Theorem ballotlemi 34800

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 6869	. . . . . 6 ⊢ (𝑑 = 𝐶 → (𝐹‘𝑑) = (𝐹‘𝐶))
2	1	fveq1d 6871	. . . . 5 ⊢ (𝑑 = 𝐶 → ((𝐹‘𝑑)‘𝑘) = ((𝐹‘𝐶)‘𝑘))
3	2	eqeq1d 2766	. . . 4 ⊢ (𝑑 = 𝐶 → (((𝐹‘𝑑)‘𝑘) = 0 ↔ ((𝐹‘𝐶)‘𝑘) = 0))
4	3	rabbidv 3423	. . 3 ⊢ (𝑑 = 𝐶 → {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0})
5	4	infeq1d 9426	. 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 6869	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (𝐹‘𝑐) = (𝐹‘𝑑))
8	7	fveq1d 6871	. . . . . . 7 ⊢ (𝑐 = 𝑑 → ((𝐹‘𝑐)‘𝑘) = ((𝐹‘𝑑)‘𝑘))
9	8	eqeq1d 2766	. . . . . 6 ⊢ (𝑐 = 𝑑 → (((𝐹‘𝑐)‘𝑘) = 0 ↔ ((𝐹‘𝑑)‘𝑘) = 0))
10	9	rabbidv 3423	. . . . 5 ⊢ (𝑐 = 𝑑 → {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0})
11	10	infeq1d 9426	. . . 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 5206	. . 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 2787	. 2 ⊢ 𝐼 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0}, ℝ, < ))
14		ltso 11265	. . 3 ⊢ < Or ℝ
15	14	infex 9443	. 2 ⊢ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ) ∈ V
16	5, 13, 15	fvmpt 6977	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144  ∀wral 3078  {crab 3416   ∖ cdif 3903   ∩ cin 3905  𝒫 cpw 4557   class class class wbr 5102   ↦ cmpt 5183  ‘cfv 6523  (class class class)co 7398  infcinf 9389  ℝcr 11074  0cc0 11075  1c1 11076   + caddc 11078   < clt 11218   − cmin 11416   / cdiv 11846  ℕcn 12212  ℤcz 12570  ...cfz 13514  ♯chash 14345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-resscn 11132  ax-pre-lttri 11149  ax-pre-lttrn 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-po 5557  df-so 5558  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-sup 9390  df-inf 9391  df-pnf 11220  df-mnf 11221  df-ltxr 11223

This theorem is referenced by:  ballotlemiex  34801  ballotlemimin  34805  ballotlemfrcn0  34829  ballotlemirc  34831