Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlemrv2 | Structured version Visualization version GIF version |
Description: Value of 𝑅 after the tie. (Contributed by Thierry Arnoux, 11-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 |
---|---|
ballotlemrv2 | ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ (𝐼‘𝐶) < 𝐽) → (𝐽 ∈ (𝑅‘𝐶) ↔ 𝐽 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ballotth.m | . . . 4 ⊢ 𝑀 ∈ ℕ | |
2 | ballotth.n | . . . 4 ⊢ 𝑁 ∈ ℕ | |
3 | ballotth.o | . . . 4 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} | |
4 | ballotth.p | . . . 4 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂))) | |
5 | ballotth.f | . . . 4 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))))) | |
6 | ballotth.e | . . . 4 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)} | |
7 | ballotth.mgtn | . . . 4 ⊢ 𝑁 < 𝑀 | |
8 | ballotth.i | . . . 4 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < )) | |
9 | ballotth.s | . . . 4 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖))) | |
10 | ballotth.r | . . . 4 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | ballotlemrv 31676 | . . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → (𝐽 ∈ (𝑅‘𝐶) ↔ if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽) ∈ 𝐶)) |
12 | 11 | 3adant3 1124 | . 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ (𝐼‘𝐶) < 𝐽) → (𝐽 ∈ (𝑅‘𝐶) ↔ if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽) ∈ 𝐶)) |
13 | fzssuz 12936 | . . . . . . . . . 10 ⊢ (1...(𝑀 + 𝑁)) ⊆ (ℤ≥‘1) | |
14 | uzssz 12252 | . . . . . . . . . 10 ⊢ (ℤ≥‘1) ⊆ ℤ | |
15 | 13, 14 | sstri 3973 | . . . . . . . . 9 ⊢ (1...(𝑀 + 𝑁)) ⊆ ℤ |
16 | 1, 2, 3, 4, 5, 6, 7, 8 | ballotlemiex 31658 | . . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)) |
17 | 16 | simpld 495 | . . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁))) |
18 | 15, 17 | sseldi 3962 | . . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ) |
19 | 18 | adantr 481 | . . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → (𝐼‘𝐶) ∈ ℤ) |
20 | 19 | zred 12075 | . . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → (𝐼‘𝐶) ∈ ℝ) |
21 | simpr 485 | . . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → 𝐽 ∈ (1...(𝑀 + 𝑁))) | |
22 | 15, 21 | sseldi 3962 | . . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → 𝐽 ∈ ℤ) |
23 | 22 | zred 12075 | . . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → 𝐽 ∈ ℝ) |
24 | 20, 23 | ltnled 10775 | . . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝐼‘𝐶) < 𝐽 ↔ ¬ 𝐽 ≤ (𝐼‘𝐶))) |
25 | 24 | biimp3a 1460 | . . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ (𝐼‘𝐶) < 𝐽) → ¬ 𝐽 ≤ (𝐼‘𝐶)) |
26 | 25 | iffalsed 4474 | . . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ (𝐼‘𝐶) < 𝐽) → if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽) = 𝐽) |
27 | 26 | eleq1d 2894 | . 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ (𝐼‘𝐶) < 𝐽) → (if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽) ∈ 𝐶 ↔ 𝐽 ∈ 𝐶)) |
28 | 12, 27 | bitrd 280 | 1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ (𝐼‘𝐶) < 𝐽) → (𝐽 ∈ (𝑅‘𝐶) ↔ 𝐽 ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 ∖ cdif 3930 ∩ cin 3932 ifcif 4463 𝒫 cpw 4535 class class class wbr 5057 ↦ cmpt 5137 “ cima 5551 ‘cfv 6348 (class class class)co 7145 infcinf 8893 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 < clt 10663 ≤ cle 10664 − cmin 10858 / cdiv 11285 ℕcn 11626 ℤcz 11969 ℤ≥cuz 12231 ...cfz 12880 ♯chash 13678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-hash 13679 |
This theorem is referenced by: (None) |
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