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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotleme | Structured version Visualization version GIF version | ||
| Description: Elements of 𝐸. (Contributed by Thierry Arnoux, 14-Dec-2016.) |
| 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)} |
| Ref | Expression |
|---|---|
| ballotleme | ⊢ (𝐶 ∈ 𝐸 ↔ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 6904 | . . . . 5 ⊢ (𝑑 = 𝐶 → (𝐹‘𝑑) = (𝐹‘𝐶)) | |
| 2 | 1 | fveq1d 6906 | . . . 4 ⊢ (𝑑 = 𝐶 → ((𝐹‘𝑑)‘𝑖) = ((𝐹‘𝐶)‘𝑖)) |
| 3 | 2 | breq2d 5153 | . . 3 ⊢ (𝑑 = 𝐶 → (0 < ((𝐹‘𝑑)‘𝑖) ↔ 0 < ((𝐹‘𝐶)‘𝑖))) |
| 4 | 3 | ralbidv 3177 | . 2 ⊢ (𝑑 = 𝐶 → (∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑑)‘𝑖) ↔ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))) |
| 5 | ballotth.e | . . 3 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} | |
| 6 | fveq2 6904 | . . . . . . 7 ⊢ (𝑐 = 𝑑 → (𝐹‘𝑐) = (𝐹‘𝑑)) | |
| 7 | 6 | fveq1d 6906 | . . . . . 6 ⊢ (𝑐 = 𝑑 → ((𝐹‘𝑐)‘𝑖) = ((𝐹‘𝑑)‘𝑖)) |
| 8 | 7 | breq2d 5153 | . . . . 5 ⊢ (𝑐 = 𝑑 → (0 < ((𝐹‘𝑐)‘𝑖) ↔ 0 < ((𝐹‘𝑑)‘𝑖))) |
| 9 | 8 | ralbidv 3177 | . . . 4 ⊢ (𝑐 = 𝑑 → (∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖) ↔ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑑)‘𝑖))) |
| 10 | 9 | cbvrabv 3446 | . . 3 ⊢ {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} = {𝑑 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑑)‘𝑖)} |
| 11 | 5, 10 | eqtri 2764 | . 2 ⊢ 𝐸 = {𝑑 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑑)‘𝑖)} |
| 12 | 4, 11 | elrab2 3694 | 1 ⊢ (𝐶 ∈ 𝐸 ↔ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3060 {crab 3435 ∖ cdif 3947 ∩ cin 3949 𝒫 cpw 4598 class class class wbr 5141 ↦ cmpt 5223 ‘cfv 6559 (class class class)co 7429 0cc0 11151 1c1 11152 + caddc 11154 < clt 11291 − cmin 11488 / cdiv 11916 ℕcn 12262 ℤcz 12609 ...cfz 13543 ♯chash 14365 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-iota 6512 df-fv 6567 |
| This theorem is referenced by: ballotlemodife 34478 ballotlem4 34479 |
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