Theorem ballotleme 30898

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 6332	. . . . 5 ⊢ (𝑑 = 𝐶 → (𝐹‘𝑑) = (𝐹‘𝐶))
2	1	fveq1d 6334	. . . 4 ⊢ (𝑑 = 𝐶 → ((𝐹‘𝑑)‘𝑖) = ((𝐹‘𝐶)‘𝑖))
3	2	breq2d 4798	. . 3 ⊢ (𝑑 = 𝐶 → (0 < ((𝐹‘𝑑)‘𝑖) ↔ 0 < ((𝐹‘𝐶)‘𝑖)))
4	3	ralbidv 3135	. 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 6332	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (𝐹‘𝑐) = (𝐹‘𝑑))
7	6	fveq1d 6334	. . . . . 6 ⊢ (𝑐 = 𝑑 → ((𝐹‘𝑐)‘𝑖) = ((𝐹‘𝑑)‘𝑖))
8	7	breq2d 4798	. . . . 5 ⊢ (𝑐 = 𝑑 → (0 < ((𝐹‘𝑐)‘𝑖) ↔ 0 < ((𝐹‘𝑑)‘𝑖)))
9	8	ralbidv 3135	. . . 4 ⊢ (𝑐 = 𝑑 → (∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖) ↔ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑑)‘𝑖)))
10	9	cbvrabv 3349	. . 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 2793	. 2 ⊢ 𝐸 = {𝑑 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑑)‘𝑖)}
12	4, 11	elrab2 3518	1 ⊢ (𝐶 ∈ 𝐸 ↔ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))

Syntax hints:   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  {crab 3065   ∖ cdif 3720   ∩ cin 3722  𝒫 cpw 4297   class class class wbr 4786   ↦ cmpt 4863  ‘cfv 6031  (class class class)co 6793  0cc0 10138  1c1 10139   + caddc 10141   < clt 10276   − cmin 10468   / cdiv 10886  ℕcn 11222  ℤcz 11579  ...cfz 12533  ♯chash 13321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039

This theorem is referenced by:  ballotlemodife  30899  ballotlem4  30900