Theorem ballotleme 33984

Description: Elements of 𝐸. (Contributed by Thierry Arnoux, 14-Dec-2016.)

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)}

Assertion

Ref	Expression
ballotleme	⊢ (𝐶 ∈ 𝐸 ↔ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))

Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂,𝑐   𝐹,𝑐,𝑖   𝐶,𝑖

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

Proof of Theorem ballotleme

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . . 5 ⊢ (𝑑 = 𝐶 → (𝐹‘𝑑) = (𝐹‘𝐶))
2	1	fveq1d 6883	. . . 4 ⊢ (𝑑 = 𝐶 → ((𝐹‘𝑑)‘𝑖) = ((𝐹‘𝐶)‘𝑖))
3	2	breq2d 5150	. . 3 ⊢ (𝑑 = 𝐶 → (0 < ((𝐹‘𝑑)‘𝑖) ↔ 0 < ((𝐹‘𝐶)‘𝑖)))
4	3	ralbidv 3169	. 2 ⊢ (𝑑 = 𝐶 → (∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑑)‘𝑖) ↔ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))
5		ballotth.e	. . 3 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
6		fveq2 6881	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (𝐹‘𝑐) = (𝐹‘𝑑))
7	6	fveq1d 6883	. . . . . 6 ⊢ (𝑐 = 𝑑 → ((𝐹‘𝑐)‘𝑖) = ((𝐹‘𝑑)‘𝑖))
8	7	breq2d 5150	. . . . 5 ⊢ (𝑐 = 𝑑 → (0 < ((𝐹‘𝑐)‘𝑖) ↔ 0 < ((𝐹‘𝑑)‘𝑖)))
9	8	ralbidv 3169	. . . 4 ⊢ (𝑐 = 𝑑 → (∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖) ↔ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑑)‘𝑖)))
10	9	cbvrabv 3434	. . 3 ⊢ {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} = {𝑑 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑑)‘𝑖)}
11	5, 10	eqtri 2752	. 2 ⊢ 𝐸 = {𝑑 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑑)‘𝑖)}
12	4, 11	elrab2 3678	1 ⊢ (𝐶 ∈ 𝐸 ↔ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3053  {crab 3424   ∖ cdif 3937   ∩ cin 3939  𝒫 cpw 4594   class class class wbr 5138   ↦ cmpt 5221  ‘cfv 6533  (class class class)co 7401  0cc0 11106  1c1 11107   + caddc 11109   < clt 11245   − cmin 11441   / cdiv 11868  ℕcn 12209  ℤcz 12555  ...cfz 13481  ♯chash 14287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541

This theorem is referenced by:  ballotlemodife  33985  ballotlem4  33986