ballotfilem2 - Intuitionistic Logic Explorer

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Theorem ballotfilem2 13142

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
ballotfi.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
ballotfi.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		ballotfi.o	. . . . . . 7 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
4	1, 2, 3	ballotfilemofi 13138	. . . . . 6 ⊢ 𝑂 ∈ Fin
5		ssrab2 3323	. . . . . 6 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
6	4, 5	elpwi2 4270	. . . . 5 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
7	4	a1i 9	. . . . . . 7 ⊢ (⊤ → 𝑂 ∈ Fin)
8		1z 9603	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℤ
9		nnaddcl 9257	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
10	1, 2, 9	mp2an 426	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 + 𝑁) ∈ ℕ
11	10	nnzi 9598	. . . . . . . . . . . . . . 15 ⊢ (𝑀 + 𝑁) ∈ ℤ
12		fzfig 10792	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (1...(𝑀 + 𝑁)) ∈ Fin)
13	8, 11, 12	mp2an 426	. . . . . . . . . . . . . 14 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
14		fidceq 7124	. . . . . . . . . . . . . 14 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑥 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → DECID 𝑥 = 𝑦)
15	13, 14	mp3an1 1361	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑦 ∈ (1...(𝑀 + 𝑁))) → DECID 𝑥 = 𝑦)
16	15	rgen2 2628	. . . . . . . . . . . 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 9890	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
19	10, 18	eleqtri 2307	. . . . . . . . . . . 12 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘1)
20		eluzfz1 10365	. . . . . . . . . . . 12 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) → 1 ∈ (1...(𝑀 + 𝑁)))
21	19, 20	mp1i 10	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝑂 → 1 ∈ (1...(𝑀 + 𝑁)))
22	3	reqabi 2720	. . . . . . . . . . . . . . 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 3402	. . . . . . . . . . . . . 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 3680	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝑂 → 𝑐 ⊆ (1...(𝑀 + 𝑁)))
28	25	simprd 114	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝑂 → 𝑐 ∈ Fin)
29	17, 21, 27, 28	elssdc 7162	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝑂 → DECID 1 ∈ 𝑐)
30		dcn 850	. . . . . . . . . 10 ⊢ (DECID 1 ∈ 𝑐 → DECID ¬ 1 ∈ 𝑐)
31	29, 30	syl 14	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝑂 → DECID ¬ 1 ∈ 𝑐)
32	31	rgen 2595	. . . . . . . 8 ⊢ ∀𝑐 ∈ 𝑂 DECID ¬ 1 ∈ 𝑐
33	32	a1i 9	. . . . . . 7 ⊢ (⊤ → ∀𝑐 ∈ 𝑂 DECID ¬ 1 ∈ 𝑐)
34	7, 33	ssfirab 7197	. . . . . 6 ⊢ (⊤ → {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ Fin)
35	34	mptru 1407	. . . . 5 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ Fin
36	6, 35	elini 3403	. . . 4 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ (𝒫 𝑂 ∩ Fin)
37		fveq2 5670	. . . . . 6 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}))
38	37	oveq1d 6065	. . . . 5 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
39		ballotfi.p	. . . . 5 ⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
40		hashcl 11144	. . . . . . . 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 13140	. . . . . . 7 ⊢ (♯‘𝑂) ∈ ℕ
43		nn0nndivcl 9562	. . . . . . 7 ⊢ (((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ₀ ∧ (♯‘𝑂) ∈ ℕ) → ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ ℝ)
44	41, 42, 43	mp2an 426	. . . . . 6 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ ℝ
45	44	elexi 2826	. . . . 5 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
46	38, 39, 45	fvmpt 5754	. . . 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 564	. . . . . . . 8 ⊢ (((𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ ¬ 1 ∈ 𝑐) ∧ (♯‘𝑐) = 𝑀) ↔ ((𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ (♯‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
49		2eluzge1 9908	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (ℤ_≥‘1)
50		fzss1 10397	. . . . . . . . . . . . . . 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 3683	. . . . . . . . . . . . 13 ⊢ 𝒫 (2...(𝑀 + 𝑁)) ⊆ 𝒫 (1...(𝑀 + 𝑁))
53		elinel1 3405	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
54	52, 53	sselid 3236	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
55		elinel2 3406	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ Fin)
56	54, 55	elind 3404	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin))
57		1lt2 9407	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 2
58		2z 9605	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℤ
59		zltnle 9623	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℤ ∧ 2 ∈ ℤ) → (1 < 2 ↔ ¬ 2 ≤ 1))
60	8, 58, 59	mp2an 426	. . . . . . . . . . . . . . . . 17 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
61	57, 60	mpbi 145	. . . . . . . . . . . . . . . 16 ⊢ ¬ 2 ≤ 1
62		elfzle1 10361	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (2...(𝑀 + 𝑁)) → 2 ≤ 1)
63	61, 62	mto 668	. . . . . . . . . . . . . . 15 ⊢ ¬ 1 ∈ (2...(𝑀 + 𝑁))
64		elelpwi 3681	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) → 1 ∈ (2...(𝑀 + 𝑁)))
65	63, 64	mto 668	. . . . . . . . . . . . . 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 677	. . . . . . . . . . . . 13 ⊢ ¬ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐)
68	67	imnani 698	. . . . . . . . . . . 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 3405	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
72		velpw 3676	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (1...(𝑀 + 𝑁)))
73		ssab 3308	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1} ↔ ∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1))
74		eqid 2232	. . . . . . . . . . . . . . . . 17 ⊢ 1 = 1
75		1ex 8269	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ V
76		eleq1 2295	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 1 → (𝑖 ∈ 𝑐 ↔ 1 ∈ 𝑐))
77		eqeq1 2239	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 1 → (𝑖 = 1 ↔ 1 = 1))
78	77	notbid 673	. . . . . . . . . . . . . . . . . . 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 2911	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) → (1 ∈ 𝑐 → ¬ 1 = 1))
81	74, 80	mt2i 649	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1) → ¬ 1 ∈ 𝑐)
82		simpr 110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((¬ 1 ∈ 𝑐 ∧ 𝑖 ∈ 𝑐) ∧ 𝑖 = 1) → 𝑖 = 1)
83		simplr 529	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((¬ 1 ∈ 𝑐 ∧ 𝑖 ∈ 𝑐) ∧ 𝑖 = 1) → 𝑖 ∈ 𝑐)
84	82, 83	eqeltrrd 2310	. . . . . . . . . . . . . . . . . . 19 ⊢ (((¬ 1 ∈ 𝑐 ∧ 𝑖 ∈ 𝑐) ∧ 𝑖 = 1) → 1 ∈ 𝑐)
85		simpll 527	. . . . . . . . . . . . . . . . . . 19 ⊢ (((¬ 1 ∈ 𝑐 ∧ 𝑖 ∈ 𝑐) ∧ 𝑖 = 1) → ¬ 1 ∈ 𝑐)
86	84, 85	pm2.65da 667	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 1 ∈ 𝑐 ∧ 𝑖 ∈ 𝑐) → ¬ 𝑖 = 1)
87	86	ex 115	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 1 ∈ 𝑐 → (𝑖 ∈ 𝑐 → ¬ 𝑖 = 1))
88	87	alrimiv 1923	. . . . . . . . . . . . . . . 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 3443	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) ↔ 𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}))
92		1le2 9446	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ≤ 2
93		1p1e2 9354	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 + 1) = 2
94		nnge1 9260	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
95	1, 94	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ≤ 𝑀
96		nnge1 9260	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
97	2, 96	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ≤ 𝑁
98		1re 8273	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 1 ∈ ℝ
99	1	nnrei 9246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑀 ∈ ℝ
100	2	nnrei 9246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑁 ∈ ℝ
101	98, 98, 99, 100	le2addi 8785	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1 ≤ 𝑀 ∧ 1 ≤ 𝑁) → (1 + 1) ≤ (𝑀 + 𝑁))
102	95, 97, 101	mp2an 426	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 + 1) ≤ (𝑀 + 𝑁)
103	93, 102	eqbrtrri 4132	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 2 ≤ (𝑀 + 𝑁)
104		2re 9307	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℝ
105	99, 100	readdcli 8287	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 + 𝑁) ∈ ℝ
106	98, 104, 105	letri 8381	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ≤ 2 ∧ 2 ≤ (𝑀 + 𝑁)) → 1 ≤ (𝑀 + 𝑁))
107	92, 103, 106	mp2an 426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ≤ (𝑀 + 𝑁)
108		eluz 9867	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) ↔ 1 ≤ (𝑀 + 𝑁)))
109	8, 11, 108	mp2an 426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) ↔ 1 ≤ (𝑀 + 𝑁))
110	107, 109	mpbir 146	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘1)
111		elfzp12 10433	. . . . . . . . . . . . . . . . . . . . . 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 936	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁)))
115	93	oveq1i 6060	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 + 1)...(𝑀 + 𝑁)) = (2...(𝑀 + 𝑁))
116	114, 115	eleqtrdi 2325	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ (2...(𝑀 + 𝑁)))
117	116	ss2abi 3310	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} ⊆ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))}
118		inab 3489	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)}
119		abid2 2355	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} = (1...(𝑀 + 𝑁))
120	119	ineq1i 3418	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
121	118, 120	eqtr3i 2255	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
122		abid2 2355	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))} = (2...(𝑀 + 𝑁))
123	117, 121, 122	3sstr3i 3278	. . . . . . . . . . . . . . . 16 ⊢ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))
124		sstr 3246	. . . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . . . 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 3676	. . . . . . . . . . . 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 3406	. . . . . . . . . . . 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 3404	. . . . . . . . . 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 458	. . . . . . . 8 ⊢ ((𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∧ (♯‘𝑐) = 𝑀) ↔ ((𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∧ ¬ 1 ∈ 𝑐) ∧ (♯‘𝑐) = 𝑀))
136	22	anbi1i 458	. . . . . . . 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 2798	. . . . . 6 ⊢ {𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀} = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
139	138	fveq2i 5673	. . . . 5 ⊢ (♯‘{𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐})
140		fzfig 10792	. . . . . . . 8 ⊢ ((2 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (2...(𝑀 + 𝑁)) ∈ Fin)
141	58, 11, 140	mp2an 426	. . . . . . 7 ⊢ (2...(𝑀 + 𝑁)) ∈ Fin
142	1	nnzi 9598	. . . . . . 7 ⊢ 𝑀 ∈ ℤ
143		hashfibc 11207	. . . . . . 7 ⊢ (((2...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑀 ∈ ℤ) → ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}))
144	141, 142, 143	mp2an 426	. . . . . 6 ⊢ ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀})
145	58	eluz1i 9861	. . . . . . . . . . 11 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘2) ↔ ((𝑀 + 𝑁) ∈ ℤ ∧ 2 ≤ (𝑀 + 𝑁)))
146	11, 103, 145	mpbir2an 951	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘2)
147		hashfz 11186	. . . . . . . . . 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 9247	. . . . . . . . . . 11 ⊢ 𝑀 ∈ ℂ
150	2	nncni 9247	. . . . . . . . . . 11 ⊢ 𝑁 ∈ ℂ
151	149, 150	addcli 8278	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℂ
152		2cn 9308	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
153		ax-1cn 8220	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
154		subadd23 8485	. . . . . . . . . 10 ⊢ (((𝑀 + 𝑁) ∈ ℂ ∧ 2 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2)))
155	151, 152, 153, 154	mp3an 1374	. . . . . . . . 9 ⊢ (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2))
156	152, 153	negsubdi2i 8559	. . . . . . . . . . 11 ⊢ -(2 − 1) = (1 − 2)
157		2m1e1 9355	. . . . . . . . . . . 12 ⊢ (2 − 1) = 1
158	157	negeqi 8467	. . . . . . . . . . 11 ⊢ -(2 − 1) = -1
159	156, 158	eqtr3i 2255	. . . . . . . . . 10 ⊢ (1 − 2) = -1
160	159	oveq2i 6061	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) + (1 − 2)) = ((𝑀 + 𝑁) + -1)
161	148, 155, 160	3eqtri 2257	. . . . . . . 8 ⊢ (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) + -1)
162	151, 153	negsubi 8551	. . . . . . . 8 ⊢ ((𝑀 + 𝑁) + -1) = ((𝑀 + 𝑁) − 1)
163	161, 162	eqtri 2253	. . . . . . 7 ⊢ (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) − 1)
164	163	oveq1i 6060	. . . . . 6 ⊢ ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (((𝑀 + 𝑁) − 1)C𝑀)
165	144, 164	eqtr3i 2255	. . . . 5 ⊢ (♯‘{𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}) = (((𝑀 + 𝑁) − 1)C𝑀)
166	139, 165	eqtr3i 2255	. . . 4 ⊢ (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (((𝑀 + 𝑁) − 1)C𝑀)
167	1, 2, 3	ballotfilem1 13139	. . . 4 ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
168	166, 167	oveq12i 6062	. . 3 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
169	47, 168	eqtri 2253	. 2 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
170		0le1 8755	. . . . 5 ⊢ 0 ≤ 1
171		0re 8274	. . . . . 6 ⊢ 0 ∈ ℝ
172	171, 98, 99	letri 8381	. . . . 5 ⊢ ((0 ≤ 1 ∧ 1 ≤ 𝑀) → 0 ≤ 𝑀)
173	170, 95, 172	mp2an 426	. . . 4 ⊢ 0 ≤ 𝑀
174	2	nngt0i 9267	. . . . . 6 ⊢ 0 < 𝑁
175	100, 174	elrpii 9989	. . . . 5 ⊢ 𝑁 ∈ ℝ⁺
176		ltaddrp 10024	. . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → 𝑀 < (𝑀 + 𝑁))
177	99, 175, 176	mp2an 426	. . . 4 ⊢ 𝑀 < (𝑀 + 𝑁)
178		0z 9588	. . . . 5 ⊢ 0 ∈ ℤ
179		elfzm11 10425	. . . . 5 ⊢ ((0 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < (𝑀 + 𝑁))))
180	178, 11, 179	mp2an 426	. . . 4 ⊢ (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < (𝑀 + 𝑁)))
181	142, 173, 177, 180	mpbir3an 1206	. . 3 ⊢ 𝑀 ∈ (0...((𝑀 + 𝑁) − 1))
182		bcm1n 11131	. . 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 426	. 2 ⊢ ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀)) = (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁))
184		pncan2 8480	. . . 4 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
185	149, 150, 184	mp2an 426	. . 3 ⊢ ((𝑀 + 𝑁) − 𝑀) = 𝑁
186	185	oveq1i 6060	. 2 ⊢ (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)) = (𝑁 / (𝑀 + 𝑁))
187	169, 183, 186	3eqtri 2257	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 716 DECID wdc 842 ∧ w3a 1005 ∀wal 1396 = wceq 1398 ⊤wtru 1399 ∈ wcel 2203 {cab 2218 ∀wral 2520 {crab 2524 ∩ cin 3210 ⊆ wss 3211 𝒫 cpw 3669 class class class wbr 4109 ↦ cmpt 4171 ‘cfv 5352 (class class class)co 6050 Fincfn 6975 ℂcc 8125 ℝcr 8126 0cc0 8127 1c1 8128 + caddc 8130 < clt 8308 ≤ cle 8309 − cmin 8444 -cneg 8445 / cdiv 8946 ℕcn 9237 2c2 9288 ℕ₀cn0 9496 ℤcz 9577 ℤ_≥cuz 9853 ℝ⁺crp 9986 ...cfz 10342 Ccbc 11109 ♯chash 11138

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-irdg 6601 df-frec 6622 df-1o 6647 df-2o 6648 df-oadd 6651 df-er 6767 df-map 6884 df-en 6976 df-dom 6977 df-fin 6978 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-n0 9497 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-fz 10343 df-seqfrec 10810 df-fac 11088 df-bc 11110 df-ihash 11139

This theorem is referenced by: (None)

W3C validator