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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlemi | Structured version Visualization version GIF version |
Description: Value of 𝐼 for a given counting 𝐶. (Contributed by Thierry Arnoux, 1-Dec-2016.) (Revised by AV, 6-Oct-2020.) |
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, < )) |
Ref | Expression |
---|---|
ballotlemi | ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6847 | . . . . . 6 ⊢ (𝑑 = 𝐶 → (𝐹‘𝑑) = (𝐹‘𝐶)) | |
2 | 1 | fveq1d 6849 | . . . . 5 ⊢ (𝑑 = 𝐶 → ((𝐹‘𝑑)‘𝑘) = ((𝐹‘𝐶)‘𝑘)) |
3 | 2 | eqeq1d 2739 | . . . 4 ⊢ (𝑑 = 𝐶 → (((𝐹‘𝑑)‘𝑘) = 0 ↔ ((𝐹‘𝐶)‘𝑘) = 0)) |
4 | 3 | rabbidv 3418 | . . 3 ⊢ (𝑑 = 𝐶 → {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}) |
5 | 4 | infeq1d 9420 | . 2 ⊢ (𝑑 = 𝐶 → inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0}, ℝ, < ) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )) |
6 | ballotth.i | . . 3 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < )) | |
7 | fveq2 6847 | . . . . . . . 8 ⊢ (𝑐 = 𝑑 → (𝐹‘𝑐) = (𝐹‘𝑑)) | |
8 | 7 | fveq1d 6849 | . . . . . . 7 ⊢ (𝑐 = 𝑑 → ((𝐹‘𝑐)‘𝑘) = ((𝐹‘𝑑)‘𝑘)) |
9 | 8 | eqeq1d 2739 | . . . . . 6 ⊢ (𝑐 = 𝑑 → (((𝐹‘𝑐)‘𝑘) = 0 ↔ ((𝐹‘𝑑)‘𝑘) = 0)) |
10 | 9 | rabbidv 3418 | . . . . 5 ⊢ (𝑐 = 𝑑 → {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0}) |
11 | 10 | infeq1d 9420 | . . . 4 ⊢ (𝑐 = 𝑑 → inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0}, ℝ, < )) |
12 | 11 | cbvmptv 5223 | . . 3 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < )) = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0}, ℝ, < )) |
13 | 6, 12 | eqtri 2765 | . 2 ⊢ 𝐼 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑑)‘𝑘) = 0}, ℝ, < )) |
14 | ltso 11242 | . . 3 ⊢ < Or ℝ | |
15 | 14 | infex 9436 | . 2 ⊢ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ) ∈ V |
16 | 5, 13, 15 | fvmpt 6953 | 1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∀wral 3065 {crab 3410 ∖ cdif 3912 ∩ cin 3914 𝒫 cpw 4565 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6501 (class class class)co 7362 infcinf 9384 ℝcr 11057 0cc0 11058 1c1 11059 + caddc 11061 < clt 11196 − cmin 11392 / cdiv 11819 ℕcn 12160 ℤcz 12506 ...cfz 13431 ♯chash 14237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-pre-lttri 11132 ax-pre-lttrn 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-ltxr 11201 |
This theorem is referenced by: ballotlemiex 33141 ballotlemimin 33145 ballotlemfrcn0 33169 ballotlemirc 33171 |
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