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Theorem ballotth 30727
Description: Bertrand's ballot problem : the probability that A is ahead throughout the counting. 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 3720 . . . . . 6 {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)} ⊆ 𝑂
31, 2eqsstri 3668 . . . . 5 𝐸𝑂
4 fzfi 12811 . . . . . . . . . . 11 (1...(𝑀 + 𝑁)) ∈ Fin
5 pwfi 8302 . . . . . . . . . . 11 ((1...(𝑀 + 𝑁)) ∈ Fin ↔ 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin)
64, 5mpbi 220 . . . . . . . . . 10 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin
7 ballotth.o . . . . . . . . . . 11 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}
8 ssrab2 3720 . . . . . . . . . . 11 {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀} ⊆ 𝒫 (1...(𝑀 + 𝑁))
97, 8eqsstri 3668 . . . . . . . . . 10 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))
10 ssfi 8221 . . . . . . . . . 10 ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))) → 𝑂 ∈ Fin)
116, 9, 10mp2an 708 . . . . . . . . 9 𝑂 ∈ Fin
12 ssfi 8221 . . . . . . . . 9 ((𝑂 ∈ Fin ∧ 𝐸𝑂) → 𝐸 ∈ Fin)
1311, 3, 12mp2an 708 . . . . . . . 8 𝐸 ∈ Fin
1413elexi 3244 . . . . . . 7 𝐸 ∈ V
1514elpw 4197 . . . . . 6 (𝐸 ∈ 𝒫 𝑂𝐸𝑂)
16 fveq2 6229 . . . . . . . 8 (𝑥 = 𝐸 → (#‘𝑥) = (#‘𝐸))
1716oveq1d 6705 . . . . . . 7 (𝑥 = 𝐸 → ((#‘𝑥) / (#‘𝑂)) = ((#‘𝐸) / (#‘𝑂)))
18 ballotth.p . . . . . . 7 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))
19 ovex 6718 . . . . . . 7 ((#‘𝐸) / (#‘𝑂)) ∈ V
2017, 18, 19fvmpt 6321 . . . . . 6 (𝐸 ∈ 𝒫 𝑂 → (𝑃𝐸) = ((#‘𝐸) / (#‘𝑂)))
2115, 20sylbir 225 . . . . 5 (𝐸𝑂 → (𝑃𝐸) = ((#‘𝐸) / (#‘𝑂)))
223, 21ax-mp 5 . . . 4 (𝑃𝐸) = ((#‘𝐸) / (#‘𝑂))
23 hashssdif 13238 . . . . . . . 8 ((𝑂 ∈ Fin ∧ 𝐸𝑂) → (#‘(𝑂𝐸)) = ((#‘𝑂) − (#‘𝐸)))
2411, 3, 23mp2an 708 . . . . . . 7 (#‘(𝑂𝐸)) = ((#‘𝑂) − (#‘𝐸))
2524eqcomi 2660 . . . . . 6 ((#‘𝑂) − (#‘𝐸)) = (#‘(𝑂𝐸))
26 hashcl 13185 . . . . . . . . 9 (𝑂 ∈ Fin → (#‘𝑂) ∈ ℕ0)
2711, 26ax-mp 5 . . . . . . . 8 (#‘𝑂) ∈ ℕ0
2827nn0cni 11342 . . . . . . 7 (#‘𝑂) ∈ ℂ
29 hashcl 13185 . . . . . . . . 9 (𝐸 ∈ Fin → (#‘𝐸) ∈ ℕ0)
3013, 29ax-mp 5 . . . . . . . 8 (#‘𝐸) ∈ ℕ0
3130nn0cni 11342 . . . . . . 7 (#‘𝐸) ∈ ℂ
32 difss 3770 . . . . . . . . . 10 (𝑂𝐸) ⊆ 𝑂
33 ssfi 8221 . . . . . . . . . 10 ((𝑂 ∈ Fin ∧ (𝑂𝐸) ⊆ 𝑂) → (𝑂𝐸) ∈ Fin)
3411, 32, 33mp2an 708 . . . . . . . . 9 (𝑂𝐸) ∈ Fin
35 hashcl 13185 . . . . . . . . 9 ((𝑂𝐸) ∈ Fin → (#‘(𝑂𝐸)) ∈ ℕ0)
3634, 35ax-mp 5 . . . . . . . 8 (#‘(𝑂𝐸)) ∈ ℕ0
3736nn0cni 11342 . . . . . . 7 (#‘(𝑂𝐸)) ∈ ℂ
3828, 31, 37subsub23i 10409 . . . . . 6 (((#‘𝑂) − (#‘𝐸)) = (#‘(𝑂𝐸)) ↔ ((#‘𝑂) − (#‘(𝑂𝐸))) = (#‘𝐸))
3925, 38mpbi 220 . . . . 5 ((#‘𝑂) − (#‘(𝑂𝐸))) = (#‘𝐸)
4039oveq1i 6700 . . . 4 (((#‘𝑂) − (#‘(𝑂𝐸))) / (#‘𝑂)) = ((#‘𝐸) / (#‘𝑂))
4122, 40eqtr4i 2676 . . 3 (𝑃𝐸) = (((#‘𝑂) − (#‘(𝑂𝐸))) / (#‘𝑂))
42 ballotth.m . . . . . . 7 𝑀 ∈ ℕ
43 ballotth.n . . . . . . 7 𝑁 ∈ ℕ
4442, 43, 7ballotlem1 30676 . . . . . 6 (#‘𝑂) = ((𝑀 + 𝑁)C𝑀)
4542nnnn0i 11338 . . . . . . . . 9 𝑀 ∈ ℕ0
46 nnaddcl 11080 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
4742, 43, 46mp2an 708 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ ℕ
4847nnnn0i 11338 . . . . . . . . 9 (𝑀 + 𝑁) ∈ ℕ0
4942nnrei 11067 . . . . . . . . . 10 𝑀 ∈ ℝ
5043nnnn0i 11338 . . . . . . . . . 10 𝑁 ∈ ℕ0
5149, 50nn0addge1i 11379 . . . . . . . . 9 𝑀 ≤ (𝑀 + 𝑁)
52 elfz2nn0 12469 . . . . . . . . 9 (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ (𝑀 ∈ ℕ0 ∧ (𝑀 + 𝑁) ∈ ℕ0𝑀 ≤ (𝑀 + 𝑁)))
5345, 48, 51, 52mpbir3an 1263 . . . . . . . 8 𝑀 ∈ (0...(𝑀 + 𝑁))
54 bccl2 13150 . . . . . . . 8 (𝑀 ∈ (0...(𝑀 + 𝑁)) → ((𝑀 + 𝑁)C𝑀) ∈ ℕ)
5553, 54ax-mp 5 . . . . . . 7 ((𝑀 + 𝑁)C𝑀) ∈ ℕ
5655nnne0i 11093 . . . . . 6 ((𝑀 + 𝑁)C𝑀) ≠ 0
5744, 56eqnetri 2893 . . . . 5 (#‘𝑂) ≠ 0
5828, 57pm3.2i 470 . . . 4 ((#‘𝑂) ∈ ℂ ∧ (#‘𝑂) ≠ 0)
59 divsubdir 10759 . . . 4 (((#‘𝑂) ∈ ℂ ∧ (#‘(𝑂𝐸)) ∈ ℂ ∧ ((#‘𝑂) ∈ ℂ ∧ (#‘𝑂) ≠ 0)) → (((#‘𝑂) − (#‘(𝑂𝐸))) / (#‘𝑂)) = (((#‘𝑂) / (#‘𝑂)) − ((#‘(𝑂𝐸)) / (#‘𝑂))))
6028, 37, 58, 59mp3an 1464 . . 3 (((#‘𝑂) − (#‘(𝑂𝐸))) / (#‘𝑂)) = (((#‘𝑂) / (#‘𝑂)) − ((#‘(𝑂𝐸)) / (#‘𝑂)))
6128, 57dividi 10796 . . . 4 ((#‘𝑂) / (#‘𝑂)) = 1
6261oveq1i 6700 . . 3 (((#‘𝑂) / (#‘𝑂)) − ((#‘(𝑂𝐸)) / (#‘𝑂))) = (1 − ((#‘(𝑂𝐸)) / (#‘𝑂)))
6341, 60, 623eqtri 2677 . 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 30726 . . . . . 6 (#‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) = (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
7069oveq1i 6700 . . . . 5 ((#‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
7170oveq1i 6700 . . . 4 (((#‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (#‘𝑂)) = (((#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (#‘𝑂))
72 rabxm 3994 . . . . . . 7 (𝑂𝐸) = ({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
7372fveq2i 6232 . . . . . 6 (#‘(𝑂𝐸)) = (#‘({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
74 ssrab2 3720 . . . . . . . . . 10 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂𝐸)
7574, 32sstri 3645 . . . . . . . . 9 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝑂
7675, 9sstri 3645 . . . . . . . 8 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
77 ssfi 8221 . . . . . . . 8 ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))) → {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∈ Fin)
786, 76, 77mp2an 708 . . . . . . 7 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∈ Fin
79 ssrab2 3720 . . . . . . . . . 10 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂𝐸)
8079, 32sstri 3645 . . . . . . . . 9 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
8180, 9sstri 3645 . . . . . . . 8 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
82 ssfi 8221 . . . . . . . 8 ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))) → {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin)
836, 81, 82mp2an 708 . . . . . . 7 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin
84 rabnc 3995 . . . . . . 7 ({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∩ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) = ∅
85 hashun 13209 . . . . . . 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 1464 . . . . . 6 (#‘({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((#‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
8773, 86eqtri 2673 . . . . 5 (#‘(𝑂𝐸)) = ((#‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
8887oveq1i 6700 . . . 4 ((#‘(𝑂𝐸)) / (#‘𝑂)) = (((#‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (#‘𝑂))
89 ssrab2 3720 . . . . . . . . 9 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
9011elexi 3244 . . . . . . . . . 10 𝑂 ∈ V
9190elpw2 4858 . . . . . . . . 9 ({𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 ↔ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂)
9289, 91mpbir 221 . . . . . . . 8 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
93 fveq2 6229 . . . . . . . . . 10 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → (#‘𝑥) = (#‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}))
9493oveq1d 6705 . . . . . . . . 9 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → ((#‘𝑥) / (#‘𝑂)) = ((#‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)))
95 ovex 6718 . . . . . . . . 9 ((#‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)) ∈ V
9694, 18, 95fvmpt 6321 . . . . . . . 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 30678 . . . . . . 7 (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))
99 nfrab1 3152 . . . . . . . . . . . 12 𝑐{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}
100 nfrab1 3152 . . . . . . . . . . . 12 𝑐{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
10199, 100dfss2f 3627 . . . . . . . . . . 11 ({𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ ∀𝑐(𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
10242, 43, 7, 18, 64, 1ballotlem4 30688 . . . . . . . . . . . . . 14 (𝑐𝑂 → (¬ 1 ∈ 𝑐 → ¬ 𝑐𝐸))
103102imdistani 726 . . . . . . . . . . . . 13 ((𝑐𝑂 ∧ ¬ 1 ∈ 𝑐) → (𝑐𝑂 ∧ ¬ 𝑐𝐸))
104 rabid 3145 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐𝑂 ∧ ¬ 1 ∈ 𝑐))
105 eldif 3617 . . . . . . . . . . . . 13 (𝑐 ∈ (𝑂𝐸) ↔ (𝑐𝑂 ∧ ¬ 𝑐𝐸))
106103, 104, 1053imtr4i 281 . . . . . . . . . . . 12 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ (𝑂𝐸))
107104simprbi 479 . . . . . . . . . . . 12 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → ¬ 1 ∈ 𝑐)
108 rabid 3145 . . . . . . . . . . . 12 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐))
109106, 107, 108sylanbrc 699 . . . . . . . . . . 11 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
110101, 109mpgbir 1766 . . . . . . . . . 10 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
111 rabss2 3718 . . . . . . . . . . 11 ((𝑂𝐸) ⊆ 𝑂 → {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐})
11232, 111ax-mp 5 . . . . . . . . . 10 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}
113110, 112eqssi 3652 . . . . . . . . 9 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} = {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
114113fveq2i 6232 . . . . . . . 8 (#‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
115114oveq1i 6700 . . . . . . 7 ((#‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)) = ((#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂))
11697, 98, 1153eqtr3i 2681 . . . . . 6 (𝑁 / (𝑀 + 𝑁)) = ((#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂))
117116oveq2i 6701 . . . . 5 (2 · (𝑁 / (𝑀 + 𝑁))) = (2 · ((#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)))
118 2cn 11129 . . . . . 6 2 ∈ ℂ
119 hashcl 13185 . . . . . . . 8 ({𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin → (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ0)
12083, 119ax-mp 5 . . . . . . 7 (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ0
121120nn0cni 11342 . . . . . 6 (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℂ
122118, 121, 28, 57divassi 10819 . . . . 5 ((2 · (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (#‘𝑂)) = (2 · ((#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (#‘𝑂)))
1231212timesi 11185 . . . . . 6 (2 · (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
124123oveq1i 6700 . . . . 5 ((2 · (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (#‘𝑂)) = (((#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (#‘𝑂))
125117, 122, 1243eqtr2i 2679 . . . 4 (2 · (𝑁 / (𝑀 + 𝑁))) = (((#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (#‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (#‘𝑂))
12671, 88, 1253eqtr4ri 2684 . . 3 (2 · (𝑁 / (𝑀 + 𝑁))) = ((#‘(𝑂𝐸)) / (#‘𝑂))
127126oveq2i 6701 . 2 (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = (1 − ((#‘(𝑂𝐸)) / (#‘𝑂)))
12847nncni 11068 . . . 4 (𝑀 + 𝑁) ∈ ℂ
12943nncni 11068 . . . . 5 𝑁 ∈ ℂ
130118, 129mulcli 10083 . . . 4 (2 · 𝑁) ∈ ℂ
13147nnne0i 11093 . . . . 5 (𝑀 + 𝑁) ≠ 0
132128, 131pm3.2i 470 . . . 4 ((𝑀 + 𝑁) ∈ ℂ ∧ (𝑀 + 𝑁) ≠ 0)
133 divsubdir 10759 . . . 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 1464 . . 3 (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁)))
1351292timesi 11185 . . . . . 6 (2 · 𝑁) = (𝑁 + 𝑁)
136135oveq2i 6701 . . . . 5 ((𝑀 + 𝑁) − (2 · 𝑁)) = ((𝑀 + 𝑁) − (𝑁 + 𝑁))
13742nncni 11068 . . . . . . 7 𝑀 ∈ ℂ
138137, 129, 129, 129addsub4i 10415 . . . . . 6 ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = ((𝑀𝑁) + (𝑁𝑁))
139129subidi 10390 . . . . . . 7 (𝑁𝑁) = 0
140139oveq2i 6701 . . . . . 6 ((𝑀𝑁) + (𝑁𝑁)) = ((𝑀𝑁) + 0)
141137, 129subcli 10395 . . . . . . 7 (𝑀𝑁) ∈ ℂ
142141addid1i 10261 . . . . . 6 ((𝑀𝑁) + 0) = (𝑀𝑁)
143138, 140, 1423eqtri 2677 . . . . 5 ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = (𝑀𝑁)
144136, 143eqtri 2673 . . . 4 ((𝑀 + 𝑁) − (2 · 𝑁)) = (𝑀𝑁)
145144oveq1i 6700 . . 3 (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = ((𝑀𝑁) / (𝑀 + 𝑁))
146128, 131dividi 10796 . . . 4 ((𝑀 + 𝑁) / (𝑀 + 𝑁)) = 1
147118, 129, 128, 131divassi 10819 . . . 4 ((2 · 𝑁) / (𝑀 + 𝑁)) = (2 · (𝑁 / (𝑀 + 𝑁)))
148146, 147oveq12i 6702 . . 3 (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁))) = (1 − (2 · (𝑁 / (𝑀 + 𝑁))))
149134, 145, 1483eqtr3ri 2682 . 2 (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = ((𝑀𝑁) / (𝑀 + 𝑁))
15063, 127, 1493eqtr2i 2679 1 (𝑃𝐸) = ((𝑀𝑁) / (𝑀 + 𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  {crab 2945  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  ifcif 4119  𝒫 cpw 4191   class class class wbr 4685  cmpt 4762  cima 5146  cfv 5926  (class class class)co 6690  Fincfn 7997  infcinf 8388  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  2c2 11108  0cn0 11330  cz 11415  ...cfz 12364  Ccbc 13129  #chash 13157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-seq 12842  df-fac 13101  df-bc 13130  df-hash 13158
This theorem is referenced by: (None)
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