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Theorem ballotfilemth 13264

Description: Lemma for ballotfi 13265. The result, with several additional hypotheses which are for use during the proof. (Contributed by Thierry Arnoux, 7-Dec-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)))
ballotth.f	⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e	⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
ballotth.mgtn	⊢ 𝑁 < 𝑀
ballotth.i	⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s	⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r	⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))

**Assertion**
Ref	Expression
ballotfilemth	⊢ (𝑃‘𝐸) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))

Distinct variable groups: 𝑀,𝑐 𝑁,𝑐 𝑂,𝑐 𝑖,𝑀 𝑖,𝑁 𝑖,𝑂 𝑘,𝑀 𝑘,𝑁 𝑘,𝑂 𝑖,𝑐,𝐹,𝑘 𝑖,𝐸,𝑘 𝑘,𝐼,𝑐 𝐸,𝑐 𝑖,𝐼,𝑐 𝑆,𝑘,𝑖,𝑐 𝑅,𝑖,𝑘 𝑥,𝑐,𝐹 𝑥,𝑀 𝑥,𝑁,𝑘,𝑖 𝑥,𝐸 𝑥,𝑂 Allowed substitution hints: 𝑃(𝑥,𝑖,𝑘,𝑐) 𝑅(𝑥,𝑐) 𝑆(𝑥) 𝐼(𝑥)

**Proof of Theorem ballotfilemth**
Step	Hyp	Ref	Expression
1		ballotth.e	. . . . . . 7 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
2	1	ssrab3 3334	. . . . . 6 ⊢ 𝐸 ⊆ 𝑂
3		ballotth.m	. . . . . . . . 9 ⊢ 𝑀 ∈ ℕ
4		ballotth.n	. . . . . . . . 9 ⊢ 𝑁 ∈ ℕ
5		ballotfilem.o	. . . . . . . . 9 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
6		ballotfilem.p	. . . . . . . . 9 ⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
7		ballotth.f	. . . . . . . . 9 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
8	3, 4, 5, 6, 7, 1	ballotfilemefi 13220	. . . . . . . 8 ⊢ 𝐸 ∈ Fin
9	8	elexi 2834	. . . . . . 7 ⊢ 𝐸 ∈ V
10	9	elpw 3694	. . . . . 6 ⊢ (𝐸 ∈ 𝒫 𝑂 ↔ 𝐸 ⊆ 𝑂)
11	2, 10	mpbir 146	. . . . 5 ⊢ 𝐸 ∈ 𝒫 𝑂
12		elin 3412	. . . . . 6 ⊢ (𝐸 ∈ (𝒫 𝑂 ∩ Fin) ↔ (𝐸 ∈ 𝒫 𝑂 ∧ 𝐸 ∈ Fin))
13		fveq2 5693	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸))
14	13	oveq1d 6094	. . . . . . 7 ⊢ (𝑥 = 𝐸 → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂)))
15		hashcl 11203	. . . . . . . . . 10 ⊢ (𝐸 ∈ Fin → (♯‘𝐸) ∈ ℕ₀)
16	8, 15	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘𝐸) ∈ ℕ₀
17	3, 4, 5	ballotfilemonn 13204	. . . . . . . . 9 ⊢ (♯‘𝑂) ∈ ℕ
18		nn0nndivcl 9612	. . . . . . . . 9 ⊢ (((♯‘𝐸) ∈ ℕ₀ ∧ (♯‘𝑂) ∈ ℕ) → ((♯‘𝐸) / (♯‘𝑂)) ∈ ℝ)
19	16, 17, 18	mp2an 430	. . . . . . . 8 ⊢ ((♯‘𝐸) / (♯‘𝑂)) ∈ ℝ
20	19	elexi 2834	. . . . . . 7 ⊢ ((♯‘𝐸) / (♯‘𝑂)) ∈ V
21	14, 6, 20	fvmpt 5779	. . . . . 6 ⊢ (𝐸 ∈ (𝒫 𝑂 ∩ Fin) → (𝑃‘𝐸) = ((♯‘𝐸) / (♯‘𝑂)))
22	12, 21	sylbir 135	. . . . 5 ⊢ ((𝐸 ∈ 𝒫 𝑂 ∧ 𝐸 ∈ Fin) → (𝑃‘𝐸) = ((♯‘𝐸) / (♯‘𝑂)))
23	11, 8, 22	mp2an 430	. . . 4 ⊢ (𝑃‘𝐸) = ((♯‘𝐸) / (♯‘𝑂))
24	3, 4, 5	ballotfilemofi 13202	. . . . . . . 8 ⊢ 𝑂 ∈ Fin
25		fihashssdif 11242	. . . . . . . 8 ⊢ ((𝑂 ∈ Fin ∧ 𝐸 ∈ Fin ∧ 𝐸 ⊆ 𝑂) → (♯‘(𝑂 ∖ 𝐸)) = ((♯‘𝑂) − (♯‘𝐸)))
26	24, 8, 2, 25	mp3an 1378	. . . . . . 7 ⊢ (♯‘(𝑂 ∖ 𝐸)) = ((♯‘𝑂) − (♯‘𝐸))
27	26	eqcomi 2242	. . . . . 6 ⊢ ((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂 ∖ 𝐸))
28		hashcl 11203	. . . . . . . . 9 ⊢ (𝑂 ∈ Fin → (♯‘𝑂) ∈ ℕ₀)
29	24, 28	ax-mp 5	. . . . . . . 8 ⊢ (♯‘𝑂) ∈ ℕ₀
30	29	nn0cni 9558	. . . . . . 7 ⊢ (♯‘𝑂) ∈ ℂ
31	16	nn0cni 9558	. . . . . . 7 ⊢ (♯‘𝐸) ∈ ℂ
32		diffifi 7192	. . . . . . . . . 10 ⊢ ((𝑂 ∈ Fin ∧ 𝐸 ∈ Fin ∧ 𝐸 ⊆ 𝑂) → (𝑂 ∖ 𝐸) ∈ Fin)
33	24, 8, 2, 32	mp3an 1378	. . . . . . . . 9 ⊢ (𝑂 ∖ 𝐸) ∈ Fin
34		hashcl 11203	. . . . . . . . 9 ⊢ ((𝑂 ∖ 𝐸) ∈ Fin → (♯‘(𝑂 ∖ 𝐸)) ∈ ℕ₀)
35	33, 34	ax-mp 5	. . . . . . . 8 ⊢ (♯‘(𝑂 ∖ 𝐸)) ∈ ℕ₀
36	35	nn0cni 9558	. . . . . . 7 ⊢ (♯‘(𝑂 ∖ 𝐸)) ∈ ℂ
37	30, 31, 36	subsub23i 8610	. . . . . 6 ⊢ (((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂 ∖ 𝐸)) ↔ ((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) = (♯‘𝐸))
38	27, 37	mpbi 145	. . . . 5 ⊢ ((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) = (♯‘𝐸)
39	38	oveq1i 6089	. . . 4 ⊢ (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂))
40	23, 39	eqtr4i 2262	. . 3 ⊢ (𝑃‘𝐸) = (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂))
41	3, 4, 5	ballotfilem1 13203	. . . . . 6 ⊢ (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
42	3	nnnn0i 9554	. . . . . . . . 9 ⊢ 𝑀 ∈ ℕ₀
43		nnaddcl 9307	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
44	3, 4, 43	mp2an 430	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℕ
45	44	nnnn0i 9554	. . . . . . . . 9 ⊢ (𝑀 + 𝑁) ∈ ℕ₀
46	3	nnrei 9296	. . . . . . . . . 10 ⊢ 𝑀 ∈ ℝ
47	4	nnnn0i 9554	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℕ₀
48	46, 47	nn0addge1i 9594	. . . . . . . . 9 ⊢ 𝑀 ≤ (𝑀 + 𝑁)
49		elfz2nn0 10502	. . . . . . . . 9 ⊢ (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ (𝑀 ∈ ℕ₀ ∧ (𝑀 + 𝑁) ∈ ℕ₀ ∧ 𝑀 ≤ (𝑀 + 𝑁)))
50	42, 45, 48, 49	mpbir3an 1210	. . . . . . . 8 ⊢ 𝑀 ∈ (0...(𝑀 + 𝑁))
51		bccl2 11189	. . . . . . . 8 ⊢ (𝑀 ∈ (0...(𝑀 + 𝑁)) → ((𝑀 + 𝑁)C𝑀) ∈ ℕ)
52	50, 51	ax-mp 5	. . . . . . 7 ⊢ ((𝑀 + 𝑁)C𝑀) ∈ ℕ
53	52	nnap0i 9318	. . . . . 6 ⊢ ((𝑀 + 𝑁)C𝑀) # 0
54	41, 53	eqbrtri 4149	. . . . 5 ⊢ (♯‘𝑂) # 0
55	30, 54	pm3.2i 272	. . . 4 ⊢ ((♯‘𝑂) ∈ ℂ ∧ (♯‘𝑂) # 0)
56		divsubdirap 9032	. . . 4 ⊢ (((♯‘𝑂) ∈ ℂ ∧ (♯‘(𝑂 ∖ 𝐸)) ∈ ℂ ∧ ((♯‘𝑂) ∈ ℂ ∧ (♯‘𝑂) # 0)) → (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂)) = (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂))))
57	30, 36, 55, 56	mp3an 1378	. . 3 ⊢ (((♯‘𝑂) − (♯‘(𝑂 ∖ 𝐸))) / (♯‘𝑂)) = (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
58	30, 54	dividapi 9069	. . . 4 ⊢ ((♯‘𝑂) / (♯‘𝑂)) = 1
59	58	oveq1i 6089	. . 3 ⊢ (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂))) = (1 − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
60	40, 57, 59	3eqtri 2263	. 2 ⊢ (𝑃‘𝐸) = (1 − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
61		ballotth.mgtn	. . . . . . 7 ⊢ 𝑁 < 𝑀
62		ballotth.i	. . . . . . 7 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
63		ballotth.s	. . . . . . 7 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
64		ballotth.r	. . . . . . 7 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
65	3, 4, 5, 6, 7, 1, 61, 62, 63, 64	ballotfilem8 13263	. . . . . 6 ⊢ (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
66	65	oveq1i 6089	. . . . 5 ⊢ ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
67	66	oveq1i 6089	. . . 4 ⊢ (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
68		rabxmdc 3554	. . . . . . . 8 ⊢ (∀𝑐 ∈ (𝑂 ∖ 𝐸)DECID 1 ∈ 𝑐 → (𝑂 ∖ 𝐸) = ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
69		eldifi 3351	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → 𝑐 ∈ 𝑂)
70		1zzd 9654	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℤ)
71	3, 4, 5, 69, 70	ballotfilemcdc 13206	. . . . . . . 8 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → DECID 1 ∈ 𝑐)
72	68, 71	mprg 2607	. . . . . . 7 ⊢ (𝑂 ∖ 𝐸) = ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
73	72	fveq2i 5696	. . . . . 6 ⊢ (♯‘(𝑂 ∖ 𝐸)) = (♯‘({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
74	3, 4, 5, 6, 7, 1	ballotfilemafi 13221	. . . . . . 7 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∈ Fin
75	3, 4, 5, 6, 7, 1	ballotfilembfi 13222	. . . . . . 7 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin
76		rabnc 3555	. . . . . . 7 ⊢ ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∩ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) = ∅
77		hashun 11228	. . . . . . 7 ⊢ (({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∈ Fin ∧ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin ∧ ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∩ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) = ∅) → (♯‘({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})))
78	74, 75, 76, 77	mp3an 1378	. . . . . 6 ⊢ (♯‘({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
79	73, 78	eqtri 2259	. . . . 5 ⊢ (♯‘(𝑂 ∖ 𝐸)) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
80	79	oveq1i 6089	. . . 4 ⊢ ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
81		ssrab2 3333	. . . . . . . . . 10 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
82	24, 81	elpwi2 4292	. . . . . . . . 9 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
83	69	anim1i 340	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) → (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐))
84	3, 4, 5, 6, 7, 1	ballotfilem4 13224	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝑂 → (¬ 1 ∈ 𝑐 → ¬ 𝑐 ∈ 𝐸))
85	84	imdistani 449	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐) → (𝑐 ∈ 𝑂 ∧ ¬ 𝑐 ∈ 𝐸))
86		eldif 3229	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↔ (𝑐 ∈ 𝑂 ∧ ¬ 𝑐 ∈ 𝐸))
87	85, 86	sylibr 134	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐) → 𝑐 ∈ (𝑂 ∖ 𝐸))
88		simpr 110	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐) → ¬ 1 ∈ 𝑐)
89	87, 88	jca 306	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐) → (𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐))
90	83, 89	impbii 126	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) ↔ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐))
91	90	rabbia2 2806	. . . . . . . . . 10 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
92	91, 75	eqeltrri 2312	. . . . . . . . 9 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ Fin
93	82, 92	elini 3413	. . . . . . . 8 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ (𝒫 𝑂 ∩ Fin)
94		fveq2 5693	. . . . . . . . . 10 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}))
95	94	oveq1d 6094	. . . . . . . . 9 ⊢ (𝑥 = {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
96		hashcl 11203	. . . . . . . . . . . 12 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ Fin → (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ₀)
97	92, 96	ax-mp 5	. . . . . . . . . . 11 ⊢ (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ₀
98		nn0nndivcl 9612	. . . . . . . . . . 11 ⊢ (((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ₀ ∧ (♯‘𝑂) ∈ ℕ) → ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ ℝ)
99	97, 17, 98	mp2an 430	. . . . . . . . . 10 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ ℝ
100	99	elexi 2834	. . . . . . . . 9 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
101	95, 6, 100	fvmpt 5779	. . . . . . . 8 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ (𝒫 𝑂 ∩ Fin) → (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
102	93, 101	ax-mp 5	. . . . . . 7 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
103	3, 4, 5, 6	ballotfilem2 13211	. . . . . . 7 ⊢ (𝑃‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))
104		nfrab1 2732	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
105		nfrab1 2732	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
106	104, 105	dfssf 3238	. . . . . . . . . . 11 ⊢ ({𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ ∀𝑐(𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
107		rabid 2727	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐))
108	85, 107, 86	3imtr4i 201	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ (𝑂 ∖ 𝐸))
109	107	simprbi 275	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → ¬ 1 ∈ 𝑐)
110		rabid 2727	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐))
111	108, 109, 110	sylanbrc 421	. . . . . . . . . . 11 ⊢ (𝑐 ∈ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
112	106, 111	mpgbir 1506	. . . . . . . . . 10 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
113		difss 3355	. . . . . . . . . . 11 ⊢ (𝑂 ∖ 𝐸) ⊆ 𝑂
114		rabss2 3331	. . . . . . . . . . 11 ⊢ ((𝑂 ∖ 𝐸) ⊆ 𝑂 → {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐})
115	113, 114	ax-mp 5	. . . . . . . . . 10 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}
116	112, 115	eqssi 3264	. . . . . . . . 9 ⊢ {𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐} = {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
117	116	fveq2i 5696	. . . . . . . 8 ⊢ (♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
118	117	oveq1i 6089	. . . . . . 7 ⊢ ((♯‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
119	102, 103, 118	3eqtr3i 2267	. . . . . 6 ⊢ (𝑁 / (𝑀 + 𝑁)) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
120	119	oveq2i 6090	. . . . 5 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = (2 · ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
121		2cn 9358	. . . . . 6 ⊢ 2 ∈ ℂ
122		hashcl 11203	. . . . . . . 8 ⊢ ({𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin → (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ₀)
123	75, 122	ax-mp 5	. . . . . . 7 ⊢ (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ₀
124	123	nn0cni 9558	. . . . . 6 ⊢ (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℂ
125	121, 124, 30, 54	divassapi 9092	. . . . 5 ⊢ ((2 · (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (2 · ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
126	124	2timesi 9417	. . . . . 6 ⊢ (2 · (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
127	126	oveq1i 6089	. . . . 5 ⊢ ((2 · (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
128	120, 125, 127	3eqtr2i 2265	. . . 4 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = (((♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
129	67, 80, 128	3eqtr4ri 2270	. . 3 ⊢ (2 · (𝑁 / (𝑀 + 𝑁))) = ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂))
130	129	oveq2i 6090	. 2 ⊢ (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = (1 − ((♯‘(𝑂 ∖ 𝐸)) / (♯‘𝑂)))
131	44	nncni 9297	. . . 4 ⊢ (𝑀 + 𝑁) ∈ ℂ
132	4	nncni 9297	. . . . 5 ⊢ 𝑁 ∈ ℂ
133	121, 132	mulcli 8325	. . . 4 ⊢ (2 · 𝑁) ∈ ℂ
134	44	nnap0i 9318	. . . . 5 ⊢ (𝑀 + 𝑁) # 0
135	131, 134	pm3.2i 272	. . . 4 ⊢ ((𝑀 + 𝑁) ∈ ℂ ∧ (𝑀 + 𝑁) # 0)
136		divsubdirap 9032	. . . 4 ⊢ (((𝑀 + 𝑁) ∈ ℂ ∧ (2 · 𝑁) ∈ ℂ ∧ ((𝑀 + 𝑁) ∈ ℂ ∧ (𝑀 + 𝑁) # 0)) → (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁))))
137	131, 133, 135, 136	mp3an 1378	. . 3 ⊢ (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁)))
138	132	2timesi 9417	. . . . . 6 ⊢ (2 · 𝑁) = (𝑁 + 𝑁)
139	138	oveq2i 6090	. . . . 5 ⊢ ((𝑀 + 𝑁) − (2 · 𝑁)) = ((𝑀 + 𝑁) − (𝑁 + 𝑁))
140	3	nncni 9297	. . . . . . 7 ⊢ 𝑀 ∈ ℂ
141	140, 132, 132, 132	addsub4i 8616	. . . . . 6 ⊢ ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = ((𝑀 − 𝑁) + (𝑁 − 𝑁))
142	132	subidi 8591	. . . . . . 7 ⊢ (𝑁 − 𝑁) = 0
143	142	oveq2i 6090	. . . . . 6 ⊢ ((𝑀 − 𝑁) + (𝑁 − 𝑁)) = ((𝑀 − 𝑁) + 0)
144	140, 132	subcli 8596	. . . . . . 7 ⊢ (𝑀 − 𝑁) ∈ ℂ
145	144	addridi 8462	. . . . . 6 ⊢ ((𝑀 − 𝑁) + 0) = (𝑀 − 𝑁)
146	141, 143, 145	3eqtri 2263	. . . . 5 ⊢ ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = (𝑀 − 𝑁)
147	139, 146	eqtri 2259	. . . 4 ⊢ ((𝑀 + 𝑁) − (2 · 𝑁)) = (𝑀 − 𝑁)
148	147	oveq1i 6089	. . 3 ⊢ (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))
149	131, 134	dividapi 9069	. . . 4 ⊢ ((𝑀 + 𝑁) / (𝑀 + 𝑁)) = 1
150	121, 132, 131, 134	divassapi 9092	. . . 4 ⊢ ((2 · 𝑁) / (𝑀 + 𝑁)) = (2 · (𝑁 / (𝑀 + 𝑁)))
151	149, 150	oveq12i 6091	. . 3 ⊢ (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁))) = (1 − (2 · (𝑁 / (𝑀 + 𝑁))))
152	137, 148, 151	3eqtr3ri 2268	. 2 ⊢ (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))
153	60, 130, 152	3eqtr2i 2265	1 ⊢ (𝑃‘𝐸) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ∀wral 2528 {crab 2532 ∖ cdif 3217 ∪ cun 3218 ∩ cin 3219 ⊆ wss 3220 ∅c0 3520 ifcif 3638 𝒫 cpw 3688 class class class wbr 4128 ↦ cmpt 4190 “ cima 4775 ‘cfv 5375 (class class class)co 6079 Fincfn 7016 infcinf 7317 ℂcc 8171 ℝcr 8172 0cc0 8173 1c1 8174 + caddc 8176 · cmul 8178 < clt 8354 ≤ cle 8355 − cmin 8491 # cap 8903 / cdiv 8996 ℕcn 9287 2c2 9338 ℕ₀cn0 9546 ℤcz 9627 ...cfz 10394 Ccbc 11168 ♯chash 11197

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 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-mulrcl 8272 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-precex 8283 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 ax-pre-mulgt0 8290 ax-pre-mulext 8291

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 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-po 4439 df-iso 4440 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-isom 5384 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-irdg 6635 df-frec 6656 df-1o 6681 df-2o 6682 df-oadd 6685 df-er 6801 df-map 6918 df-en 7017 df-dom 7018 df-fin 7019 df-sup 7318 df-inf 7319 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-reap 8897 df-ap 8904 df-div 8997 df-inn 9288 df-2 9346 df-n0 9547 df-z 9628 df-uz 9905 df-q 10003 df-rp 10038 df-fz 10395 df-fzo 10533 df-seqfrec 10868 df-fac 11147 df-bc 11169 df-ihash 11198

This theorem is referenced by: ballotfi 13265

W3C validator