Theorem ballotfilemimin 13193

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 10382	. . . . . . 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 10378	. . . . . . 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 13188	. . . . . . . . 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 10379	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
15		zltlem1 9652	. . . . . . 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 488	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) ∧ ((𝐹‘𝐶)‘𝑗) = 0) → (𝐼‘𝐶) ∈ ℤ)
20	19	zred 9718	. . . . 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 9718	. . . . 5 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) ∧ ((𝐹‘𝐶)‘𝑗) = 0) → 𝑗 ∈ ℝ)
24		1zzd 9621	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℤ)
25	14, 24	zsubcld 9723	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℤ)
26	25	zred 9718	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℝ)
27		nnaddcl 9274	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
28	4, 5, 27	mp2an 426	. . . . . . . . . . . . 13 ⊢ (𝑀 + 𝑁) ∈ ℕ
29	28	a1i 9	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℕ)
30	29	nnred 9267	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℝ)
31		elfzle2 10382	. . . . . . . . . . . . 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 9717	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℤ)
34		zlem1lt 9651	. . . . . . . . . . . . 13 ⊢ (((𝐼‘𝐶) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁)))
35	14, 33, 34	syl2anc 411	. . . . . . . . . . . 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 8408	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁))
38		eluz 9885	. . . . . . . . . . 11 ⊢ ((((𝐼‘𝐶) − 1) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)) ↔ ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
39	25, 33, 38	syl2anc 411	. . . . . . . . . 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 10419	. . . . . . . . 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 3241	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑗 ∈ (1...((𝐼‘𝐶) − 1)) → 𝑗 ∈ (1...(𝑀 + 𝑁))))
44		fveqeq2 5684	. . . . . . . . . 10 ⊢ (𝑙 = 𝑗 → (((𝐹‘𝐶)‘𝑙) = 0 ↔ ((𝐹‘𝐶)‘𝑗) = 0))
45	44	elrab 2976	. . . . . . . . 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 13187	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))
47		fveqeq2 5684	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑘 → (((𝐹‘𝐶)‘𝑙) = 0 ↔ ((𝐹‘𝐶)‘𝑘) = 0))
48	47	cbvrabv 2814	. . . . . . . . . . . . 13 ⊢ {𝑙 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑙) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}
49	48	infeq1i 7317	. . . . . . . . . . . 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 2285	. . . . . . . . . . 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 13192	. . . . . . . . . 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 4136	. . . . . . . . 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 8411	. . . 4 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) ∧ ((𝐹‘𝐶)‘𝑗) = 0) → ¬ 𝑗 < (𝐼‘𝐶))
59	18, 58	pm2.65da 667	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...((𝐼‘𝐶) − 1))) → ¬ ((𝐹‘𝐶)‘𝑗) = 0)
60	59	nrexdv 2637	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ¬ ∃𝑗 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑗) = 0)
61		fveqeq2 5684	. . 3 ⊢ (𝑗 = 𝑘 → (((𝐹‘𝐶)‘𝑗) = 0 ↔ ((𝐹‘𝐶)‘𝑘) = 0))
62	61	cbvrexv 2781	. 2 ⊢ (∃𝑗 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑗) = 0 ↔ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
63	60, 62	sylnib 683	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ¬ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  {crab 2526   ∖ cdif 3211   ∩ cin 3213   ⊆ wss 3214  𝒫 cpw 3674   class class class wbr 4114   ↦ cmpt 4176  ‘cfv 5357  (class class class)co 6058  Fincfn 6988  infcinf 7287  ℝcr 8142  0cc0 8143  1c1 8144   + caddc 8146   < clt 8324   ≤ cle 8325   − cmin 8460   / cdiv 8963  ℕcn 9254  ℤcz 9594  ℤ_≥cuz 9871  ...cfz 10361  ♯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-isom 5366  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-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  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-fzo 10499  df-ihash 11164

This theorem is referenced by:  ballotfilemic  13194  ballotfilem1c  13195