Theorem ballotlemi 32179

Description: Value of 𝐼 for a given counting 𝐶. (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
ballotlemi	⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))

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

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

Proof of Theorem ballotlemi

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6717	. . . . . 6 ⊢ (𝑑 = 𝐶 → (𝐹‘𝑑) = (𝐹‘𝐶))
2	1	fveq1d 6719	. . . . 5 ⊢ (𝑑 = 𝐶 → ((𝐹‘𝑑)‘𝑘) = ((𝐹‘𝐶)‘𝑘))
3	2	eqeq1d 2739	. . . 4 ⊢ (𝑑 = 𝐶 → (((𝐹‘𝑑)‘𝑘) = 0 ↔ ((𝐹‘𝐶)‘𝑘) = 0))
4	3	rabbidv 3390	. . 3 ⊢ (𝑑 = 𝐶 → {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0})
5	4	infeq1d 9093	. 2 ⊢ (𝑑 = 𝐶 → inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0}, ℝ, < ) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))
6		ballotth.i	. . 3 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
7		fveq2 6717	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (𝐹‘𝑐) = (𝐹‘𝑑))
8	7	fveq1d 6719	. . . . . . 7 ⊢ (𝑐 = 𝑑 → ((𝐹‘𝑐)‘𝑘) = ((𝐹‘𝑑)‘𝑘))
9	8	eqeq1d 2739	. . . . . 6 ⊢ (𝑐 = 𝑑 → (((𝐹‘𝑐)‘𝑘) = 0 ↔ ((𝐹‘𝑑)‘𝑘) = 0))
10	9	rabbidv 3390	. . . . 5 ⊢ (𝑐 = 𝑑 → {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0})
11	10	infeq1d 9093	. . . 4 ⊢ (𝑐 = 𝑑 → inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0}, ℝ, < ))
12	11	cbvmptv 5158	. . 3 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < )) = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0}, ℝ, < ))
13	6, 12	eqtri 2765	. 2 ⊢ 𝐼 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0}, ℝ, < ))
14		ltso 10913	. . 3 ⊢ < Or ℝ
15	14	infex 9109	. 2 ⊢ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ) ∈ V
16	5, 13, 15	fvmpt 6818	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110  ∀wral 3061  {crab 3065   ∖ cdif 3863   ∩ cin 3865  𝒫 cpw 4513   class class class wbr 5053   ↦ cmpt 5135  ‘cfv 6380  (class class class)co 7213  infcinf 9057  ℝcr 10728  0cc0 10729  1c1 10730   + caddc 10732   < clt 10867   − cmin 11062   / cdiv 11489  ℕcn 11830  ℤcz 12176  ...cfz 13095  ♯chash 13896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-resscn 10786  ax-pre-lttri 10803  ax-pre-lttrn 10804

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-po 5468  df-so 5469  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-sup 9058  df-inf 9059  df-pnf 10869  df-mnf 10870  df-ltxr 10872

This theorem is referenced by:  ballotlemiex  32180  ballotlemimin  32184  ballotlemfrcn0  32208  ballotlemirc  32210