ballotfilem2 - Intuitionistic Logic Explorer

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Theorem ballotfilem2 13172

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 13163	. . . . . 6 ⊢ 𝑂 ∈ Fin
5		ssrab2 3327	. . . . . 6 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
6	4, 5	elpwi2 4275	. . . . 5 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
7	4	a1i 9	. . . . . . 7 ⊢ (⊤ → 𝑂 ∈ Fin)
8		1z 9620	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℤ
9		nnaddcl 9274	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
10	1, 2, 9	mp2an 426	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 + 𝑁) ∈ ℕ
11	10	nnzi 9615	. . . . . . . . . . . . . . 15 ⊢ (𝑀 + 𝑁) ∈ ℤ
12		fzfig 10816	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (1...(𝑀 + 𝑁)) ∈ Fin)
13	8, 11, 12	mp2an 426	. . . . . . . . . . . . . 14 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
14		fidceq 7137	. . . . . . . . . . . . . 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 2630	. . . . . . . . . . . 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 9908	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
19	10, 18	eleqtri 2309	. . . . . . . . . . . 12 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘1)
20		eluzfz1 10385	. . . . . . . . . . . 12 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘1) → 1 ∈ (1...(𝑀 + 𝑁)))
21	19, 20	mp1i 10	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝑂 → 1 ∈ (1...(𝑀 + 𝑁)))
22	3	reqabi 2722	. . . . . . . . . . . . . . 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 3406	. . . . . . . . . . . . . 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 3685	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝑂 → 𝑐 ⊆ (1...(𝑀 + 𝑁)))
28	25	simprd 114	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝑂 → 𝑐 ∈ Fin)
29	17, 21, 27, 28	elssdc 7175	. . . . . . . . . 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 2597	. . . . . . . 8 ⊢ ∀𝑐 ∈ 𝑂 DECID ¬ 1 ∈ 𝑐
33	32	a1i 9	. . . . . . 7 ⊢ (⊤ → ∀𝑐 ∈ 𝑂 DECID ¬ 1 ∈ 𝑐)
34	7, 33	ssfirab 7210	. . . . . 6 ⊢ (⊤ → {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ Fin)
35	34	mptru 1407	. . . . 5 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ Fin
36	6, 35	elini 3407	. . . 4 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ (𝒫 𝑂 ∩ Fin)
37		fveq2 5675	. . . . . 6 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}))
38	37	oveq1d 6073	. . . . 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 11169	. . . . . . . 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 13165	. . . . . . 7 ⊢ (♯‘𝑂) ∈ ℕ
43		nn0nndivcl 9579	. . . . . . 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 2828	. . . . 5 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
46	38, 39, 45	fvmpt 5759	. . . 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 9926	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (ℤ_≥‘1)
50		fzss1 10418	. . . . . . . . . . . . . . 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 3688	. . . . . . . . . . . . 13 ⊢ 𝒫 (2...(𝑀 + 𝑁)) ⊆ 𝒫 (1...(𝑀 + 𝑁))
53		elinel1 3409	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
54	52, 53	sselid 3240	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
55		elinel2 3410	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ Fin)
56	54, 55	elind 3408	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin))
57		1lt2 9424	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 2
58		2z 9622	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℤ
59		zltnle 9640	. . . . . . . . . . . . . . . . . 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 10381	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (2...(𝑀 + 𝑁)) → 2 ≤ 1)
63	61, 62	mto 668	. . . . . . . . . . . . . . 15 ⊢ ¬ 1 ∈ (2...(𝑀 + 𝑁))
64		elelpwi 3686	. . . . . . . . . . . . . . 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 3409	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
72		velpw 3681	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (1...(𝑀 + 𝑁)))
73		ssab 3312	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1} ↔ ∀𝑖(𝑖 ∈ 𝑐 → ¬ 𝑖 = 1))
74		eqid 2234	. . . . . . . . . . . . . . . . 17 ⊢ 1 = 1
75		1ex 8285	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ V
76		eleq1 2297	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 1 → (𝑖 ∈ 𝑐 ↔ 1 ∈ 𝑐))
77		eqeq1 2241	. . . . . . . . . . . . . . . . . . . 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 2913	. . . . . . . . . . . . . . . . 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 2312	. . . . . . . . . . . . . . . . . . 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 3447	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) ↔ 𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}))
92		1le2 9463	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ≤ 2
93		1p1e2 9371	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 + 1) = 2
94		nnge1 9277	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
95	1, 94	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ≤ 𝑀
96		nnge1 9277	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
97	2, 96	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ≤ 𝑁
98		1re 8289	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 1 ∈ ℝ
99	1	nnrei 9263	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑀 ∈ ℝ
100	2	nnrei 9263	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑁 ∈ ℝ
101	98, 98, 99, 100	le2addi 8802	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1 ≤ 𝑀 ∧ 1 ≤ 𝑁) → (1 + 1) ≤ (𝑀 + 𝑁))
102	95, 97, 101	mp2an 426	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 + 1) ≤ (𝑀 + 𝑁)
103	93, 102	eqbrtrri 4137	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 2 ≤ (𝑀 + 𝑁)
104		2re 9324	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℝ
105	99, 100	readdcli 8303	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 + 𝑁) ∈ ℝ
106	98, 104, 105	letri 8397	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ≤ 2 ∧ 2 ≤ (𝑀 + 𝑁)) → 1 ≤ (𝑀 + 𝑁))
107	92, 103, 106	mp2an 426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ≤ (𝑀 + 𝑁)
108		eluz 9885	. . . . . . . . . . . . . . . . . . . . . . . 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 10455	. . . . . . . . . . . . . . . . . . . . . 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 6068	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 + 1)...(𝑀 + 𝑁)) = (2...(𝑀 + 𝑁))
116	114, 115	eleqtrdi 2327	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ (2...(𝑀 + 𝑁)))
117	116	ss2abi 3314	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} ⊆ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))}
118		inab 3493	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)}
119		abid2 2357	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} = (1...(𝑀 + 𝑁))
120	119	ineq1i 3422	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑖 ∣ 𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
121	118, 120	eqtr3i 2257	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
122		abid2 2357	. . . . . . . . . . . . . . . . 17 ⊢ {𝑖 ∣ 𝑖 ∈ (2...(𝑀 + 𝑁))} = (2...(𝑀 + 𝑁))
123	117, 121, 122	3sstr3i 3282	. . . . . . . . . . . . . . . 16 ⊢ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))
124		sstr 3250	. . . . . . . . . . . . . . . 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 3681	. . . . . . . . . . . 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 3410	. . . . . . . . . . . 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 3408	. . . . . . . . . 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 2800	. . . . . 6 ⊢ {𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀} = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
139	138	fveq2i 5678	. . . . 5 ⊢ (♯‘{𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐})
140		fzfig 10816	. . . . . . . 8 ⊢ ((2 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (2...(𝑀 + 𝑁)) ∈ Fin)
141	58, 11, 140	mp2an 426	. . . . . . 7 ⊢ (2...(𝑀 + 𝑁)) ∈ Fin
142	1	nnzi 9615	. . . . . . 7 ⊢ 𝑀 ∈ ℤ
143		hashfibc 11232	. . . . . . 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 9879	. . . . . . . . . . 11 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘2) ↔ ((𝑀 + 𝑁) ∈ ℤ ∧ 2 ≤ (𝑀 + 𝑁)))
146	11, 103, 145	mpbir2an 951	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ (ℤ_≥‘2)
147		hashfz 11211	. . . . . . . . . 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 9264	. . . . . . . . . . 11 ⊢ 𝑀 ∈ ℂ
150	2	nncni 9264	. . . . . . . . . . 11 ⊢ 𝑁 ∈ ℂ
151	149, 150	addcli 8294	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℂ
152		2cn 9325	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
153		ax-1cn 8236	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
154		subadd23 8501	. . . . . . . . . 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 8575	. . . . . . . . . . 11 ⊢ -(2 − 1) = (1 − 2)
157		2m1e1 9372	. . . . . . . . . . . 12 ⊢ (2 − 1) = 1
158	157	negeqi 8483	. . . . . . . . . . 11 ⊢ -(2 − 1) = -1
159	156, 158	eqtr3i 2257	. . . . . . . . . 10 ⊢ (1 − 2) = -1
160	159	oveq2i 6069	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) + (1 − 2)) = ((𝑀 + 𝑁) + -1)
161	148, 155, 160	3eqtri 2259	. . . . . . . 8 ⊢ (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) + -1)
162	151, 153	negsubi 8567	. . . . . . . 8 ⊢ ((𝑀 + 𝑁) + -1) = ((𝑀 + 𝑁) − 1)
163	161, 162	eqtri 2255	. . . . . . 7 ⊢ (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) − 1)
164	163	oveq1i 6068	. . . . . 6 ⊢ ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (((𝑀 + 𝑁) − 1)C𝑀)
165	144, 164	eqtr3i 2257	. . . . 5 ⊢ (♯‘{𝑐 ∈ (𝒫 (2...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}) = (((𝑀 + 𝑁) − 1)C𝑀)
166	139, 165	eqtr3i 2257	. . . 4 ⊢ (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (((𝑀 + 𝑁) − 1)C𝑀)
167	1, 2, 3	ballotfilem1 13164	. . . 4 ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
168	166, 167	oveq12i 6070	. . 3 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
169	47, 168	eqtri 2255	. 2 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
170		0le1 8772	. . . . 5 ⊢ 0 ≤ 1
171		0re 8290	. . . . . 6 ⊢ 0 ∈ ℝ
172	171, 98, 99	letri 8397	. . . . 5 ⊢ ((0 ≤ 1 ∧ 1 ≤ 𝑀) → 0 ≤ 𝑀)
173	170, 95, 172	mp2an 426	. . . 4 ⊢ 0 ≤ 𝑀
174	2	nngt0i 9284	. . . . . 6 ⊢ 0 < 𝑁
175	100, 174	elrpii 10007	. . . . 5 ⊢ 𝑁 ∈ ℝ⁺
176		ltaddrp 10042	. . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → 𝑀 < (𝑀 + 𝑁))
177	99, 175, 176	mp2an 426	. . . 4 ⊢ 𝑀 < (𝑀 + 𝑁)
178		0z 9605	. . . . 5 ⊢ 0 ∈ ℤ
179		elfzm11 10447	. . . . 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 11156	. . 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 8496	. . . 4 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
185	149, 150, 184	mp2an 426	. . 3 ⊢ ((𝑀 + 𝑁) − 𝑀) = 𝑁
186	185	oveq1i 6068	. 2 ⊢ (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)) = (𝑁 / (𝑀 + 𝑁))
187	169, 183, 186	3eqtri 2259	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 2205 {cab 2220 ∀wral 2522 {crab 2526 ∩ cin 3213 ⊆ wss 3214 𝒫 cpw 3674 class class class wbr 4114 ↦ cmpt 4176 ‘cfv 5357 (class class class)co 6058 Fincfn 6988 ℂcc 8141 ℝcr 8142 0cc0 8143 1c1 8144 + caddc 8146 < clt 8324 ≤ cle 8325 − cmin 8460 -cneg 8461 / cdiv 8963 ℕcn 9254 2c2 9305 ℕ₀cn0 9513 ℤcz 9594 ℤ_≥cuz 9871 ℝ⁺crp 10004 ...cfz 10361 Ccbc 11134 ♯chash 11163

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-2o 6661 df-oadd 6664 df-er 6780 df-map 6897 df-en 6989 df-dom 6990 df-fin 6991 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-fz 10362 df-seqfrec 10834 df-fac 11113 df-bc 11135 df-ihash 11164

This theorem is referenced by: ballotfilemth 13225

W3C validator