Theorem ballotlemi 34514

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 6822	. . . . . 6 ⊢ (𝑑 = 𝐶 → (𝐹‘𝑑) = (𝐹‘𝐶))
2	1	fveq1d 6824	. . . . 5 ⊢ (𝑑 = 𝐶 → ((𝐹‘𝑑)‘𝑘) = ((𝐹‘𝐶)‘𝑘))
3	2	eqeq1d 2733	. . . 4 ⊢ (𝑑 = 𝐶 → (((𝐹‘𝑑)‘𝑘) = 0 ↔ ((𝐹‘𝐶)‘𝑘) = 0))
4	3	rabbidv 3402	. . 3 ⊢ (𝑑 = 𝐶 → {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0})
5	4	infeq1d 9362	. 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 6822	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → (𝐹‘𝑐) = (𝐹‘𝑑))
8	7	fveq1d 6824	. . . . . . 7 ⊢ (𝑐 = 𝑑 → ((𝐹‘𝑐)‘𝑘) = ((𝐹‘𝑑)‘𝑘))
9	8	eqeq1d 2733	. . . . . 6 ⊢ (𝑐 = 𝑑 → (((𝐹‘𝑐)‘𝑘) = 0 ↔ ((𝐹‘𝑑)‘𝑘) = 0))
10	9	rabbidv 3402	. . . . 5 ⊢ (𝑐 = 𝑑 → {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0})
11	10	infeq1d 9362	. . . 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 5193	. . 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 2754	. 2 ⊢ 𝐼 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0}, ℝ, < ))
14		ltso 11193	. . 3 ⊢ < Or ℝ
15	14	infex 9379	. 2 ⊢ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ) ∈ V
16	5, 13, 15	fvmpt 6929	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395   ∖ cdif 3894   ∩ cin 3896  𝒫 cpw 4547   class class class wbr 5089   ↦ cmpt 5170  ‘cfv 6481  (class class class)co 7346  infcinf 9325  ℝcr 11005  0cc0 11006  1c1 11007   + caddc 11009   < clt 11146   − cmin 11344   / cdiv 11774  ℕcn 12125  ℤcz 12468  ...cfz 13407  ♯chash 14237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-resscn 11063  ax-pre-lttri 11080  ax-pre-lttrn 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-po 5522  df-so 5523  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-sup 9326  df-inf 9327  df-pnf 11148  df-mnf 11149  df-ltxr 11151

This theorem is referenced by:  ballotlemiex  34515  ballotlemimin  34519  ballotlemfrcn0  34543  ballotlemirc  34545