Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlemgval | Structured version Visualization version GIF version |
Description: Expand the value of ↑. (Contributed by Thierry Arnoux, 21-Apr-2017.) |
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)))) |
Ref | Expression |
---|---|
ballotlemgval | ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 ↑ 𝑉) = ((#‘(𝑉 ∩ 𝑈)) − (#‘(𝑉 ∖ 𝑈)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq2 3770 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝑣 ∩ 𝑢) = (𝑣 ∩ 𝑈)) | |
2 | 1 | fveq2d 6107 | . . 3 ⊢ (𝑢 = 𝑈 → (#‘(𝑣 ∩ 𝑢)) = (#‘(𝑣 ∩ 𝑈))) |
3 | difeq2 3684 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝑣 ∖ 𝑢) = (𝑣 ∖ 𝑈)) | |
4 | 3 | fveq2d 6107 | . . 3 ⊢ (𝑢 = 𝑈 → (#‘(𝑣 ∖ 𝑢)) = (#‘(𝑣 ∖ 𝑈))) |
5 | 2, 4 | oveq12d 6567 | . 2 ⊢ (𝑢 = 𝑈 → ((#‘(𝑣 ∩ 𝑢)) − (#‘(𝑣 ∖ 𝑢))) = ((#‘(𝑣 ∩ 𝑈)) − (#‘(𝑣 ∖ 𝑈)))) |
6 | ineq1 3769 | . . . 4 ⊢ (𝑣 = 𝑉 → (𝑣 ∩ 𝑈) = (𝑉 ∩ 𝑈)) | |
7 | 6 | fveq2d 6107 | . . 3 ⊢ (𝑣 = 𝑉 → (#‘(𝑣 ∩ 𝑈)) = (#‘(𝑉 ∩ 𝑈))) |
8 | difeq1 3683 | . . . 4 ⊢ (𝑣 = 𝑉 → (𝑣 ∖ 𝑈) = (𝑉 ∖ 𝑈)) | |
9 | 8 | fveq2d 6107 | . . 3 ⊢ (𝑣 = 𝑉 → (#‘(𝑣 ∖ 𝑈)) = (#‘(𝑉 ∖ 𝑈))) |
10 | 7, 9 | oveq12d 6567 | . 2 ⊢ (𝑣 = 𝑉 → ((#‘(𝑣 ∩ 𝑈)) − (#‘(𝑣 ∖ 𝑈))) = ((#‘(𝑉 ∩ 𝑈)) − (#‘(𝑉 ∖ 𝑈)))) |
11 | ballotlemg | . 2 ⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((#‘(𝑣 ∩ 𝑢)) − (#‘(𝑣 ∖ 𝑢)))) | |
12 | ovex 6577 | . 2 ⊢ ((#‘(𝑉 ∩ 𝑈)) − (#‘(𝑉 ∖ 𝑈))) ∈ V | |
13 | 5, 10, 11, 12 | ovmpt2 6694 | 1 ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 ↑ 𝑉) = ((#‘(𝑉 ∩ 𝑈)) − (#‘(𝑉 ∖ 𝑈)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∖ cdif 3537 ∩ cin 3539 ifcif 4036 𝒫 cpw 4108 class class class wbr 4583 ↦ cmpt 4643 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Fincfn 7841 infcinf 8230 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 ℤcz 11254 ...cfz 12197 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: ballotlemgun 29913 ballotlemfg 29914 ballotlemfrc 29915 |
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