Theorem ballotlemimin 34690

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
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
ballotlemimin	⊢ (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 ballotlemimin

Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfzle2 13473	. . . . . 6 ⊢ (𝑘 ∈ (1...((𝐼‘𝐶) − 1)) → 𝑘 ≤ ((𝐼‘𝐶) − 1))
2	1	adantl 482	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑘 ∈ (1...((𝐼‘𝐶) − 1))) → 𝑘 ≤ ((𝐼‘𝐶) − 1))
3		elfzelz 13469	. . . . . 6 ⊢ (𝑘 ∈ (1...((𝐼‘𝐶) − 1)) → 𝑘 ∈ ℤ)
4		ballotth.m	. . . . . . . . 9 ⊢ 𝑀 ∈ ℕ
5		ballotth.n	. . . . . . . . 9 ⊢ 𝑁 ∈ ℕ
6		ballotth.o	. . . . . . . . 9 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
7		ballotth.p	. . . . . . . . 9 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
8		ballotth.f	. . . . . . . . 9 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
9		ballotth.e	. . . . . . . . 9 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
10		ballotth.mgtn	. . . . . . . . 9 ⊢ 𝑁 < 𝑀
11		ballotth.i	. . . . . . . . 9 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
12	4, 5, 6, 7, 8, 9, 10, 11	ballotlemiex 34686	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
13	12	simpld 495	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
14	13	elfzelzd 13470	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
15		zltlem1 12571	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ (𝐼‘𝐶) ∈ ℤ) → (𝑘 < (𝐼‘𝐶) ↔ 𝑘 ≤ ((𝐼‘𝐶) − 1)))
16	3, 14, 15	syl2anr 603	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑘 ∈ (1...((𝐼‘𝐶) − 1))) → (𝑘 < (𝐼‘𝐶) ↔ 𝑘 ≤ ((𝐼‘𝐶) − 1)))
17	2, 16	mpbird 258	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑘 ∈ (1...((𝐼‘𝐶) − 1))) → 𝑘 < (𝐼‘𝐶))
18	17	adantr 481	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑘 ∈ (1...((𝐼‘𝐶) − 1))) ∧ ((𝐹‘𝐶)‘𝑘) = 0) → 𝑘 < (𝐼‘𝐶))
19		1zzd 12549	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℤ)
20	14, 19	zsubcld 12629	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℤ)
21	20	zred 12624	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℝ)
22		nnaddcl 12188	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
23	4, 5, 22	mp2an 698	. . . . . . . . . . . . 13 ⊢ (𝑀 + 𝑁) ∈ ℕ
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℕ)
25	24	nnred 12180	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℝ)
26		elfzle2 13473	. . . . . . . . . . . . 13 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
27	13, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
28	24	nnzd 12541	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℤ)
29		zlem1lt 12570	. . . . . . . . . . . . 13 ⊢ (((𝐼‘𝐶) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁)))
30	14, 28, 29	syl2anc 590	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁)))
31	27, 30	mpbid 233	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁))
32	21, 25, 31	ltled 11285	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁))
33		eluz 12793	. . . . . . . . . . 11 ⊢ ((((𝐼‘𝐶) − 1) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)) ↔ ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
34	20, 28, 33	syl2anc 590	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)) ↔ ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
35	32, 34	mpbird 258	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)))
36		fzss2 13509	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)) → (1...((𝐼‘𝐶) − 1)) ⊆ (1...(𝑀 + 𝑁)))
37	35, 36	syl 17	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (1...((𝐼‘𝐶) − 1)) ⊆ (1...(𝑀 + 𝑁)))
38	37	sseld 3914	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑘 ∈ (1...((𝐼‘𝐶) − 1)) → 𝑘 ∈ (1...(𝑀 + 𝑁))))
39		rabid 3412	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ↔ (𝑘 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘𝑘) = 0))
40	4, 5, 6, 7, 8, 9, 10, 11	ballotlemsup 34689	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}𝑦 < 𝑤)))
41		ltso 11217	. . . . . . . . . . . 12 ⊢ < Or ℝ
42	41	a1i 11	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}𝑦 < 𝑤)) → < Or ℝ)
43		id 22	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}𝑦 < 𝑤)) → ∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}𝑦 < 𝑤)))
44	42, 43	inflb 9393	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}𝑦 < 𝑤)) → (𝑘 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} → ¬ 𝑘 < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )))
45	40, 44	syl 17	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑘 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} → ¬ 𝑘 < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )))
46	4, 5, 6, 7, 8, 9, 10, 11	ballotlemi 34685	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))
47	46	breq2d 5084	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑘 < (𝐼‘𝐶) ↔ 𝑘 < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )))
48	47	notbid 319	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (¬ 𝑘 < (𝐼‘𝐶) ↔ ¬ 𝑘 < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )))
49	45, 48	sylibrd 260	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑘 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} → ¬ 𝑘 < (𝐼‘𝐶)))
50	39, 49	biimtrrid 244	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑘 ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘𝑘) = 0) → ¬ 𝑘 < (𝐼‘𝐶)))
51	38, 50	syland 609	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑘 ∈ (1...((𝐼‘𝐶) − 1)) ∧ ((𝐹‘𝐶)‘𝑘) = 0) → ¬ 𝑘 < (𝐼‘𝐶)))
52	51	imp 407	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑘 ∈ (1...((𝐼‘𝐶) − 1)) ∧ ((𝐹‘𝐶)‘𝑘) = 0)) → ¬ 𝑘 < (𝐼‘𝐶))
53		biid 262	. . . . 5 ⊢ (𝑘 < (𝐼‘𝐶) ↔ 𝑘 < (𝐼‘𝐶))
54	52, 53	sylnib 329	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑘 ∈ (1...((𝐼‘𝐶) − 1)) ∧ ((𝐹‘𝐶)‘𝑘) = 0)) → ¬ 𝑘 < (𝐼‘𝐶))
55	54	anassrs 468	. . 3 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑘 ∈ (1...((𝐼‘𝐶) − 1))) ∧ ((𝐹‘𝐶)‘𝑘) = 0) → ¬ 𝑘 < (𝐼‘𝐶))
56	18, 55	pm2.65da 822	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑘 ∈ (1...((𝐼‘𝐶) − 1))) → ¬ ((𝐹‘𝐶)‘𝑘) = 0)
57	56	nrexdv 3134	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ¬ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  {crab 3391   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4529   class class class wbr 5072   ↦ cmpt 5153   Or wor 5525  ‘cfv 6485  (class class class)co 7356  infcinf 9344  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032   < clt 11170   ≤ cle 11171   − cmin 11368   / cdiv 11798  ℕcn 12165  ℤcz 12515  ℤ_≥cuz 12779  ...cfz 13452  ♯chash 14283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-dju 9816  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-hash 14284

This theorem is referenced by:  ballotlemic  34691  ballotlem1c  34692