Theorem ballotlemgval 31891

Description: Expand the value of ↑. (Contributed by Thierry Arnoux, 21-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))
ballotlemg	⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))

Assertion

Ref	Expression
ballotlemgval	⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))

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   𝑖,𝐸,𝑘   𝑘,𝐼,𝑐   𝐸,𝑐   𝑖,𝐼,𝑐   𝑆,𝑘,𝑖,𝑐   𝑅,𝑖   𝑣,𝑢,𝐼   𝑢,𝑅,𝑣   𝑢,𝑆,𝑣   𝑢,𝑈,𝑣   𝑢,𝑉,𝑣

Allowed substitution hints:   𝑃(𝑥,𝑣,𝑢,𝑖,𝑘,𝑐)   𝑅(𝑥,𝑘,𝑐)   𝑆(𝑥)   𝑈(𝑥,𝑖,𝑘,𝑐)   𝐸(𝑥,𝑣,𝑢)   ↑ (𝑥,𝑣,𝑢,𝑖,𝑘,𝑐)   𝐹(𝑥,𝑣,𝑢)   𝐼(𝑥)   𝑀(𝑥,𝑣,𝑢)   𝑁(𝑥,𝑣,𝑢)   𝑂(𝑥,𝑣,𝑢)   𝑉(𝑥,𝑖,𝑘,𝑐)

Proof of Theorem ballotlemgval

Step	Hyp	Ref	Expression
1		ineq2 4133	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑣 ∩ 𝑢) = (𝑣 ∩ 𝑈))
2	1	fveq2d 6649	. . 3 ⊢ (𝑢 = 𝑈 → (♯‘(𝑣 ∩ 𝑢)) = (♯‘(𝑣 ∩ 𝑈)))
3		difeq2 4044	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑣 ∖ 𝑢) = (𝑣 ∖ 𝑈))
4	3	fveq2d 6649	. . 3 ⊢ (𝑢 = 𝑈 → (♯‘(𝑣 ∖ 𝑢)) = (♯‘(𝑣 ∖ 𝑈)))
5	2, 4	oveq12d 7153	. 2 ⊢ (𝑢 = 𝑈 → ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))) = ((♯‘(𝑣 ∩ 𝑈)) − (♯‘(𝑣 ∖ 𝑈))))
6		ineq1 4131	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣 ∩ 𝑈) = (𝑉 ∩ 𝑈))
7	6	fveq2d 6649	. . 3 ⊢ (𝑣 = 𝑉 → (♯‘(𝑣 ∩ 𝑈)) = (♯‘(𝑉 ∩ 𝑈)))
8		difeq1 4043	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣 ∖ 𝑈) = (𝑉 ∖ 𝑈))
9	8	fveq2d 6649	. . 3 ⊢ (𝑣 = 𝑉 → (♯‘(𝑣 ∖ 𝑈)) = (♯‘(𝑉 ∖ 𝑈)))
10	7, 9	oveq12d 7153	. 2 ⊢ (𝑣 = 𝑉 → ((♯‘(𝑣 ∩ 𝑈)) − (♯‘(𝑣 ∖ 𝑈))) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))
11		ballotlemg	. 2 ⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
12		ovex 7168	. 2 ⊢ ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) ∈ V
13	5, 10, 11, 12	ovmpo 7289	1 ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110   ∖ cdif 3878   ∩ cin 3880  ifcif 4425  𝒫 cpw 4497   class class class wbr 5030   ↦ cmpt 5110   “ cima 5522  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  Fincfn 8492  infcinf 8889  ℝcr 10525  0cc0 10526  1c1 10527   + caddc 10529   < clt 10664   ≤ cle 10665   − cmin 10859   / cdiv 11286  ℕcn 11625  ℤcz 11969  ...cfz 12885  ♯chash 13686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140

This theorem is referenced by:  ballotlemgun  31892  ballotlemfg  31893  ballotlemfrc  31894