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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlemsi | Structured version Visualization version GIF version |
Description: The image by 𝑆 of the first tie pick is the first pick. (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))) |
Ref | Expression |
---|---|
ballotlemsi | ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶)‘(𝐼‘𝐶)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ballotth.m | . . . . 5 ⊢ 𝑀 ∈ ℕ | |
2 | ballotth.n | . . . . 5 ⊢ 𝑁 ∈ ℕ | |
3 | ballotth.o | . . . . 5 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} | |
4 | ballotth.p | . . . . 5 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂))) | |
5 | ballotth.f | . . . . 5 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))))) | |
6 | ballotth.e | . . . . 5 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} | |
7 | ballotth.mgtn | . . . . 5 ⊢ 𝑁 < 𝑀 | |
8 | ballotth.i | . . . . 5 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < )) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | ballotlemiex 34291 | . . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)) |
10 | 9 | simpld 493 | . . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁))) |
11 | ballotth.s | . . . 4 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖))) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 11 | ballotlemsv 34299 | . . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘(𝐼‘𝐶)) = if((𝐼‘𝐶) ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − (𝐼‘𝐶)), (𝐼‘𝐶))) |
13 | 10, 12 | mpdan 685 | . 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶)‘(𝐼‘𝐶)) = if((𝐼‘𝐶) ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − (𝐼‘𝐶)), (𝐼‘𝐶))) |
14 | elfzelz 13550 | . . . . . 6 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℤ) | |
15 | 14 | zred 12713 | . . . . 5 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℝ) |
16 | 10, 15 | syl 17 | . . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℝ) |
17 | 16 | leidd 11826 | . . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ≤ (𝐼‘𝐶)) |
18 | 17 | iftrued 4540 | . 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → if((𝐼‘𝐶) ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − (𝐼‘𝐶)), (𝐼‘𝐶)) = (((𝐼‘𝐶) + 1) − (𝐼‘𝐶))) |
19 | 16 | recnd 11288 | . . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℂ) |
20 | 1cnd 11255 | . . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℂ) | |
21 | 19, 20 | pncan2d 11619 | . 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (((𝐼‘𝐶) + 1) − (𝐼‘𝐶)) = 1) |
22 | 13, 18, 21 | 3eqtrd 2769 | 1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶)‘(𝐼‘𝐶)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∀wral 3050 {crab 3418 ∖ cdif 3943 ∩ cin 3945 ifcif 4532 𝒫 cpw 4606 class class class wbr 5152 ↦ cmpt 5235 ‘cfv 6553 (class class class)co 7423 infcinf 9480 ℝcr 11153 0cc0 11154 1c1 11155 + caddc 11157 < clt 11294 ≤ cle 11295 − cmin 11490 / cdiv 11917 ℕcn 12259 ℤcz 12605 ...cfz 13533 ♯chash 14342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-oadd 8499 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9481 df-inf 9482 df-dju 9940 df-card 9978 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-n0 12520 df-z 12606 df-uz 12870 df-fz 13534 df-hash 14343 |
This theorem is referenced by: ballotlemfrci 34317 |
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