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Theorem ballotth 34308
Description: Bertrand's ballot problem : the probability that A is ahead throughout the counting. The proof formalized here is a proof "by reflection", as opposed to other known proofs "by induction" or "by permutation". This is Metamath 100 proof #30. (Contributed by Thierry Arnoux, 7-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
ballotth (𝑃𝐸) = ((𝑀𝑁) / (𝑀 + 𝑁))
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   𝑥,𝐸   𝑥,𝑂
Allowed substitution hints:   𝑃(𝑥,𝑖,𝑘,𝑐)   𝑅(𝑥,𝑐)   𝑆(𝑥)   𝐼(𝑥)

Proof of Theorem ballotth
StepHypRef Expression
1 ballotth.e . . . . . 6 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
2 ssrab2 4073 . . . . . 6 {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)} ⊆ 𝑂
31, 2eqsstri 4011 . . . . 5 𝐸𝑂
4 fzfi 13978 . . . . . . . . . . 11 (1...(𝑀 + 𝑁)) ∈ Fin
5 pwfi 9206 . . . . . . . . . . 11 ((1...(𝑀 + 𝑁)) ∈ Fin ↔ 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin)
64, 5mpbi 229 . . . . . . . . . 10 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin
7 ballotth.o . . . . . . . . . . 11 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
8 ssrab2 4073 . . . . . . . . . . 11 {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} ⊆ 𝒫 (1...(𝑀 + 𝑁))
97, 8eqsstri 4011 . . . . . . . . . 10 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))
10 ssfi 9201 . . . . . . . . . 10 ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))) → 𝑂 ∈ Fin)
116, 9, 10mp2an 690 . . . . . . . . 9 𝑂 ∈ Fin
12 ssfi 9201 . . . . . . . . 9 ((𝑂 ∈ Fin ∧ 𝐸𝑂) → 𝐸 ∈ Fin)
1311, 3, 12mp2an 690 . . . . . . . 8 𝐸 ∈ Fin
1413elexi 3482 . . . . . . 7 𝐸 ∈ V
1514elpw 4608 . . . . . 6 (𝐸 ∈ 𝒫 𝑂𝐸𝑂)
16 fveq2 6896 . . . . . . . 8 (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸))
1716oveq1d 7434 . . . . . . 7 (𝑥 = 𝐸 → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂)))
18 ballotth.p . . . . . . 7 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
19 ovex 7452 . . . . . . 7 ((♯‘𝐸) / (♯‘𝑂)) ∈ V
2017, 18, 19fvmpt 7004 . . . . . 6 (𝐸 ∈ 𝒫 𝑂 → (𝑃𝐸) = ((♯‘𝐸) / (♯‘𝑂)))
2115, 20sylbir 234 . . . . 5 (𝐸𝑂 → (𝑃𝐸) = ((♯‘𝐸) / (♯‘𝑂)))
223, 21ax-mp 5 . . . 4 (𝑃𝐸) = ((♯‘𝐸) / (♯‘𝑂))
23 hashssdif 14415 . . . . . . . 8 ((𝑂 ∈ Fin ∧ 𝐸𝑂) → (♯‘(𝑂𝐸)) = ((♯‘𝑂) − (♯‘𝐸)))
2411, 3, 23mp2an 690 . . . . . . 7 (♯‘(𝑂𝐸)) = ((♯‘𝑂) − (♯‘𝐸))
2524eqcomi 2734 . . . . . 6 ((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂𝐸))
26 hashcl 14359 . . . . . . . . 9 (𝑂 ∈ Fin → (♯‘𝑂) ∈ ℕ0)
2711, 26ax-mp 5 . . . . . . . 8 (♯‘𝑂) ∈ ℕ0
2827nn0cni 12522 . . . . . . 7 (♯‘𝑂) ∈ ℂ
29 hashcl 14359 . . . . . . . . 9 (𝐸 ∈ Fin → (♯‘𝐸) ∈ ℕ0)
3013, 29ax-mp 5 . . . . . . . 8 (♯‘𝐸) ∈ ℕ0
3130nn0cni 12522 . . . . . . 7 (♯‘𝐸) ∈ ℂ
32 difss 4128 . . . . . . . . . 10 (𝑂𝐸) ⊆ 𝑂
33 ssfi 9201 . . . . . . . . . 10 ((𝑂 ∈ Fin ∧ (𝑂𝐸) ⊆ 𝑂) → (𝑂𝐸) ∈ Fin)
3411, 32, 33mp2an 690 . . . . . . . . 9 (𝑂𝐸) ∈ Fin
35 hashcl 14359 . . . . . . . . 9 ((𝑂𝐸) ∈ Fin → (♯‘(𝑂𝐸)) ∈ ℕ0)
3634, 35ax-mp 5 . . . . . . . 8 (♯‘(𝑂𝐸)) ∈ ℕ0
3736nn0cni 12522 . . . . . . 7 (♯‘(𝑂𝐸)) ∈ ℂ
3828, 31, 37subsub23i 11587 . . . . . 6 (((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂𝐸)) ↔ ((♯‘𝑂) − (♯‘(𝑂𝐸))) = (♯‘𝐸))
3925, 38mpbi 229 . . . . 5 ((♯‘𝑂) − (♯‘(𝑂𝐸))) = (♯‘𝐸)
4039oveq1i 7429 . . . 4 (((♯‘𝑂) − (♯‘(𝑂𝐸))) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂))
4122, 40eqtr4i 2756 . . 3 (𝑃𝐸) = (((♯‘𝑂) − (♯‘(𝑂𝐸))) / (♯‘𝑂))
42 ballotth.m . . . . . . 7 𝑀 ∈ ℕ
43 ballotth.n . . . . . . 7 𝑁 ∈ ℕ
4442, 43, 7ballotlem1 34257 . . . . . 6 (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
4542nnnn0i 12518 . . . . . . . . 9 𝑀 ∈ ℕ0
46 nnaddcl 12273 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
4742, 43, 46mp2an 690 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ ℕ
4847nnnn0i 12518 . . . . . . . . 9 (𝑀 + 𝑁) ∈ ℕ0
4942nnrei 12259 . . . . . . . . . 10 𝑀 ∈ ℝ
5043nnnn0i 12518 . . . . . . . . . 10 𝑁 ∈ ℕ0
5149, 50nn0addge1i 12558 . . . . . . . . 9 𝑀 ≤ (𝑀 + 𝑁)
52 elfz2nn0 13632 . . . . . . . . 9 (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ (𝑀 ∈ ℕ0 ∧ (𝑀 + 𝑁) ∈ ℕ0𝑀 ≤ (𝑀 + 𝑁)))
5345, 48, 51, 52mpbir3an 1338 . . . . . . . 8 𝑀 ∈ (0...(𝑀 + 𝑁))
54 bccl2 14326 . . . . . . . 8 (𝑀 ∈ (0...(𝑀 + 𝑁)) → ((𝑀 + 𝑁)C𝑀) ∈ ℕ)
5553, 54ax-mp 5 . . . . . . 7 ((𝑀 + 𝑁)C𝑀) ∈ ℕ
5655nnne0i 12290 . . . . . 6 ((𝑀 + 𝑁)C𝑀) ≠ 0
5744, 56eqnetri 3000 . . . . 5 (♯‘𝑂) ≠ 0
5828, 57pm3.2i 469 . . . 4 ((♯‘𝑂) ∈ ℂ ∧ (♯‘𝑂) ≠ 0)
59 divsubdir 11946 . . . 4 (((♯‘𝑂) ∈ ℂ ∧ (♯‘(𝑂𝐸)) ∈ ℂ ∧ ((♯‘𝑂) ∈ ℂ ∧ (♯‘𝑂) ≠ 0)) → (((♯‘𝑂) − (♯‘(𝑂𝐸))) / (♯‘𝑂)) = (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂𝐸)) / (♯‘𝑂))))
6028, 37, 58, 59mp3an 1457 . . 3 (((♯‘𝑂) − (♯‘(𝑂𝐸))) / (♯‘𝑂)) = (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂𝐸)) / (♯‘𝑂)))
6128, 57dividi 11985 . . . 4 ((♯‘𝑂) / (♯‘𝑂)) = 1
6261oveq1i 7429 . . 3 (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂𝐸)) / (♯‘𝑂))) = (1 − ((♯‘(𝑂𝐸)) / (♯‘𝑂)))
6341, 60, 623eqtri 2757 . 2 (𝑃𝐸) = (1 − ((♯‘(𝑂𝐸)) / (♯‘𝑂)))
64 ballotth.f . . . . . . 7 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
65 ballotth.mgtn . . . . . . 7 𝑁 < 𝑀
66 ballotth.i . . . . . . 7 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
67 ballotth.s . . . . . . 7 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
68 ballotth.r . . . . . . 7 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
6942, 43, 7, 18, 64, 1, 65, 66, 67, 68ballotlem8 34307 . . . . . 6 (♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
7069oveq1i 7429 . . . . 5 ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
7170oveq1i 7429 . . . 4 (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
72 rabxm 4388 . . . . . . 7 (𝑂𝐸) = ({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
7372fveq2i 6899 . . . . . 6 (♯‘(𝑂𝐸)) = (♯‘({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
74 ssrab2 4073 . . . . . . . . . 10 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂𝐸)
7574, 32sstri 3986 . . . . . . . . 9 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝑂
7675, 9sstri 3986 . . . . . . . 8 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
77 ssfi 9201 . . . . . . . 8 ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))) → {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∈ Fin)
786, 76, 77mp2an 690 . . . . . . 7 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∈ Fin
79 ssrab2 4073 . . . . . . . . . 10 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂𝐸)
8079, 32sstri 3986 . . . . . . . . 9 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
8180, 9sstri 3986 . . . . . . . 8 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
82 ssfi 9201 . . . . . . . 8 ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))) → {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin)
836, 81, 82mp2an 690 . . . . . . 7 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin
84 rabnc 4389 . . . . . . 7 ({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∩ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) = ∅
85 hashun 14385 . . . . . . 7 (({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∈ Fin ∧ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin ∧ ({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∩ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) = ∅) → (♯‘({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})))
8678, 83, 84, 85mp3an 1457 . . . . . 6 (♯‘({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
8773, 86eqtri 2753 . . . . 5 (♯‘(𝑂𝐸)) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
8887oveq1i 7429 . . . 4 ((♯‘(𝑂𝐸)) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
89 ssrab2 4073 . . . . . . . . 9 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
9011elexi 3482 . . . . . . . . . 10 𝑂 ∈ V
9190elpw2 5348 . . . . . . . . 9 ({𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 ↔ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂)
9289, 91mpbir 230 . . . . . . . 8 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
93 fveq2 6896 . . . . . . . . . 10 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}))
9493oveq1d 7434 . . . . . . . . 9 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
95 ovex 7452 . . . . . . . . 9 ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
9694, 18, 95fvmpt 7004 . . . . . . . 8 ({𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 → (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
9792, 96ax-mp 5 . . . . . . 7 (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
9842, 43, 7, 18ballotlem2 34259 . . . . . . 7 (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))
99 nfrab1 3438 . . . . . . . . . . . 12 𝑐{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}
100 nfrab1 3438 . . . . . . . . . . . 12 𝑐{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
10199, 100dfssf 3967 . . . . . . . . . . 11 ({𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ ∀𝑐(𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
10242, 43, 7, 18, 64, 1ballotlem4 34269 . . . . . . . . . . . . . 14 (𝑐𝑂 → (¬ 1 ∈ 𝑐 → ¬ 𝑐𝐸))
103102imdistani 567 . . . . . . . . . . . . 13 ((𝑐𝑂 ∧ ¬ 1 ∈ 𝑐) → (𝑐𝑂 ∧ ¬ 𝑐𝐸))
104 rabid 3439 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐𝑂 ∧ ¬ 1 ∈ 𝑐))
105 eldif 3954 . . . . . . . . . . . . 13 (𝑐 ∈ (𝑂𝐸) ↔ (𝑐𝑂 ∧ ¬ 𝑐𝐸))
106103, 104, 1053imtr4i 291 . . . . . . . . . . . 12 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ (𝑂𝐸))
107104simprbi 495 . . . . . . . . . . . 12 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → ¬ 1 ∈ 𝑐)
108 rabid 3439 . . . . . . . . . . . 12 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐))
109106, 107, 108sylanbrc 581 . . . . . . . . . . 11 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
110101, 109mpgbir 1793 . . . . . . . . . 10 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
111 rabss2 4071 . . . . . . . . . . 11 ((𝑂𝐸) ⊆ 𝑂 → {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐})
11232, 111ax-mp 5 . . . . . . . . . 10 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}
113110, 112eqssi 3993 . . . . . . . . 9 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} = {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
114113fveq2i 6899 . . . . . . . 8 (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
115114oveq1i 7429 . . . . . . 7 ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
11697, 98, 1153eqtr3i 2761 . . . . . 6 (𝑁 / (𝑀 + 𝑁)) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
117116oveq2i 7430 . . . . 5 (2 · (𝑁 / (𝑀 + 𝑁))) = (2 · ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
118 2cn 12325 . . . . . 6 2 ∈ ℂ
119 hashcl 14359 . . . . . . . 8 ({𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin → (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ0)
12083, 119ax-mp 5 . . . . . . 7 (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ0
121120nn0cni 12522 . . . . . 6 (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℂ
122118, 121, 28, 57divassi 12008 . . . . 5 ((2 · (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (2 · ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
1231212timesi 12388 . . . . . 6 (2 · (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
124123oveq1i 7429 . . . . 5 ((2 · (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
125117, 122, 1243eqtr2i 2759 . . . 4 (2 · (𝑁 / (𝑀 + 𝑁))) = (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
12671, 88, 1253eqtr4ri 2764 . . 3 (2 · (𝑁 / (𝑀 + 𝑁))) = ((♯‘(𝑂𝐸)) / (♯‘𝑂))
127126oveq2i 7430 . 2 (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = (1 − ((♯‘(𝑂𝐸)) / (♯‘𝑂)))
12847nncni 12260 . . . 4 (𝑀 + 𝑁) ∈ ℂ
12943nncni 12260 . . . . 5 𝑁 ∈ ℂ
130118, 129mulcli 11258 . . . 4 (2 · 𝑁) ∈ ℂ
13147nnne0i 12290 . . . . 5 (𝑀 + 𝑁) ≠ 0
132128, 131pm3.2i 469 . . . 4 ((𝑀 + 𝑁) ∈ ℂ ∧ (𝑀 + 𝑁) ≠ 0)
133 divsubdir 11946 . . . 4 (((𝑀 + 𝑁) ∈ ℂ ∧ (2 · 𝑁) ∈ ℂ ∧ ((𝑀 + 𝑁) ∈ ℂ ∧ (𝑀 + 𝑁) ≠ 0)) → (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁))))
134128, 130, 132, 133mp3an 1457 . . 3 (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁)))
1351292timesi 12388 . . . . . 6 (2 · 𝑁) = (𝑁 + 𝑁)
136135oveq2i 7430 . . . . 5 ((𝑀 + 𝑁) − (2 · 𝑁)) = ((𝑀 + 𝑁) − (𝑁 + 𝑁))
13742nncni 12260 . . . . . . 7 𝑀 ∈ ℂ
138137, 129, 129, 129addsub4i 11593 . . . . . 6 ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = ((𝑀𝑁) + (𝑁𝑁))
139129subidi 11568 . . . . . . 7 (𝑁𝑁) = 0
140139oveq2i 7430 . . . . . 6 ((𝑀𝑁) + (𝑁𝑁)) = ((𝑀𝑁) + 0)
141137, 129subcli 11573 . . . . . . 7 (𝑀𝑁) ∈ ℂ
142141addridi 11438 . . . . . 6 ((𝑀𝑁) + 0) = (𝑀𝑁)
143138, 140, 1423eqtri 2757 . . . . 5 ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = (𝑀𝑁)
144136, 143eqtri 2753 . . . 4 ((𝑀 + 𝑁) − (2 · 𝑁)) = (𝑀𝑁)
145144oveq1i 7429 . . 3 (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = ((𝑀𝑁) / (𝑀 + 𝑁))
146128, 131dividi 11985 . . . 4 ((𝑀 + 𝑁) / (𝑀 + 𝑁)) = 1
147118, 129, 128, 131divassi 12008 . . . 4 ((2 · 𝑁) / (𝑀 + 𝑁)) = (2 · (𝑁 / (𝑀 + 𝑁)))
148146, 147oveq12i 7431 . . 3 (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁))) = (1 − (2 · (𝑁 / (𝑀 + 𝑁))))
149134, 145, 1483eqtr3ri 2762 . 2 (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = ((𝑀𝑁) / (𝑀 + 𝑁))
15063, 127, 1493eqtr2i 2759 1 (𝑃𝐸) = ((𝑀𝑁) / (𝑀 + 𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2929  wral 3050  {crab 3418  cdif 3941  cun 3942  cin 3943  wss 3944  c0 4322  ifcif 4530  𝒫 cpw 4604   class class class wbr 5149  cmpt 5232  cima 5681  cfv 6549  (class class class)co 7419  Fincfn 8964  infcinf 9471  cc 11143  cr 11144  0cc0 11145  1c1 11146   + caddc 11148   · cmul 11150   < clt 11285  cle 11286  cmin 11481   / cdiv 11908  cn 12250  2c2 12305  0cn0 12510  cz 12596  ...cfz 13524  Ccbc 14305  chash 14333
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222
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 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9472  df-inf 9473  df-dju 9931  df-card 9969  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-div 11909  df-nn 12251  df-2 12313  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-fz 13525  df-seq 14008  df-fac 14277  df-bc 14306  df-hash 14334
This theorem is referenced by: (None)
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