Theorem ballotlemrval 34499

Description: Value of 𝑅. (Contributed by Thierry Arnoux, 14-Apr-2017.)

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, < ))
ballotth.s	⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r	⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))

Assertion

Ref	Expression
ballotlemrval	⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶))

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,𝑐   𝐸,𝑐   𝑖,𝐼,𝑐   𝑆,𝑘,𝑖,𝑐

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

Proof of Theorem ballotlemrval

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6907	. . 3 ⊢ (𝑑 = 𝐶 → (𝑆‘𝑑) = (𝑆‘𝐶))
2		id 22	. . 3 ⊢ (𝑑 = 𝐶 → 𝑑 = 𝐶)
3	1, 2	imaeq12d 6081	. 2 ⊢ (𝑑 = 𝐶 → ((𝑆‘𝑑) “ 𝑑) = ((𝑆‘𝐶) “ 𝐶))
4		ballotth.r	. . 3 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
5		fveq2 6907	. . . . 5 ⊢ (𝑐 = 𝑑 → (𝑆‘𝑐) = (𝑆‘𝑑))
6		id 22	. . . . 5 ⊢ (𝑐 = 𝑑 → 𝑐 = 𝑑)
7	5, 6	imaeq12d 6081	. . . 4 ⊢ (𝑐 = 𝑑 → ((𝑆‘𝑐) “ 𝑐) = ((𝑆‘𝑑) “ 𝑑))
8	7	cbvmptv 5261	. . 3 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑑) “ 𝑑))
9	4, 8	eqtri 2763	. 2 ⊢ 𝑅 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑑) “ 𝑑))
10		fvex 6920	. . 3 ⊢ (𝑆‘𝐶) ∈ V
11		imaexg 7936	. . 3 ⊢ ((𝑆‘𝐶) ∈ V → ((𝑆‘𝐶) “ 𝐶) ∈ V)
12	10, 11	ax-mp 5	. 2 ⊢ ((𝑆‘𝐶) “ 𝐶) ∈ V
13	3, 9, 12	fvmpt 7016	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433  Vcvv 3478   ∖ cdif 3960   ∩ cin 3962  ifcif 4531  𝒫 cpw 4605   class class class wbr 5148   ↦ cmpt 5231   “ cima 5692  ‘cfv 6563  (class class class)co 7431  infcinf 9479  ℝcr 11152  0cc0 11153  1c1 11154   + caddc 11156   < clt 11293   ≤ cle 11294   − cmin 11490   / cdiv 11918  ℕcn 12264  ℤcz 12611  ...cfz 13544  ♯chash 14366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571

This theorem is referenced by:  ballotlemscr  34500  ballotlemrv  34501  ballotlemro  34504  ballotlemrinv0  34514