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Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlemrval | Structured version Visualization version GIF version |
Description: Value of 𝑅. (Contributed by Thierry Arnoux, 14-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)) |
Ref | Expression |
---|---|
ballotlemrval | ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6669 | . . 3 ⊢ (𝑑 = 𝐶 → (𝑆‘𝑑) = (𝑆‘𝐶)) | |
2 | id 22 | . . 3 ⊢ (𝑑 = 𝐶 → 𝑑 = 𝐶) | |
3 | 1, 2 | imaeq12d 5929 | . 2 ⊢ (𝑑 = 𝐶 → ((𝑆‘𝑑) “ 𝑑) = ((𝑆‘𝐶) “ 𝐶)) |
4 | ballotth.r | . . 3 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) | |
5 | fveq2 6669 | . . . . 5 ⊢ (𝑐 = 𝑑 → (𝑆‘𝑐) = (𝑆‘𝑑)) | |
6 | id 22 | . . . . 5 ⊢ (𝑐 = 𝑑 → 𝑐 = 𝑑) | |
7 | 5, 6 | imaeq12d 5929 | . . . 4 ⊢ (𝑐 = 𝑑 → ((𝑆‘𝑐) “ 𝑐) = ((𝑆‘𝑑) “ 𝑑)) |
8 | 7 | cbvmptv 5168 | . . 3 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑑) “ 𝑑)) |
9 | 4, 8 | eqtri 2844 | . 2 ⊢ 𝑅 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑑) “ 𝑑)) |
10 | fvex 6682 | . . 3 ⊢ (𝑆‘𝐶) ∈ V | |
11 | imaexg 7619 | . . 3 ⊢ ((𝑆‘𝐶) ∈ V → ((𝑆‘𝐶) “ 𝐶) ∈ V) | |
12 | 10, 11 | ax-mp 5 | . 2 ⊢ ((𝑆‘𝐶) “ 𝐶) ∈ V |
13 | 3, 9, 12 | fvmpt 6767 | 1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 Vcvv 3494 ∖ cdif 3932 ∩ cin 3934 ifcif 4466 𝒫 cpw 4538 class class class wbr 5065 ↦ cmpt 5145 “ cima 5557 ‘cfv 6354 (class class class)co 7155 infcinf 8904 ℝcr 10535 0cc0 10536 1c1 10537 + caddc 10539 < clt 10674 ≤ cle 10675 − cmin 10869 / cdiv 11296 ℕcn 11637 ℤcz 11980 ...cfz 12891 ♯chash 13689 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fv 6362 |
This theorem is referenced by: ballotlemscr 31776 ballotlemrv 31777 ballotlemro 31780 ballotlemrinv0 31790 |
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