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Theorem ballotth 31694
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 4053 . . . . . 6 {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)} ⊆ 𝑂
31, 2eqsstri 3998 . . . . 5 𝐸𝑂
4 fzfi 13328 . . . . . . . . . . 11 (1...(𝑀 + 𝑁)) ∈ Fin
5 pwfi 8807 . . . . . . . . . . 11 ((1...(𝑀 + 𝑁)) ∈ Fin ↔ 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin)
64, 5mpbi 231 . . . . . . . . . 10 𝒫 (1...(𝑀 + 𝑁)) ∈ Fin
7 ballotth.o . . . . . . . . . . 11 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
8 ssrab2 4053 . . . . . . . . . . 11 {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} ⊆ 𝒫 (1...(𝑀 + 𝑁))
97, 8eqsstri 3998 . . . . . . . . . 10 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))
10 ssfi 8726 . . . . . . . . . 10 ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))) → 𝑂 ∈ Fin)
116, 9, 10mp2an 688 . . . . . . . . 9 𝑂 ∈ Fin
12 ssfi 8726 . . . . . . . . 9 ((𝑂 ∈ Fin ∧ 𝐸𝑂) → 𝐸 ∈ Fin)
1311, 3, 12mp2an 688 . . . . . . . 8 𝐸 ∈ Fin
1413elexi 3511 . . . . . . 7 𝐸 ∈ V
1514elpw 4542 . . . . . 6 (𝐸 ∈ 𝒫 𝑂𝐸𝑂)
16 fveq2 6663 . . . . . . . 8 (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸))
1716oveq1d 7160 . . . . . . 7 (𝑥 = 𝐸 → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂)))
18 ballotth.p . . . . . . 7 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
19 ovex 7178 . . . . . . 7 ((♯‘𝐸) / (♯‘𝑂)) ∈ V
2017, 18, 19fvmpt 6761 . . . . . 6 (𝐸 ∈ 𝒫 𝑂 → (𝑃𝐸) = ((♯‘𝐸) / (♯‘𝑂)))
2115, 20sylbir 236 . . . . 5 (𝐸𝑂 → (𝑃𝐸) = ((♯‘𝐸) / (♯‘𝑂)))
223, 21ax-mp 5 . . . 4 (𝑃𝐸) = ((♯‘𝐸) / (♯‘𝑂))
23 hashssdif 13761 . . . . . . . 8 ((𝑂 ∈ Fin ∧ 𝐸𝑂) → (♯‘(𝑂𝐸)) = ((♯‘𝑂) − (♯‘𝐸)))
2411, 3, 23mp2an 688 . . . . . . 7 (♯‘(𝑂𝐸)) = ((♯‘𝑂) − (♯‘𝐸))
2524eqcomi 2827 . . . . . 6 ((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂𝐸))
26 hashcl 13705 . . . . . . . . 9 (𝑂 ∈ Fin → (♯‘𝑂) ∈ ℕ0)
2711, 26ax-mp 5 . . . . . . . 8 (♯‘𝑂) ∈ ℕ0
2827nn0cni 11897 . . . . . . 7 (♯‘𝑂) ∈ ℂ
29 hashcl 13705 . . . . . . . . 9 (𝐸 ∈ Fin → (♯‘𝐸) ∈ ℕ0)
3013, 29ax-mp 5 . . . . . . . 8 (♯‘𝐸) ∈ ℕ0
3130nn0cni 11897 . . . . . . 7 (♯‘𝐸) ∈ ℂ
32 difss 4105 . . . . . . . . . 10 (𝑂𝐸) ⊆ 𝑂
33 ssfi 8726 . . . . . . . . . 10 ((𝑂 ∈ Fin ∧ (𝑂𝐸) ⊆ 𝑂) → (𝑂𝐸) ∈ Fin)
3411, 32, 33mp2an 688 . . . . . . . . 9 (𝑂𝐸) ∈ Fin
35 hashcl 13705 . . . . . . . . 9 ((𝑂𝐸) ∈ Fin → (♯‘(𝑂𝐸)) ∈ ℕ0)
3634, 35ax-mp 5 . . . . . . . 8 (♯‘(𝑂𝐸)) ∈ ℕ0
3736nn0cni 11897 . . . . . . 7 (♯‘(𝑂𝐸)) ∈ ℂ
3828, 31, 37subsub23i 10964 . . . . . 6 (((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂𝐸)) ↔ ((♯‘𝑂) − (♯‘(𝑂𝐸))) = (♯‘𝐸))
3925, 38mpbi 231 . . . . 5 ((♯‘𝑂) − (♯‘(𝑂𝐸))) = (♯‘𝐸)
4039oveq1i 7155 . . . 4 (((♯‘𝑂) − (♯‘(𝑂𝐸))) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂))
4122, 40eqtr4i 2844 . . 3 (𝑃𝐸) = (((♯‘𝑂) − (♯‘(𝑂𝐸))) / (♯‘𝑂))
42 ballotth.m . . . . . . 7 𝑀 ∈ ℕ
43 ballotth.n . . . . . . 7 𝑁 ∈ ℕ
4442, 43, 7ballotlem1 31643 . . . . . 6 (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
4542nnnn0i 11893 . . . . . . . . 9 𝑀 ∈ ℕ0
46 nnaddcl 11648 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
4742, 43, 46mp2an 688 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ ℕ
4847nnnn0i 11893 . . . . . . . . 9 (𝑀 + 𝑁) ∈ ℕ0
4942nnrei 11635 . . . . . . . . . 10 𝑀 ∈ ℝ
5043nnnn0i 11893 . . . . . . . . . 10 𝑁 ∈ ℕ0
5149, 50nn0addge1i 11933 . . . . . . . . 9 𝑀 ≤ (𝑀 + 𝑁)
52 elfz2nn0 12986 . . . . . . . . 9 (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ (𝑀 ∈ ℕ0 ∧ (𝑀 + 𝑁) ∈ ℕ0𝑀 ≤ (𝑀 + 𝑁)))
5345, 48, 51, 52mpbir3an 1333 . . . . . . . 8 𝑀 ∈ (0...(𝑀 + 𝑁))
54 bccl2 13671 . . . . . . . 8 (𝑀 ∈ (0...(𝑀 + 𝑁)) → ((𝑀 + 𝑁)C𝑀) ∈ ℕ)
5553, 54ax-mp 5 . . . . . . 7 ((𝑀 + 𝑁)C𝑀) ∈ ℕ
5655nnne0i 11665 . . . . . 6 ((𝑀 + 𝑁)C𝑀) ≠ 0
5744, 56eqnetri 3083 . . . . 5 (♯‘𝑂) ≠ 0
5828, 57pm3.2i 471 . . . 4 ((♯‘𝑂) ∈ ℂ ∧ (♯‘𝑂) ≠ 0)
59 divsubdir 11322 . . . 4 (((♯‘𝑂) ∈ ℂ ∧ (♯‘(𝑂𝐸)) ∈ ℂ ∧ ((♯‘𝑂) ∈ ℂ ∧ (♯‘𝑂) ≠ 0)) → (((♯‘𝑂) − (♯‘(𝑂𝐸))) / (♯‘𝑂)) = (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂𝐸)) / (♯‘𝑂))))
6028, 37, 58, 59mp3an 1452 . . 3 (((♯‘𝑂) − (♯‘(𝑂𝐸))) / (♯‘𝑂)) = (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂𝐸)) / (♯‘𝑂)))
6128, 57dividi 11361 . . . 4 ((♯‘𝑂) / (♯‘𝑂)) = 1
6261oveq1i 7155 . . 3 (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂𝐸)) / (♯‘𝑂))) = (1 − ((♯‘(𝑂𝐸)) / (♯‘𝑂)))
6341, 60, 623eqtri 2845 . 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 31693 . . . . . 6 (♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
7069oveq1i 7155 . . . . 5 ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
7170oveq1i 7155 . . . 4 (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
72 rabxm 4337 . . . . . . 7 (𝑂𝐸) = ({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
7372fveq2i 6666 . . . . . 6 (♯‘(𝑂𝐸)) = (♯‘({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
74 ssrab2 4053 . . . . . . . . . 10 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂𝐸)
7574, 32sstri 3973 . . . . . . . . 9 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝑂
7675, 9sstri 3973 . . . . . . . 8 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
77 ssfi 8726 . . . . . . . 8 ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))) → {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∈ Fin)
786, 76, 77mp2an 688 . . . . . . 7 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∈ Fin
79 ssrab2 4053 . . . . . . . . . 10 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂𝐸)
8079, 32sstri 3973 . . . . . . . . 9 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
8180, 9sstri 3973 . . . . . . . 8 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
82 ssfi 8726 . . . . . . . 8 ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))) → {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin)
836, 81, 82mp2an 688 . . . . . . 7 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin
84 rabnc 4338 . . . . . . 7 ({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∩ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) = ∅
85 hashun 13731 . . . . . . 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 1452 . . . . . 6 (♯‘({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
8773, 86eqtri 2841 . . . . 5 (♯‘(𝑂𝐸)) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
8887oveq1i 7155 . . . 4 ((♯‘(𝑂𝐸)) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
89 ssrab2 4053 . . . . . . . . 9 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
9011elexi 3511 . . . . . . . . . 10 𝑂 ∈ V
9190elpw2 5239 . . . . . . . . 9 ({𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 ↔ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂)
9289, 91mpbir 232 . . . . . . . 8 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
93 fveq2 6663 . . . . . . . . . 10 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}))
9493oveq1d 7160 . . . . . . . . 9 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
95 ovex 7178 . . . . . . . . 9 ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
9694, 18, 95fvmpt 6761 . . . . . . . 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 31645 . . . . . . 7 (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))
99 nfrab1 3382 . . . . . . . . . . . 12 𝑐{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}
100 nfrab1 3382 . . . . . . . . . . . 12 𝑐{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
10199, 100dfss2f 3955 . . . . . . . . . . 11 ({𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ ∀𝑐(𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
10242, 43, 7, 18, 64, 1ballotlem4 31655 . . . . . . . . . . . . . 14 (𝑐𝑂 → (¬ 1 ∈ 𝑐 → ¬ 𝑐𝐸))
103102imdistani 569 . . . . . . . . . . . . 13 ((𝑐𝑂 ∧ ¬ 1 ∈ 𝑐) → (𝑐𝑂 ∧ ¬ 𝑐𝐸))
104 rabid 3376 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐𝑂 ∧ ¬ 1 ∈ 𝑐))
105 eldif 3943 . . . . . . . . . . . . 13 (𝑐 ∈ (𝑂𝐸) ↔ (𝑐𝑂 ∧ ¬ 𝑐𝐸))
106103, 104, 1053imtr4i 293 . . . . . . . . . . . 12 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ (𝑂𝐸))
107104simprbi 497 . . . . . . . . . . . 12 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → ¬ 1 ∈ 𝑐)
108 rabid 3376 . . . . . . . . . . . 12 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐))
109106, 107, 108sylanbrc 583 . . . . . . . . . . 11 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
110101, 109mpgbir 1791 . . . . . . . . . 10 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
111 rabss2 4051 . . . . . . . . . . 11 ((𝑂𝐸) ⊆ 𝑂 → {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐})
11232, 111ax-mp 5 . . . . . . . . . 10 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}
113110, 112eqssi 3980 . . . . . . . . 9 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} = {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
114113fveq2i 6666 . . . . . . . 8 (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
115114oveq1i 7155 . . . . . . 7 ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
11697, 98, 1153eqtr3i 2849 . . . . . 6 (𝑁 / (𝑀 + 𝑁)) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
117116oveq2i 7156 . . . . 5 (2 · (𝑁 / (𝑀 + 𝑁))) = (2 · ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
118 2cn 11700 . . . . . 6 2 ∈ ℂ
119 hashcl 13705 . . . . . . . 8 ({𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin → (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ0)
12083, 119ax-mp 5 . . . . . . 7 (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ0
121120nn0cni 11897 . . . . . 6 (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℂ
122118, 121, 28, 57divassi 11384 . . . . 5 ((2 · (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (2 · ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
1231212timesi 11763 . . . . . 6 (2 · (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
124123oveq1i 7155 . . . . 5 ((2 · (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
125117, 122, 1243eqtr2i 2847 . . . 4 (2 · (𝑁 / (𝑀 + 𝑁))) = (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
12671, 88, 1253eqtr4ri 2852 . . 3 (2 · (𝑁 / (𝑀 + 𝑁))) = ((♯‘(𝑂𝐸)) / (♯‘𝑂))
127126oveq2i 7156 . 2 (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = (1 − ((♯‘(𝑂𝐸)) / (♯‘𝑂)))
12847nncni 11636 . . . 4 (𝑀 + 𝑁) ∈ ℂ
12943nncni 11636 . . . . 5 𝑁 ∈ ℂ
130118, 129mulcli 10636 . . . 4 (2 · 𝑁) ∈ ℂ
13147nnne0i 11665 . . . . 5 (𝑀 + 𝑁) ≠ 0
132128, 131pm3.2i 471 . . . 4 ((𝑀 + 𝑁) ∈ ℂ ∧ (𝑀 + 𝑁) ≠ 0)
133 divsubdir 11322 . . . 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 1452 . . 3 (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁)))
1351292timesi 11763 . . . . . 6 (2 · 𝑁) = (𝑁 + 𝑁)
136135oveq2i 7156 . . . . 5 ((𝑀 + 𝑁) − (2 · 𝑁)) = ((𝑀 + 𝑁) − (𝑁 + 𝑁))
13742nncni 11636 . . . . . . 7 𝑀 ∈ ℂ
138137, 129, 129, 129addsub4i 10970 . . . . . 6 ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = ((𝑀𝑁) + (𝑁𝑁))
139129subidi 10945 . . . . . . 7 (𝑁𝑁) = 0
140139oveq2i 7156 . . . . . 6 ((𝑀𝑁) + (𝑁𝑁)) = ((𝑀𝑁) + 0)
141137, 129subcli 10950 . . . . . . 7 (𝑀𝑁) ∈ ℂ
142141addid1i 10815 . . . . . 6 ((𝑀𝑁) + 0) = (𝑀𝑁)
143138, 140, 1423eqtri 2845 . . . . 5 ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = (𝑀𝑁)
144136, 143eqtri 2841 . . . 4 ((𝑀 + 𝑁) − (2 · 𝑁)) = (𝑀𝑁)
145144oveq1i 7155 . . 3 (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = ((𝑀𝑁) / (𝑀 + 𝑁))
146128, 131dividi 11361 . . . 4 ((𝑀 + 𝑁) / (𝑀 + 𝑁)) = 1
147118, 129, 128, 131divassi 11384 . . . 4 ((2 · 𝑁) / (𝑀 + 𝑁)) = (2 · (𝑁 / (𝑀 + 𝑁)))
148146, 147oveq12i 7157 . . 3 (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁))) = (1 − (2 · (𝑁 / (𝑀 + 𝑁))))
149134, 145, 1483eqtr3ri 2850 . 2 (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = ((𝑀𝑁) / (𝑀 + 𝑁))
15063, 127, 1493eqtr2i 2847 1 (𝑃𝐸) = ((𝑀𝑁) / (𝑀 + 𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  {crab 3139  cdif 3930  cun 3931  cin 3932  wss 3933  c0 4288  ifcif 4463  𝒫 cpw 4535   class class class wbr 5057  cmpt 5137  cima 5551  cfv 6348  (class class class)co 7145  Fincfn 8497  infcinf 8893  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530   < clt 10663  cle 10664  cmin 10858   / cdiv 11285  cn 11626  2c2 11680  0cn0 11885  cz 11969  ...cfz 12880  Ccbc 13650  chash 13678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-seq 13358  df-fac 13622  df-bc 13651  df-hash 13679
This theorem is referenced by: (None)
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