Theorem ballotlemgval 31101

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 4007	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑣 ∩ 𝑢) = (𝑣 ∩ 𝑈))
2	1	fveq2d 6416	. . 3 ⊢ (𝑢 = 𝑈 → (♯‘(𝑣 ∩ 𝑢)) = (♯‘(𝑣 ∩ 𝑈)))
3		difeq2 3921	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑣 ∖ 𝑢) = (𝑣 ∖ 𝑈))
4	3	fveq2d 6416	. . 3 ⊢ (𝑢 = 𝑈 → (♯‘(𝑣 ∖ 𝑢)) = (♯‘(𝑣 ∖ 𝑈)))
5	2, 4	oveq12d 6897	. 2 ⊢ (𝑢 = 𝑈 → ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))) = ((♯‘(𝑣 ∩ 𝑈)) − (♯‘(𝑣 ∖ 𝑈))))
6		ineq1 4006	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣 ∩ 𝑈) = (𝑉 ∩ 𝑈))
7	6	fveq2d 6416	. . 3 ⊢ (𝑣 = 𝑉 → (♯‘(𝑣 ∩ 𝑈)) = (♯‘(𝑉 ∩ 𝑈)))
8		difeq1 3920	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣 ∖ 𝑈) = (𝑉 ∖ 𝑈))
9	8	fveq2d 6416	. . 3 ⊢ (𝑣 = 𝑉 → (♯‘(𝑣 ∖ 𝑈)) = (♯‘(𝑉 ∖ 𝑈)))
10	7, 9	oveq12d 6897	. 2 ⊢ (𝑣 = 𝑉 → ((♯‘(𝑣 ∩ 𝑈)) − (♯‘(𝑣 ∖ 𝑈))) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))
11		ballotlemg	. 2 ⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
12		ovex 6911	. 2 ⊢ ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))) ∈ V
13	5, 10, 11, 12	ovmpt2 7031	1 ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈))))

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∀wral 3090  {crab 3094   ∖ cdif 3767   ∩ cin 3769  ifcif 4278  𝒫 cpw 4350   class class class wbr 4844   ↦ cmpt 4923   “ cima 5316  ‘cfv 6102  (class class class)co 6879   ↦ cmpt2 6881  Fincfn 8196  infcinf 8590  ℝcr 10224  0cc0 10225  1c1 10226   + caddc 10228   < clt 10364   ≤ cle 10365   − cmin 10557   / cdiv 10977  ℕcn 11313  ℤcz 11665  ...cfz 12579  ♯chash 13369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884

This theorem is referenced by:  ballotlemgun  31102  ballotlemfg  31103  ballotlemfrc  31104