Theorem ballotlemi 30562

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 6191	. . . . . 6 ⊢ (𝑑 = 𝐶 → (𝐹‘𝑑) = (𝐹‘𝐶))
2	1	fveq1d 6193	. . . . 5 ⊢ (𝑑 = 𝐶 → ((𝐹‘𝑑)‘𝑘) = ((𝐹‘𝐶)‘𝑘))
3	2	eqeq1d 2624	. . . 4 ⊢ (𝑑 = 𝐶 → (((𝐹‘𝑑)‘𝑘) = 0 ↔ ((𝐹‘𝐶)‘𝑘) = 0))
4	3	rabbidv 3189	. . 3 ⊢ (𝑑 = 𝐶 → {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0})
5	4	infeq1d 8383	. 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 6191	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (𝐹‘𝑐) = (𝐹‘𝑑))
8	7	fveq1d 6193	. . . . . . 7 ⊢ (𝑐 = 𝑑 → ((𝐹‘𝑐)‘𝑘) = ((𝐹‘𝑑)‘𝑘))
9	8	eqeq1d 2624	. . . . . 6 ⊢ (𝑐 = 𝑑 → (((𝐹‘𝑐)‘𝑘) = 0 ↔ ((𝐹‘𝑑)‘𝑘) = 0))
10	9	rabbidv 3189	. . . . 5 ⊢ (𝑐 = 𝑑 → {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0})
11	10	infeq1d 8383	. . . 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 4750	. . 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 2644	. 2 ⊢ 𝐼 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0}, ℝ, < ))
14		ltso 10118	. . 3 ⊢ < Or ℝ
15	14	infex 8399	. 2 ⊢ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ) ∈ V
16	5, 13, 15	fvmpt 6282	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ∖ cdif 3571   ∩ cin 3573  𝒫 cpw 4158   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  infcinf 8347  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   − cmin 10266   / cdiv 10684  ℕcn 11020  ℤcz 11377  ...cfz 12326  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-ltxr 10079

This theorem is referenced by:  ballotlemiex  30563  ballotlemimin  30567  ballotlemfrcn0  30591  ballotlemirc  30593