ballotlem1c - Mathbox for Thierry Arnoux

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Theorem ballotlem1c 34839

Description: If the first vote is for A, the vote on the first tie is for B. (Contributed by Thierry Arnoux, 4-Apr-2017.)

**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)))
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, < ))

**Assertion**
Ref	Expression
ballotlem1c	⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → ¬ (𝐼‘𝐶) ∈ 𝐶)

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 𝐶,𝑖,𝑘 𝑖,𝐸,𝑘 𝐶,𝑘 𝑘,𝐼 𝑘,𝑐,𝐸 𝑖,𝐼 Allowed substitution hints: 𝐶(𝑥,𝑐) 𝑃(𝑥,𝑖,𝑘,𝑐) 𝐸(𝑥) 𝐹(𝑥) 𝐼(𝑥,𝑐) 𝑀(𝑥) 𝑁(𝑥) 𝑂(𝑥)

**Proof of Theorem ballotlem1c**
Step	Hyp	Ref	Expression
1		ballotth.m	. . 3 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . 3 ⊢ 𝑁 ∈ ℕ
3		ballotth.o	. . 3 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
4		ballotth.p	. . 3 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
5		ballotth.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		eldifi 4093	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
7	6	ad2antrr 738	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → 𝐶 ∈ 𝑂)
8		ballotth.e	. . . . . . . . . 10 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
9		ballotth.mgtn	. . . . . . . . . 10 ⊢ 𝑁 < 𝑀
10		ballotth.i	. . . . . . . . . 10 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
11	1, 2, 3, 4, 5, 8, 9, 10	ballotlemiex 34833	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
12	11	simpld 499	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
13		elfznn 13577	. . . . . . . 8 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℕ)
14	12, 13	syl 18	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℕ)
15	14	adantr 485	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → (𝐼‘𝐶) ∈ ℕ)
16	1, 2, 3, 4, 5, 8, 9, 10	ballotlemii 34835	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → (𝐼‘𝐶) ≠ 1)
17		eluz2b3 12942	. . . . . 6 ⊢ ((𝐼‘𝐶) ∈ (ℤ_≥‘2) ↔ ((𝐼‘𝐶) ∈ ℕ ∧ (𝐼‘𝐶) ≠ 1))
18	15, 16, 17	sylanbrc 594	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → (𝐼‘𝐶) ∈ (ℤ_≥‘2))
19		uz2m1nn 12943	. . . . 5 ⊢ ((𝐼‘𝐶) ∈ (ℤ_≥‘2) → ((𝐼‘𝐶) − 1) ∈ ℕ)
20	18, 19	syl 18	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → ((𝐼‘𝐶) − 1) ∈ ℕ)
21	20	adantr 485	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐼‘𝐶) − 1) ∈ ℕ)
22		fveq2 6879	. . . . 5 ⊢ (𝑖 = 1 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘1))
23	22	breq2d 5122	. . . 4 ⊢ (𝑖 = 1 → (0 ≤ ((𝐹‘𝐶)‘𝑖) ↔ 0 ≤ ((𝐹‘𝐶)‘1)))
24		elnnuz 12898	. . . . . . 7 ⊢ (((𝐼‘𝐶) − 1) ∈ ℕ ↔ ((𝐼‘𝐶) − 1) ∈ (ℤ_≥‘1))
25	24	biimpi 219	. . . . . 6 ⊢ (((𝐼‘𝐶) − 1) ∈ ℕ → ((𝐼‘𝐶) − 1) ∈ (ℤ_≥‘1))
26		eluzfz1 13555	. . . . . 6 ⊢ (((𝐼‘𝐶) − 1) ∈ (ℤ_≥‘1) → 1 ∈ (1...((𝐼‘𝐶) − 1)))
27	20, 25, 26	3syl 19	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → 1 ∈ (1...((𝐼‘𝐶) − 1)))
28	27	adantr 485	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ (1...((𝐼‘𝐶) − 1)))
29		0le1 11733	. . . . . . 7 ⊢ 0 ≤ 1
30		1e0p1 12754	. . . . . . 7 ⊢ 1 = (0 + 1)
31	29, 30	breqtri 5137	. . . . . 6 ⊢ 0 ≤ (0 + 1)
32		1nn 12240	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
33	32	a1i 11	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℕ)
34	1, 2, 3, 4, 5, 6, 33	ballotlemfp1 34823	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((¬ 1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) − 1)) ∧ (1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) + 1))))
35	34	simprd 500	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (1 ∈ 𝐶 → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) + 1)))
36	35	imp 411	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘1) = (((𝐹‘𝐶)‘(1 − 1)) + 1))
37		1m1e0 12309	. . . . . . . . . 10 ⊢ (1 − 1) = 0
38	37	fveq2i 6882	. . . . . . . . 9 ⊢ ((𝐹‘𝐶)‘(1 − 1)) = ((𝐹‘𝐶)‘0)
39	38	oveq1i 7418	. . . . . . . 8 ⊢ (((𝐹‘𝐶)‘(1 − 1)) + 1) = (((𝐹‘𝐶)‘0) + 1)
40	39	a1i 11	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → (((𝐹‘𝐶)‘(1 − 1)) + 1) = (((𝐹‘𝐶)‘0) + 1))
41	1, 2, 3, 4, 5	ballotlemfval0 34827	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝑂 → ((𝐹‘𝐶)‘0) = 0)
42	6, 41	syl 18	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘𝐶)‘0) = 0)
43	42	adantr 485	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → ((𝐹‘𝐶)‘0) = 0)
44	43	oveq1d 7423	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → (((𝐹‘𝐶)‘0) + 1) = (0 + 1))
45	36, 40, 44	3eqtrrd 2809	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → (0 + 1) = ((𝐹‘𝐶)‘1))
46	31, 45	breqtrid 5149	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → 0 ≤ ((𝐹‘𝐶)‘1))
47	46	adantr 485	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → 0 ≤ ((𝐹‘𝐶)‘1))
48	23, 28, 47	rspcedvdw 3593	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → ∃𝑖 ∈ (1...((𝐼‘𝐶) − 1))0 ≤ ((𝐹‘𝐶)‘𝑖))
49		df-neg 11440	. . . . . 6 ⊢ -1 = (0 − 1)
50	1, 2, 3, 4, 5, 6, 14	ballotlemfp1 34823	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((¬ (𝐼‘𝐶) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) − 1)) ∧ ((𝐼‘𝐶) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) + 1))))
51	50	simprd 500	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) + 1)))
52	51	imp 411	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) + 1))
53	11	simprd 500	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
54	53	adantr 485	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
55	52, 54	eqtr3d 2806	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) + 1) = 0)
56		0cnd 11195	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → 0 ∈ ℂ)
57		1cnd 11198	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ ℂ)
58	6	adantr 485	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → 𝐶 ∈ 𝑂)
59	14	nnzd 12613	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
60	59	adantr 485	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → (𝐼‘𝐶) ∈ ℤ)
61		1zzd 12621	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → 1 ∈ ℤ)
62	60, 61	zsubcld 12701	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐼‘𝐶) − 1) ∈ ℤ)
63	1, 2, 3, 4, 5, 58, 62	ballotlemfelz 34822	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) ∈ ℤ)
64	63	zcnd 12697	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) ∈ ℂ)
65	56, 57, 64	subadd2d 11584	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((0 − 1) = ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) ↔ (((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) + 1) = 0))
66	55, 65	mpbird 260	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → (0 − 1) = ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)))
67	49, 66	eqtrid 2816	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → -1 = ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)))
68		neg1lt0 12202	. . . . 5 ⊢ -1 < 0
69	67, 68	eqbrtrrdi 5152	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) < 0)
70	69	adantlr 727	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → ((𝐹‘𝐶)‘((𝐼‘𝐶) − 1)) < 0)
71	1, 2, 3, 4, 5, 7, 21, 48, 70	ballotlemfcc 34825	. 2 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
72	1, 2, 3, 4, 5, 8, 9, 10	ballotlemimin 34837	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ¬ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
73	72	ad2antrr 738	. 2 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) ∧ (𝐼‘𝐶) ∈ 𝐶) → ¬ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
74	71, 73	pm2.65da 828	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → ¬ (𝐼‘𝐶) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 {crab 3423 ∖ cdif 3910 ∩ cin 3912 𝒫 cpw 4564 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6533 (class class class)co 7408 infcinf 9397 ℝcr 11095 0cc0 11096 1c1 11097 + caddc 11099 < clt 11239 ≤ cle 11240 − cmin 11437 -cneg 11438 / cdiv 11867 ℕcn 12229 2c2 12291 ℤcz 12587 ℤ_≥cuz 12858 ...cfz 13531 ♯chash 14362

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-dju 9883 df-card 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-nn 12230 df-2 12299 df-n0 12501 df-z 12588 df-uz 12859 df-fz 13532 df-hash 14363

This theorem is referenced by: ballotlem7 34867

W3C validator