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Theorem ballotlem7 33522
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 6584 . 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 33520 . 2 𝑅 = 𝑅
13 rabid 3452 . . . . . 6 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐))
143, 4, 5, 6, 7, 8, 9, 10, 11, 1ballotlemrc 33517 . . . . . . . 8 (𝑐 ∈ (𝑂𝐸) → (𝑅𝑐) ∈ (𝑂𝐸))
1514adantr 481 . . . . . . 7 ((𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐) → (𝑅𝑐) ∈ (𝑂𝐸))
163, 4, 5, 6, 7, 8, 9, 10ballotlem1c 33494 . . . . . . . . . 10 ((𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐) → ¬ (𝐼𝑐) ∈ 𝑐)
1716ex 413 . . . . . . . . 9 (𝑐 ∈ (𝑂𝐸) → (1 ∈ 𝑐 → ¬ (𝐼𝑐) ∈ 𝑐))
183, 4, 5, 6, 7, 8, 9, 10, 11, 1ballotlem1ri 33521 . . . . . . . . . 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 407 . . . . . . 7 ((𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐) → ¬ 1 ∈ (𝑅𝑐))
2215, 21jca 512 . . . . . 6 ((𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐) → ((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐)))
2313, 22sylbi 216 . . . . 5 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} → ((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐)))
2423rgen 3063 . . . 4 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐))
25 eleq2 2822 . . . . . . . 8 (𝑏 = (𝑅𝑐) → (1 ∈ 𝑏 ↔ 1 ∈ (𝑅𝑐)))
2625notbid 317 . . . . . . 7 (𝑏 = (𝑅𝑐) → (¬ 1 ∈ 𝑏 ↔ ¬ 1 ∈ (𝑅𝑐)))
2726elrab 3682 . . . . . 6 ((𝑅𝑐) ∈ {𝑏 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑏} ↔ ((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐)))
28 eleq2 2822 . . . . . . . . 9 (𝑏 = 𝑐 → (1 ∈ 𝑏 ↔ 1 ∈ 𝑐))
2928notbid 317 . . . . . . . 8 (𝑏 = 𝑐 → (¬ 1 ∈ 𝑏 ↔ ¬ 1 ∈ 𝑐))
3029cbvrabv 3442 . . . . . . 7 {𝑏 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
3130eleq2i 2825 . . . . . 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 3093 . . . 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 6901 . . . . . . 7 (𝑆𝑐) ∈ V
37 imaexg 7902 . . . . . . 7 ((𝑆𝑐) ∈ V → ((𝑆𝑐) “ 𝑐) ∈ V)
3836, 37ax-mp 5 . . . . . 6 ((𝑆𝑐) “ 𝑐) ∈ V
3938, 1dmmpti 6691 . . . . 5 dom 𝑅 = (𝑂𝐸)
4035, 39sseqtrri 4018 . . . 4 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ dom 𝑅
41 nfrab1 3451 . . . . 5 𝑐{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}
42 nfrab1 3451 . . . . 5 𝑐{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
43 nfmpt1 5255 . . . . . 6 𝑐(𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
441, 43nfcxfr 2901 . . . . 5 𝑐𝑅
4541, 42, 44funimass4f 31848 . . . 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 3452 . . . . . 6 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐))
4914adantr 481 . . . . . . 7 ((𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐) → (𝑅𝑐) ∈ (𝑂𝐸))
503, 4, 5, 6, 7, 8, 9, 10ballotlemic 33493 . . . . . . . . . 10 ((𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐) → (𝐼𝑐) ∈ 𝑐)
5150ex 413 . . . . . . . . 9 (𝑐 ∈ (𝑂𝐸) → (¬ 1 ∈ 𝑐 → (𝐼𝑐) ∈ 𝑐))
5251, 18sylibrd 258 . . . . . . . 8 (𝑐 ∈ (𝑂𝐸) → (¬ 1 ∈ 𝑐 → 1 ∈ (𝑅𝑐)))
5352imp 407 . . . . . . 7 ((𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐) → 1 ∈ (𝑅𝑐))
5449, 53jca 512 . . . . . 6 ((𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐) → ((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐)))
5548, 54sylbi 216 . . . . 5 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} → ((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐)))
5655rgen 3063 . . . 4 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐))
5725elrab 3682 . . . . . 6 ((𝑅𝑐) ∈ {𝑏 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑏} ↔ ((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐)))
5828cbvrabv 3442 . . . . . . 7 {𝑏 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}
5958eleq2i 2825 . . . . . 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 3093 . . . 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 4018 . . . 4 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ dom 𝑅
6542, 41, 44funimass4f 31848 . . . 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 31841 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 396   = wceq 1541  wcel 2106  wral 3061  {crab 3432  Vcvv 3474  cdif 3944  cin 3946  wss 3947  ifcif 4527  𝒫 cpw 4601   class class class wbr 5147  cmpt 5230  dom cdm 5675  cres 5677  cima 5678  Fun wfun 6534  1-1-ontowf1o 6539  cfv 6540  (class class class)co 7405  infcinf 9432  cr 11105  0cc0 11106  1c1 11107   + caddc 11109   < clt 11244  cle 11245  cmin 11440   / cdiv 11867  cn 12208  cz 12554  ...cfz 13480  chash 14286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-fz 13481  df-hash 14287
This theorem is referenced by:  ballotlem8  33523
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