Theorem ballotfilemimin 13251

Description: (𝐼‘𝐶) is the first tie. (Contributed by Thierry Arnoux, 1-Dec-2016.) (Revised by AV, 6-Oct-2020.)

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, < ))

Assertion

Ref	Expression
ballotfilemimin	⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ¬ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)

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 ballotfilemimin

Dummy variables 𝑗 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfzle2 10434	. . . . . . 7 ⊢ (𝑗 ∈ (1...((𝐼‘𝐶) − 1)) → 𝑗 ≤ ((𝐼‘𝐶) − 1))
2	1	adantl 277	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) → 𝑗 ≤ ((𝐼‘𝐶) − 1))
3		elfzelz 10430	. . . . . . 7 ⊢ (𝑗 ∈ (1...((𝐼‘𝐶) − 1)) → 𝑗 ∈ ℤ)
4		ballotth.m	. . . . . . . . . 10 ⊢ 𝑀 ∈ ℕ
5		ballotth.n	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℕ
6		ballotfilem.o	. . . . . . . . . 10 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
7		ballotfilem.p	. . . . . . . . . 10 ⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
8		ballotth.f	. . . . . . . . . 10 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
9		ballotth.e	. . . . . . . . . 10 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
10		ballotth.mgtn	. . . . . . . . . 10 ⊢ 𝑁 < 𝑀
11		ballotth.i	. . . . . . . . . 10 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
12	4, 5, 6, 7, 8, 9, 10, 11	ballotfilemiex 13246	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
13	12	simpld 112	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
14	13	elfzelzd 10431	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
15		zltlem1 9704	. . . . . . 7 ⊢ ((𝑗 ∈ ℤ ∧ (𝐼‘𝐶) ∈ ℤ) → (𝑗 < (𝐼‘𝐶) ↔ 𝑗 ≤ ((𝐼‘𝐶) − 1)))
16	3, 14, 15	syl2anr 290	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) → (𝑗 < (𝐼‘𝐶) ↔ 𝑗 ≤ ((𝐼‘𝐶) − 1)))
17	2, 16	mpbird 167	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) → 𝑗 < (𝐼‘𝐶))
18	17	adantr 276	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) ∧ ((𝐹‘𝐶)‘𝑗) = 0) → 𝑗 < (𝐼‘𝐶))
19	14	ad2antrr 492	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) ∧ ((𝐹‘𝐶)‘𝑗) = 0) → (𝐼‘𝐶) ∈ ℤ)
20	19	zred 9770	. . . . 5 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) ∧ ((𝐹‘𝐶)‘𝑗) = 0) → (𝐼‘𝐶) ∈ ℝ)
21	3	adantl 277	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) → 𝑗 ∈ ℤ)
22	21	adantr 276	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) ∧ ((𝐹‘𝐶)‘𝑗) = 0) → 𝑗 ∈ ℤ)
23	22	zred 9770	. . . . 5 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) ∧ ((𝐹‘𝐶)‘𝑗) = 0) → 𝑗 ∈ ℝ)
24		1zzd 9673	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℤ)
25	14, 24	zsubcld 9775	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℤ)
26	25	zred 9770	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℝ)
27		nnaddcl 9325	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
28	4, 5, 27	mp2an 430	. . . . . . . . . . . . 13 ⊢ (𝑀 + 𝑁) ∈ ℕ
29	28	a1i 9	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℕ)
30	29	nnred 9318	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℝ)
31		elfzle2 10434	. . . . . . . . . . . . 13 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
32	13, 31	syl 14	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
33	29	nnzd 9769	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℤ)
34		zlem1lt 9703	. . . . . . . . . . . . 13 ⊢ (((𝐼‘𝐶) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁)))
35	14, 33, 34	syl2anc 415	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁)))
36	32, 35	mpbid 147	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁))
37	26, 30, 36	ltled 8445	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁))
38		eluz 9937	. . . . . . . . . . 11 ⊢ ((((𝐼‘𝐶) − 1) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)) ↔ ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
39	25, 33, 38	syl2anc 415	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)) ↔ ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
40	37, 39	mpbird 167	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)))
41		fzss2 10472	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)) → (1...((𝐼‘𝐶) − 1)) ⊆ (1...(𝑀 + 𝑁)))
42	40, 41	syl 14	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (1...((𝐼‘𝐶) − 1)) ⊆ (1...(𝑀 + 𝑁)))
43	42	sseld 3247	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑗 ∈ (1...((𝐼‘𝐶) − 1)) → 𝑗 ∈ (1...(𝑀 + 𝑁))))
44		fveqeq2 5704	. . . . . . . . . 10 ⊢ (𝑙 = 𝑗 → (((𝐹‘𝐶)‘𝑙) = 0 ↔ ((𝐹‘𝐶)‘𝑗) = 0))
45	44	elrab 2982	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝑙 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑙) = 0} ↔ (𝑗 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘𝑗) = 0))
46	4, 5, 6, 7, 8, 9, 10, 11	ballotfilemi 13245	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))
47		fveqeq2 5704	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑘 → (((𝐹‘𝐶)‘𝑙) = 0 ↔ ((𝐹‘𝐶)‘𝑘) = 0))
48	47	cbvrabv 2820	. . . . . . . . . . . . 13 ⊢ {𝑙 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑙) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}
49	48	infeq1i 7353	. . . . . . . . . . . 12 ⊢ inf({𝑙 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑙) = 0}, ℝ, < ) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )
50	46, 49	eqtr4di 2289	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑙 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑙) = 0}, ℝ, < ))
51	50	adantr 276	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ {𝑙 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑙) = 0}) → (𝐼‘𝐶) = inf({𝑙 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑙) = 0}, ℝ, < ))
52	4, 5, 6, 7, 8, 9, 10, 11, 48	ballotfilemsle 13250	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ {𝑙 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑙) = 0}) → inf({𝑙 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑙) = 0}, ℝ, < ) ≤ 𝑗)
53	51, 52	eqbrtrd 4152	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ {𝑙 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑙) = 0}) → (𝐼‘𝐶) ≤ 𝑗)
54	45, 53	sylan2br 288	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑗 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘𝑗) = 0)) → (𝐼‘𝐶) ≤ 𝑗)
55	54	ex 115	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑗 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘𝑗) = 0) → (𝐼‘𝐶) ≤ 𝑗))
56	43, 55	syland 293	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑗 ∈ (1...((𝐼‘𝐶) − 1)) ∧ ((𝐹‘𝐶)‘𝑗) = 0) → (𝐼‘𝐶) ≤ 𝑗))
57	56	impl 380	. . . . 5 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) ∧ ((𝐹‘𝐶)‘𝑗) = 0) → (𝐼‘𝐶) ≤ 𝑗)
58	20, 23, 57	lensymd 8448	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) ∧ ((𝐹‘𝐶)‘𝑗) = 0) → ¬ 𝑗 < (𝐼‘𝐶))
59	18, 58	pm2.65da 671	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) → ¬ ((𝐹‘𝐶)‘𝑗) = 0)
60	59	nrexdv 2643	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ¬ ∃𝑗 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑗) = 0)
61		fveqeq2 5704	. . 3 ⊢ (𝑗 = 𝑘 → (((𝐹‘𝐶)‘𝑗) = 0 ↔ ((𝐹‘𝐶)‘𝑘) = 0))
62	61	cbvrexv 2787	. 2 ⊢ (∃𝑗 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑗) = 0 ↔ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
63	60, 62	sylnib 687	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ¬ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532   ∖ cdif 3217   ∩ cin 3219   ⊆ wss 3220  𝒫 cpw 3688   class class class wbr 4130   ↦ cmpt 4192  ‘cfv 5377  (class class class)co 6085  Fincfn 7022  infcinf 7323  ℝcr 8178  0cc0 8179  1c1 8180   + caddc 8182   < clt 8360   ≤ cle 8361   − cmin 8497   / cdiv 9003  ℕcn 9305  ℤcz 9646  ℤ_≥cuz 9923  ...cfz 10413  ♯chash 11216

This proof depends on 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-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-fz 10414  df-fzo 10552  df-ihash 11217

This theorem is used by:  ballotfilemic  13252  ballotfilem1c  13253