	84, 85	pm2.65da 671	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 1 ∈ 𝑐 ∧ 𝑖 ∈ 𝑐) → ¬ 𝑖 = 1)
87	86	ex 115	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 1 ∈ 𝑐 → (𝑖 ∈ 𝑐 → ¬ 𝑖 = 1))
88	87	alrimiv 1927	. . . . . . . . . . . . . . . 16 ⊢ (¬ 1 ∈ 𝑐 → ∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1))
89	81, 88	impbii 126	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) ↔ ¬ 1 ∈ 𝑐)
90	73, 89	bitr2i 185	. . . . . . . . . . . . . 14 ⊢ (¬ 1 ∈ 𝑐 ↔ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1})
91		ssin 3453	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) ↔ 𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}))
92		1le2 9492	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ≤ 2
93		1p1e2 9400	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 + 1) = 2
94		nnge1 9306	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
95	1, 94	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ≤ 𝑀
96		nnge1 9306	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
97	2, 96	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ≤ 𝑁
98		1re 8315	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 1 ∈ ℝ
99	1	nnrei 9292	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑀 ∈ ℝ
100	2	nnrei 9292	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑁 ∈ ℝ
101	98, 98, 99, 100	le2addi 8829	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1 ≤ 𝑀 ∧ 1 ≤ 𝑁) → (1 + 1) ≤ (𝑀 + 𝑁))
102	95, 97, 101	mp2an 430	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 + 1) ≤ (𝑀 + 𝑁)
103	93, 102	eqbrtrri 4148	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 2 ≤ (𝑀 + 𝑁)
104		2re 9353	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℝ
105	99, 100	readdcli 8329	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 + 𝑁) ∈ ℝ
106	98, 104, 105	letri 8423	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ≤ 2 ∧ 2 ≤ (𝑀 + 𝑁)) → 1 ≤ (𝑀 + 𝑁))
107	92, 103, 106	mp2an 430	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ≤ (𝑀 + 𝑁)
108		eluz 9914	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) ↔ 1 ≤ (𝑀 + 𝑁)))
109	8, 11, 108	mp2an 430	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) ↔ 1 ≤ (𝑀 + 𝑁))
110	107, 109	mpbir 146	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘1)
111		elfzp12 10484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↔ (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁)))))
112	110, 111	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↔ (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁))))
113	112	biimpi 120	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) → (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁))))
114	113	orcanai 940	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁)))
115	93	oveq1i 6085	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 + 1)...(𝑀 + 𝑁)) = (2...(𝑀 + 𝑁))
116	114, 115	eleqtrdi 2331	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ (2...(𝑀 + 𝑁)))
117	116	ss2abi 3320	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} ⊆ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))}
118		inab 3499	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)}
119		abid2 2361	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} = (1...(𝑀 + 𝑁))
120	119	ineq1i 3428	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
121	118, 120	eqtr3i 2261	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
122		abid2 2361	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))} = (2...(𝑀 + 𝑁))
123	117, 121, 122	3sstr3i 3288	. . . . . . . . . . . . . . . 16 ⊢ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))
124		sstr 3256	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ∧ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
125	123, 124	mpan2 429	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
126	91, 125	sylbi 121	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
127	72, 90, 126	syl2anb 291	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
128	71, 127	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ ¬ 1 ∈ 𝑐) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
129		velpw 3692	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (2...(𝑀 + 𝑁)))
130	128, 129	sylibr 134	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ ¬ 1 ∈ 𝑐) → 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
131		elinel2 3416	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ Fin)
132	131	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ ¬ 1 ∈ 𝑐) → 𝑐 ∈ Fin)
133	130, 132	elind 3414	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ ¬ 1 ∈ 𝑐) → 𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin))
134	70, 133	impbii 126	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ↔ (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ ¬ 1 ∈ 𝑐))
135	134	anbi1i 462	. . . . . . . 8 ⊢ ((𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∧ (♯‘𝑐) = 𝑀) ↔ ((𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ ¬ 1 ∈ 𝑐) ∧ (♯‘𝑐) = 𝑀))
136	22	anbi1i 462	. . . . . . . 8 ⊢ ((𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐) ↔ ((𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ (♯‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
137	48, 135, 136	3bitr4i 212	. . . . . . 7 ⊢ ((𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∧ (♯‘𝑐) = 𝑀) ↔ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐))
138	137	rabbia2 2806	. . . . . 6 ⊢ {𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀} = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
139	138	fveq2i 5693	. . . . 5 ⊢ (♯‘{𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐})
140		fzfig 10845	. . . . . . . 8 ⊢ ((2 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (2...(𝑀 + 𝑁)) ∈ Fin)
141	58, 11, 140	mp2an 430	. . . . . . 7 ⊢ (2...(𝑀 + 𝑁)) ∈ Fin
142	1	nnzi 9644	. . . . . . 7 ⊢ 𝑀 ∈ ℤ
143		hashfibc 11261	. . . . . . 7 ⊢ (((2...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑀 ∈ ℤ) → ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}))
144	141, 142, 143	mp2an 430	. . . . . 6 ⊢ ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀})
145	58	eluz1i 9908	. . . . . . . . . . 11 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘2) ↔ ((𝑀 + 𝑁) ∈ ℤ ∧ 2 ≤ (𝑀 + 𝑁)))
146	11, 103, 145	mpbir2an 955	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘2)
147		hashfz 11240	. . . . . . . . . 10 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘2) → (♯‘(2...(𝑀 + 𝑁))) = (((𝑀 + 𝑁) − 2) + 1))
148	146, 147	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘(2...(𝑀 + 𝑁))) = (((𝑀 + 𝑁) − 2) + 1)
149	1	nncni 9293	. . . . . . . . . . 11 ⊢ 𝑀 ∈ ℂ
150	2	nncni 9293	. . . . . . . . . . 11 ⊢ 𝑁 ∈ ℂ
151	149, 150	addcli 8320	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℂ
152		2cn 9354	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
153		ax-1cn 8262	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
154		subadd23 8528	. . . . . . . . . 10 ⊢ (((𝑀 + 𝑁) ∈ ℂ ∧ 2 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2)))
155	151, 152, 153, 154	mp3an 1378	. . . . . . . . 9 ⊢ (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2))
156	152, 153	negsubdi2i 8602	. . . . . . . . . . 11 ⊢ -(2 − 1) = (1 − 2)
157		2m1e1 9401	. . . . . . . . . . . 12 ⊢ (2 − 1) = 1
158	157	negeqi 8510	. . . . . . . . . . 11 ⊢ -(2 − 1) = -1
159	156, 158	eqtr3i 2261	. . . . . . . . . 10 ⊢ (1 − 2) = -1
160	159	oveq2i 6086	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) + (1 − 2)) = ((𝑀 + 𝑁) + -1)
161	148, 155, 160	3eqtri 2263	. . . . . . . 8 ⊢ (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) + -1)
162	151, 153	negsubi 8594	. . . . . . . 8 ⊢ ((𝑀 + 𝑁) + -1) = ((𝑀 + 𝑁) − 1)
163	161, 162	eqtri 2259	. . . . . . 7 ⊢ (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) − 1)
164	163	oveq1i 6085	. . . . . 6 ⊢ ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (((𝑀 + 𝑁) − 1)C𝑀)
165	144, 164	eqtr3i 2261	. . . . 5 ⊢ (♯‘{𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}) = (((𝑀 + 𝑁) − 1)C𝑀)
166	139, 165	eqtr3i 2261	. . . 4 ⊢ (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (((𝑀 + 𝑁) − 1)C𝑀)
167	1, 2, 3	ballotfilem1 13198	. . . 4 ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
168	166, 167	oveq12i 6087	. . 3 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
169	47, 168	eqtri 2259	. 2 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
170		0le1 8799	. . . . 5 ⊢ 0 ≤ 1
171		0re 8316	. . . . . 6 ⊢ 0 ∈ ℝ
172	171, 98, 99	letri 8423	. . . . 5 ⊢ ((0 ≤ 1 ∧ 1 ≤ 𝑀) → 0 ≤ 𝑀)
173	170, 95, 172	mp2an 430	. . . 4 ⊢ 0 ≤ 𝑀
174	2	nngt0i 9313	. . . . . 6 ⊢ 0 < 𝑁
175	100, 174	elrpii 10036	. . . . 5 ⊢ 𝑁 ∈ ℝ⁺
176		ltaddrp 10071	. . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → 𝑀 < (𝑀 + 𝑁))
177	99, 175, 176	mp2an 430	. . . 4 ⊢ 𝑀 < (𝑀 + 𝑁)
178		0z 9634	. . . . 5 ⊢ 0 ∈ ℤ
179		elfzm11 10476	. . . . 5 ⊢ ((0 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < (𝑀 + 𝑁))))
180	178, 11, 179	mp2an 430	. . . 4 ⊢ (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < (𝑀 + 𝑁)))
181	142, 173, 177, 180	mpbir3an 1210	. . 3 ⊢ 𝑀 ∈ (0...((𝑀 + 𝑁) − 1))
182		bcm1n 11185	. . 3 ⊢ ((𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ∧ (𝑀 + 𝑁) ∈ ℕ) → ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀)) = (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)))
183	181, 10, 182	mp2an 430	. 2 ⊢ ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀)) = (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁))
184		pncan2 8523	. . . 4 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
185	149, 150, 184	mp2an 430	. . 3 ⊢ ((𝑀 + 𝑁) − 𝑀) = 𝑁
186	185	oveq1i 6085	. 2 ⊢ (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)) = (𝑁 / (𝑀 + 𝑁))
187	169, 183, 186	3eqtri 2263	1 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 DECID wdc 846 ∧ w3a 1009 ∀wal 1400 = wceq 1402 ⊤wtru 1403 ∈ wcel 2209 {cab 2224 ∀wral 2528 {crab 2532 ∩ cin 3219 ⊆ wss 3220 𝒫 cpw 3685 class class class wbr 4125 ↦ cmpt 4187 ‘cfv 5372 (class class class)co 6075 Fincfn 7012 ℂcc 8167 ℝcr 8168 0cc0 8169 1c1 8170 + caddc 8172 < clt 8350 ≤ cle 8351 − cmin 8487 -cneg 8488 / cdiv 8992 ℕcn 9283 2c2 9334 ℕ₀cn0 9542 ℤcz 9623 ℤ_≥cuz 9900 ℝ⁺crp 10033 ...cfz 10390 Ccbc 11163 ♯chash 11192

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-13 2211 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-frec 6652 df-1o 6677 df-2o 6678 df-oadd 6681 df-er 6797 df-map 6914 df-en 7013 df-dom 7014 df-fin 7015 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-n0 9543 df-z 9624 df-uz 9901 df-q 9999 df-rp 10034 df-fz 10391 df-seqfrec 10863 df-fac 11142 df-bc 11164 df-ihash 11193

This theorem is referenced by: ballotfilemth 13259

W3C validator