Theorem ballotleme 29885

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 6103	. . . . 5 ⊢ (𝑑 = 𝐶 → (𝐹‘𝑑) = (𝐹‘𝐶))
2	1	fveq1d 6105	. . . 4 ⊢ (𝑑 = 𝐶 → ((𝐹‘𝑑)‘𝑖) = ((𝐹‘𝐶)‘𝑖))
3	2	breq2d 4595	. . 3 ⊢ (𝑑 = 𝐶 → (0 < ((𝐹‘𝑑)‘𝑖) ↔ 0 < ((𝐹‘𝐶)‘𝑖)))
4	3	ralbidv 2969	. 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 6103	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (𝐹‘𝑐) = (𝐹‘𝑑))
7	6	fveq1d 6105	. . . . . 6 ⊢ (𝑐 = 𝑑 → ((𝐹‘𝑐)‘𝑖) = ((𝐹‘𝑑)‘𝑖))
8	7	breq2d 4595	. . . . 5 ⊢ (𝑐 = 𝑑 → (0 < ((𝐹‘𝑐)‘𝑖) ↔ 0 < ((𝐹‘𝑑)‘𝑖)))
9	8	ralbidv 2969	. . . 4 ⊢ (𝑐 = 𝑑 → (∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖) ↔ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑑)‘𝑖)))
10	9	cbvrabv 3172	. . 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 2632	. 2 ⊢ 𝐸 = {𝑑 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑑)‘𝑖)}
12	4, 11	elrab2 3333	1 ⊢ (𝐶 ∈ 𝐸 ↔ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ∖ cdif 3537   ∩ cin 3539  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   − cmin 10145   / cdiv 10563  ℕcn 10897  ℤcz 11254  ...cfz 12197  #chash 12979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812

This theorem is referenced by:  ballotlemodife  29886  ballotlem4  29887