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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 6905 | . . 3 ⊢ (𝑑 = 𝐶 → (𝑆‘𝑑) = (𝑆‘𝐶)) | |
| 2 | id 22 | . . 3 ⊢ (𝑑 = 𝐶 → 𝑑 = 𝐶) | |
| 3 | 1, 2 | imaeq12d 6078 | . 2 ⊢ (𝑑 = 𝐶 → ((𝑆‘𝑑) “ 𝑑) = ((𝑆‘𝐶) “ 𝐶)) | 
| 4 | ballotth.r | . . 3 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) | |
| 5 | fveq2 6905 | . . . . 5 ⊢ (𝑐 = 𝑑 → (𝑆‘𝑐) = (𝑆‘𝑑)) | |
| 6 | id 22 | . . . . 5 ⊢ (𝑐 = 𝑑 → 𝑐 = 𝑑) | |
| 7 | 5, 6 | imaeq12d 6078 | . . . 4 ⊢ (𝑐 = 𝑑 → ((𝑆‘𝑐) “ 𝑐) = ((𝑆‘𝑑) “ 𝑑)) | 
| 8 | 7 | cbvmptv 5254 | . . 3 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑑) “ 𝑑)) | 
| 9 | 4, 8 | eqtri 2764 | . 2 ⊢ 𝑅 = (𝑑 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑑) “ 𝑑)) | 
| 10 | fvex 6918 | . . 3 ⊢ (𝑆‘𝐶) ∈ V | |
| 11 | imaexg 7936 | . . 3 ⊢ ((𝑆‘𝐶) ∈ V → ((𝑆‘𝐶) “ 𝐶) ∈ V) | |
| 12 | 10, 11 | ax-mp 5 | . 2 ⊢ ((𝑆‘𝐶) “ 𝐶) ∈ V | 
| 13 | 3, 9, 12 | fvmpt 7015 | 1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∀wral 3060 {crab 3435 Vcvv 3479 ∖ cdif 3947 ∩ cin 3949 ifcif 4524 𝒫 cpw 4599 class class class wbr 5142 ↦ cmpt 5224 “ cima 5687 ‘cfv 6560 (class class class)co 7432 infcinf 9482 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 < clt 11296 ≤ cle 11297 − cmin 11493 / cdiv 11921 ℕcn 12267 ℤcz 12615 ...cfz 13548 ♯chash 14370 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 | 
| This theorem is referenced by: ballotlemscr 34522 ballotlemrv 34523 ballotlemro 34526 ballotlemrinv0 34536 | 
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