Theorem ballotlemrval 34675

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 6834	. . 3 ⊢ (𝑑 = 𝐶 → (𝑆‘𝑑) = (𝑆‘𝐶))
2		id 22	. . 3 ⊢ (𝑑 = 𝐶 → 𝑑 = 𝐶)
3	1, 2	imaeq12d 6020	. 2 ⊢ (𝑑 = 𝐶 → ((𝑆‘𝑑) “ 𝑑) = ((𝑆‘𝐶) “ 𝐶))
4		ballotth.r	. . 3 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
5		fveq2 6834	. . . . 5 ⊢ (𝑐 = 𝑑 → (𝑆‘𝑐) = (𝑆‘𝑑))
6		id 22	. . . . 5 ⊢ (𝑐 = 𝑑 → 𝑐 = 𝑑)
7	5, 6	imaeq12d 6020	. . . 4 ⊢ (𝑐 = 𝑑 → ((𝑆‘𝑐) “ 𝑐) = ((𝑆‘𝑑) “ 𝑑))
8	7	cbvmptv 5202	. . 3 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑑) “ 𝑑))
9	4, 8	eqtri 2759	. 2 ⊢ 𝑅 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑑) “ 𝑑))
10		fvex 6847	. . 3 ⊢ (𝑆‘𝐶) ∈ V
11		imaexg 7855	. . 3 ⊢ ((𝑆‘𝐶) ∈ V → ((𝑆‘𝐶) “ 𝐶) ∈ V)
12	10, 11	ax-mp 5	. 2 ⊢ ((𝑆‘𝐶) “ 𝐶) ∈ V
13	3, 9, 12	fvmpt 6941	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ∀wral 3051  {crab 3399  Vcvv 3440   ∖ cdif 3898   ∩ cin 3900  ifcif 4479  𝒫 cpw 4554   class class class wbr 5098   ↦ cmpt 5179   “ cima 5627  ‘cfv 6492  (class class class)co 7358  infcinf 9344  ℝcr 11025  0cc0 11026  1c1 11027   + caddc 11029   < clt 11166   ≤ cle 11167   − cmin 11364   / cdiv 11794  ℕcn 12145  ℤcz 12488  ...cfz 13423  ♯chash 14253

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500

This theorem is referenced by:  ballotlemscr  34676  ballotlemrv  34677  ballotlemro  34680  ballotlemrinv0  34690