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Theorem ballotlem7 34064
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 6581 . 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 34062 . 2 𝑅 = 𝑅
13 rabid 3446 . . . . . 6 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐))
143, 4, 5, 6, 7, 8, 9, 10, 11, 1ballotlemrc 34059 . . . . . . . 8 (𝑐 ∈ (𝑂𝐸) → (𝑅𝑐) ∈ (𝑂𝐸))
1514adantr 480 . . . . . . 7 ((𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐) → (𝑅𝑐) ∈ (𝑂𝐸))
163, 4, 5, 6, 7, 8, 9, 10ballotlem1c 34036 . . . . . . . . . 10 ((𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐) → ¬ (𝐼𝑐) ∈ 𝑐)
1716ex 412 . . . . . . . . 9 (𝑐 ∈ (𝑂𝐸) → (1 ∈ 𝑐 → ¬ (𝐼𝑐) ∈ 𝑐))
183, 4, 5, 6, 7, 8, 9, 10, 11, 1ballotlem1ri 34063 . . . . . . . . . 10 (𝑐 ∈ (𝑂𝐸) → (1 ∈ (𝑅𝑐) ↔ (𝐼𝑐) ∈ 𝑐))
1918notbid 318 . . . . . . . . 9 (𝑐 ∈ (𝑂𝐸) → (¬ 1 ∈ (𝑅𝑐) ↔ ¬ (𝐼𝑐) ∈ 𝑐))
2017, 19sylibrd 259 . . . . . . . 8 (𝑐 ∈ (𝑂𝐸) → (1 ∈ 𝑐 → ¬ 1 ∈ (𝑅𝑐)))
2120imp 406 . . . . . . 7 ((𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐) → ¬ 1 ∈ (𝑅𝑐))
2215, 21jca 511 . . . . . 6 ((𝑐 ∈ (𝑂𝐸) ∧ 1 ∈ 𝑐) → ((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐)))
2313, 22sylbi 216 . . . . 5 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} → ((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐)))
2423rgen 3057 . . . 4 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐))
25 eleq2 2816 . . . . . . . 8 (𝑏 = (𝑅𝑐) → (1 ∈ 𝑏 ↔ 1 ∈ (𝑅𝑐)))
2625notbid 318 . . . . . . 7 (𝑏 = (𝑅𝑐) → (¬ 1 ∈ 𝑏 ↔ ¬ 1 ∈ (𝑅𝑐)))
2726elrab 3678 . . . . . 6 ((𝑅𝑐) ∈ {𝑏 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑏} ↔ ((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐)))
28 eleq2 2816 . . . . . . . . 9 (𝑏 = 𝑐 → (1 ∈ 𝑏 ↔ 1 ∈ 𝑐))
2928notbid 318 . . . . . . . 8 (𝑏 = 𝑐 → (¬ 1 ∈ 𝑏 ↔ ¬ 1 ∈ 𝑐))
3029cbvrabv 3436 . . . . . . 7 {𝑏 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
3130eleq2i 2819 . . . . . 6 ((𝑅𝑐) ∈ {𝑏 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑏} ↔ (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
3227, 31bitr3i 277 . . . . 5 (((𝑅𝑐) ∈ (𝑂𝐸) ∧ ¬ 1 ∈ (𝑅𝑐)) ↔ (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
3332ralbii 3087 . . . 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 4072 . . . . 5 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂𝐸)
36 fvex 6898 . . . . . . 7 (𝑆𝑐) ∈ V
37 imaexg 7903 . . . . . . 7 ((𝑆𝑐) ∈ V → ((𝑆𝑐) “ 𝑐) ∈ V)
3836, 37ax-mp 5 . . . . . 6 ((𝑆𝑐) “ 𝑐) ∈ V
3938, 1dmmpti 6688 . . . . 5 dom 𝑅 = (𝑂𝐸)
4035, 39sseqtrri 4014 . . . 4 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ dom 𝑅
41 nfrab1 3445 . . . . 5 𝑐{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}
42 nfrab1 3445 . . . . 5 𝑐{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
43 nfmpt1 5249 . . . . . 6 𝑐(𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
441, 43nfcxfr 2895 . . . . 5 𝑐𝑅
4541, 42, 44funimass4f 32370 . . . 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 689 . . 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 3446 . . . . . 6 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐))
4914adantr 480 . . . . . . 7 ((𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐) → (𝑅𝑐) ∈ (𝑂𝐸))
503, 4, 5, 6, 7, 8, 9, 10ballotlemic 34035 . . . . . . . . . 10 ((𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐) → (𝐼𝑐) ∈ 𝑐)
5150ex 412 . . . . . . . . 9 (𝑐 ∈ (𝑂𝐸) → (¬ 1 ∈ 𝑐 → (𝐼𝑐) ∈ 𝑐))
5251, 18sylibrd 259 . . . . . . . 8 (𝑐 ∈ (𝑂𝐸) → (¬ 1 ∈ 𝑐 → 1 ∈ (𝑅𝑐)))
5352imp 406 . . . . . . 7 ((𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐) → 1 ∈ (𝑅𝑐))
5449, 53jca 511 . . . . . 6 ((𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐) → ((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐)))
5548, 54sylbi 216 . . . . 5 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} → ((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐)))
5655rgen 3057 . . . 4 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐))
5725elrab 3678 . . . . . 6 ((𝑅𝑐) ∈ {𝑏 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑏} ↔ ((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐)))
5828cbvrabv 3436 . . . . . . 7 {𝑏 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}
5958eleq2i 2819 . . . . . 6 ((𝑅𝑐) ∈ {𝑏 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑏} ↔ (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐})
6057, 59bitr3i 277 . . . . 5 (((𝑅𝑐) ∈ (𝑂𝐸) ∧ 1 ∈ (𝑅𝑐)) ↔ (𝑅𝑐) ∈ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐})
6160ralbii 3087 . . . 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 4072 . . . . 5 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂𝐸)
6463, 39sseqtrri 4014 . . . 4 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ dom 𝑅
6542, 41, 44funimass4f 32370 . . . 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 689 . . 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 32363 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 395   = wceq 1533  wcel 2098  wral 3055  {crab 3426  Vcvv 3468  cdif 3940  cin 3942  wss 3943  ifcif 4523  𝒫 cpw 4597   class class class wbr 5141  cmpt 5224  dom cdm 5669  cres 5671  cima 5672  Fun wfun 6531  1-1-ontowf1o 6536  cfv 6537  (class class class)co 7405  infcinf 9438  cr 11111  0cc0 11112  1c1 11113   + caddc 11115   < clt 11252  cle 11253  cmin 11448   / cdiv 11875  cn 12216  cz 12562  ...cfz 13490  chash 14295
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-oadd 8471  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12981  df-fz 13491  df-hash 14296
This theorem is referenced by:  ballotlem8  34065
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