Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ballotlem7 Structured version   Visualization version   GIF version

Theorem ballotlem7 34369
Description: 𝑅 is a bijection between two subsets of (𝑂𝐸): one where a vote for A is picked first, and one where a vote for B is picked first. (Contributed by Thierry Arnoux, 12-Dec-2016.)
Hypotheses
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 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
ballotth.mgtn 𝑁 < 𝑀
ballotth.i 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
Assertion
Ref Expression
ballotlem7 (𝑅 ↾ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}):{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}–1-1-onto→{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂   𝑘,𝑀   𝑘,𝑁   𝑘,𝑂   𝑖,𝑐,𝐹,𝑘   𝑖,𝐸,𝑘   𝑘,𝐼,𝑐   𝐸,𝑐   𝑖,𝐼,𝑐   𝑆,𝑘,𝑖,𝑐   𝑅,𝑖,𝑘   𝑥,𝑐,𝐹   𝑥,𝑀   𝑥,𝑁,𝑘,𝑖
Allowed substitution hints:   𝑃(𝑥,𝑖,𝑘,𝑐)   𝑅(𝑥,𝑐)   𝑆(𝑥)   𝐸(𝑥)   𝐼(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlem7
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 ballotth.r . . 3 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
21funmpt2 6598 . 2 Fun 𝑅
3 ballotth.m . . 3 𝑀 ∈ ℕ
4 ballotth.n . . 3 𝑁 ∈ ℕ
5 ballotth.o . . 3 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
6 ballotth.p . . 3 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
7 ballotth.f . . 3 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
8 ballotth.e . . 3 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
9 ballotth.mgtn . . 3 𝑁 < 𝑀
10 ballotth.i . . 3 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
11 ballotth.s . . 3 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
123, 4, 5, 6, 7, 8, 9, 10, 11, 1ballotlemrinv 34367 . 2 𝑅 = 𝑅
13 rabid 3440 . . . . . 6 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐))
143, 4, 5, 6, 7, 8, 9, 10, 11, 1ballotlemrc 34364 . . . . . . . 8 (𝑐 ∈ (𝑂𝐸) → (𝑅𝑐) ∈ (𝑂𝐸))
1514adantr 479 . . . . . . 7 ((𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐) → (𝑅𝑐) ∈ (𝑂𝐸))
163, 4, 5, 6, 7, 8, 9, 10ballotlem1c 34341 . . . . . . . . . 10 ((𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐) → ¬ (𝐼𝑐) ∈ 𝑐)
1716ex 411 . . . . . . . . 9 (𝑐 ∈ (𝑂𝐸) → (1 ∈ 𝑐 → ¬ (𝐼𝑐) ∈ 𝑐))
183, 4, 5, 6, 7, 8, 9, 10, 11, 1ballotlem1ri 34368 . . . . . . . . . 10 (𝑐 ∈ (𝑂𝐸) → (1 ∈ (𝑅𝑐) ↔ (𝐼𝑐) ∈ 𝑐))
1918notbid 317 . . . . . . . . 9 (𝑐 ∈ (𝑂𝐸) → (¬ 1 ∈ (𝑅𝑐) ↔ ¬ (𝐼𝑐) ∈ 𝑐))
2017, 19sylibrd 258 . . . . . . . 8 (𝑐 ∈ (𝑂𝐸) → (1 ∈ 𝑐 → ¬ 1 ∈ (𝑅𝑐)))
2120imp 405 . . . . . . 7 ((𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐) → ¬ 1 ∈ (𝑅𝑐))
2215, 21jca 510 . . . . . 6 ((𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐) → ((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐)))
2313, 22sylbi 216 . . . . 5 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} → ((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐)))
2423rgen 3053 . . . 4 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐))
25 eleq2 2815 . . . . . . . 8 (𝑏 = (𝑅𝑐) → (1 ∈ 𝑏 ↔ 1 ∈ (𝑅𝑐)))
2625notbid 317 . . . . . . 7 (𝑏 = (𝑅𝑐) → (¬ 1 ∈ 𝑏 ↔ ¬ 1 ∈ (𝑅𝑐)))
2726elrab 3681 . . . . . 6 ((𝑅𝑐) ∈ {𝑏 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑏} ↔ ((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐)))
28 eleq2 2815 . . . . . . . . 9 (𝑏 = 𝑐 → (1 ∈ 𝑏 ↔ 1 ∈ 𝑐))
2928notbid 317 . . . . . . . 8 (𝑏 = 𝑐 → (¬ 1 ∈ 𝑏 ↔ ¬ 1 ∈ 𝑐))
3029cbvrabv 3430 . . . . . . 7 {𝑏 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
3130eleq2i 2818 . . . . . 6 ((𝑅𝑐) ∈ {𝑏 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑏} ↔ (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
3227, 31bitr3i 276 . . . . 5 (((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐)) ↔ (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
3332ralbii 3083 . . . 4 (∀𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐)) ↔ ∀𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
3424, 33mpbi 229 . . 3 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
35 ssrab2 4076 . . . . 5 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂𝐸)
36 fvex 6914 . . . . . . 7 (𝑆𝑐) ∈ V
37 imaexg 7926 . . . . . . 7 ((𝑆𝑐) ∈ V → ((𝑆𝑐) “ 𝑐) ∈ V)
3836, 37ax-mp 5 . . . . . 6 ((𝑆𝑐) “ 𝑐) ∈ V
3938, 1dmmpti 6705 . . . . 5 dom 𝑅 = (𝑂𝐸)
4035, 39sseqtrri 4017 . . . 4 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ dom 𝑅
41 nfrab1 3439 . . . . 5 𝑐{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}
42 nfrab1 3439 . . . . 5 𝑐{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
43 nfmpt1 5261 . . . . . 6 𝑐(𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
441, 43nfcxfr 2890 . . . . 5 𝑐𝑅
4541, 42, 44funimass4f 32554 . . . 4 ((Fun 𝑅 ∧ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ dom 𝑅) → ((𝑅 “ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ ∀𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
462, 40, 45mp2an 690 . . 3 ((𝑅 “ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ ∀𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
4734, 46mpbir 230 . 2 (𝑅 “ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
48 rabid 3440 . . . . . 6 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐))
4914adantr 479 . . . . . . 7 ((𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐) → (𝑅𝑐) ∈ (𝑂𝐸))
503, 4, 5, 6, 7, 8, 9, 10ballotlemic 34340 . . . . . . . . . 10 ((𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐) → (𝐼𝑐) ∈ 𝑐)
5150ex 411 . . . . . . . . 9 (𝑐 ∈ (𝑂𝐸) → (¬ 1 ∈ 𝑐 → (𝐼𝑐) ∈ 𝑐))
5251, 18sylibrd 258 . . . . . . . 8 (𝑐 ∈ (𝑂𝐸) → (¬ 1 ∈ 𝑐 → 1 ∈ (𝑅𝑐)))
5352imp 405 . . . . . . 7 ((𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐) → 1 ∈ (𝑅𝑐))
5449, 53jca 510 . . . . . 6 ((𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐) → ((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐)))
5548, 54sylbi 216 . . . . 5 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} → ((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐)))
5655rgen 3053 . . . 4 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐))
5725elrab 3681 . . . . . 6 ((𝑅𝑐) ∈ {𝑏 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑏} ↔ ((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐)))
5828cbvrabv 3430 . . . . . . 7 {𝑏 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}
5958eleq2i 2818 . . . . . 6 ((𝑅𝑐) ∈ {𝑏 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑏} ↔ (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐})
6057, 59bitr3i 276 . . . . 5 (((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐)) ↔ (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐})
6160ralbii 3083 . . . 4 (∀𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐)) ↔ ∀𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐})
6256, 61mpbi 229 . . 3 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}
63 ssrab2 4076 . . . . 5 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂𝐸)
6463, 39sseqtrri 4017 . . . 4 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ dom 𝑅
6542, 41, 44funimass4f 32554 . . . 4 ((Fun 𝑅 ∧ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ dom 𝑅) → ((𝑅 “ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ↔ ∀𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}))
662, 64, 65mp2an 690 . . 3 ((𝑅 “ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ↔ ∀𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐})
6762, 66mpbir 230 . 2 (𝑅 “ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}
682, 12, 47, 67, 40, 64rinvf1o 32547 1 (𝑅 ↾ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}):{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}–1-1-onto→{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  {crab 3419  Vcvv 3462  cdif 3944  cin 3946  wss 3947  ifcif 4533  𝒫 cpw 4607   class class class wbr 5153  cmpt 5236  dom cdm 5682  cres 5684  cima 5685  Fun wfun 6548  1-1-ontowf1o 6553  cfv 6554  (class class class)co 7424  infcinf 9484  cr 11157  0cc0 11158  1c1 11159   + caddc 11161   < clt 11298  cle 11299  cmin 11494   / cdiv 11921  cn 12264  cz 12610  ...cfz 13538  chash 14347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-oadd 8500  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-inf 9486  df-dju 9944  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-n0 12525  df-z 12611  df-uz 12875  df-rp 13029  df-fz 13539  df-hash 14348
This theorem is referenced by:  ballotlem8  34370
  Copyright terms: Public domain W3C validator