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Theorem ballotth 30932
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 3836 . . . . . 6 {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)} ⊆ 𝑂
31, 2eqsstri 3784 . . . . 5 𝐸𝑂
4 fzfi 12972 . . . . . . . . . . 11 (1...(𝑀 + 𝑁)) ∈ Fin
5 pwfi 8415 . . . . . . . . . . 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 3836 . . . . . . . . . . 11 {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} ⊆ 𝒫 (1...(𝑀 + 𝑁))
97, 8eqsstri 3784 . . . . . . . . . 10 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))
10 ssfi 8334 . . . . . . . . . 10 ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑂 ⊆ 𝒫 (1...(𝑀 + 𝑁))) → 𝑂 ∈ Fin)
116, 9, 10mp2an 672 . . . . . . . . 9 𝑂 ∈ Fin
12 ssfi 8334 . . . . . . . . 9 ((𝑂 ∈ Fin ∧ 𝐸𝑂) → 𝐸 ∈ Fin)
1311, 3, 12mp2an 672 . . . . . . . 8 𝐸 ∈ Fin
1413elexi 3365 . . . . . . 7 𝐸 ∈ V
1514elpw 4303 . . . . . 6 (𝐸 ∈ 𝒫 𝑂𝐸𝑂)
16 fveq2 6330 . . . . . . . 8 (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸))
1716oveq1d 6806 . . . . . . 7 (𝑥 = 𝐸 → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂)))
18 ballotth.p . . . . . . 7 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
19 ovex 6821 . . . . . . 7 ((♯‘𝐸) / (♯‘𝑂)) ∈ V
2017, 18, 19fvmpt 6422 . . . . . 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 13395 . . . . . . . 8 ((𝑂 ∈ Fin ∧ 𝐸𝑂) → (♯‘(𝑂𝐸)) = ((♯‘𝑂) − (♯‘𝐸)))
2411, 3, 23mp2an 672 . . . . . . 7 (♯‘(𝑂𝐸)) = ((♯‘𝑂) − (♯‘𝐸))
2524eqcomi 2780 . . . . . 6 ((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂𝐸))
26 hashcl 13342 . . . . . . . . 9 (𝑂 ∈ Fin → (♯‘𝑂) ∈ ℕ0)
2711, 26ax-mp 5 . . . . . . . 8 (♯‘𝑂) ∈ ℕ0
2827nn0cni 11504 . . . . . . 7 (♯‘𝑂) ∈ ℂ
29 hashcl 13342 . . . . . . . . 9 (𝐸 ∈ Fin → (♯‘𝐸) ∈ ℕ0)
3013, 29ax-mp 5 . . . . . . . 8 (♯‘𝐸) ∈ ℕ0
3130nn0cni 11504 . . . . . . 7 (♯‘𝐸) ∈ ℂ
32 difss 3888 . . . . . . . . . 10 (𝑂𝐸) ⊆ 𝑂
33 ssfi 8334 . . . . . . . . . 10 ((𝑂 ∈ Fin ∧ (𝑂𝐸) ⊆ 𝑂) → (𝑂𝐸) ∈ Fin)
3411, 32, 33mp2an 672 . . . . . . . . 9 (𝑂𝐸) ∈ Fin
35 hashcl 13342 . . . . . . . . 9 ((𝑂𝐸) ∈ Fin → (♯‘(𝑂𝐸)) ∈ ℕ0)
3634, 35ax-mp 5 . . . . . . . 8 (♯‘(𝑂𝐸)) ∈ ℕ0
3736nn0cni 11504 . . . . . . 7 (♯‘(𝑂𝐸)) ∈ ℂ
3828, 31, 37subsub23i 10571 . . . . . 6 (((♯‘𝑂) − (♯‘𝐸)) = (♯‘(𝑂𝐸)) ↔ ((♯‘𝑂) − (♯‘(𝑂𝐸))) = (♯‘𝐸))
3925, 38mpbi 220 . . . . 5 ((♯‘𝑂) − (♯‘(𝑂𝐸))) = (♯‘𝐸)
4039oveq1i 6801 . . . 4 (((♯‘𝑂) − (♯‘(𝑂𝐸))) / (♯‘𝑂)) = ((♯‘𝐸) / (♯‘𝑂))
4122, 40eqtr4i 2796 . . 3 (𝑃𝐸) = (((♯‘𝑂) − (♯‘(𝑂𝐸))) / (♯‘𝑂))
42 ballotth.m . . . . . . 7 𝑀 ∈ ℕ
43 ballotth.n . . . . . . 7 𝑁 ∈ ℕ
4442, 43, 7ballotlem1 30881 . . . . . 6 (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
4542nnnn0i 11500 . . . . . . . . 9 𝑀 ∈ ℕ0
46 nnaddcl 11242 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
4742, 43, 46mp2an 672 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ ℕ
4847nnnn0i 11500 . . . . . . . . 9 (𝑀 + 𝑁) ∈ ℕ0
4942nnrei 11229 . . . . . . . . . 10 𝑀 ∈ ℝ
5043nnnn0i 11500 . . . . . . . . . 10 𝑁 ∈ ℕ0
5149, 50nn0addge1i 11541 . . . . . . . . 9 𝑀 ≤ (𝑀 + 𝑁)
52 elfz2nn0 12631 . . . . . . . . 9 (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ (𝑀 ∈ ℕ0 ∧ (𝑀 + 𝑁) ∈ ℕ0𝑀 ≤ (𝑀 + 𝑁)))
5345, 48, 51, 52mpbir3an 1426 . . . . . . . 8 𝑀 ∈ (0...(𝑀 + 𝑁))
54 bccl2 13307 . . . . . . . 8 (𝑀 ∈ (0...(𝑀 + 𝑁)) → ((𝑀 + 𝑁)C𝑀) ∈ ℕ)
5553, 54ax-mp 5 . . . . . . 7 ((𝑀 + 𝑁)C𝑀) ∈ ℕ
5655nnne0i 11255 . . . . . 6 ((𝑀 + 𝑁)C𝑀) ≠ 0
5744, 56eqnetri 3013 . . . . 5 (♯‘𝑂) ≠ 0
5828, 57pm3.2i 456 . . . 4 ((♯‘𝑂) ∈ ℂ ∧ (♯‘𝑂) ≠ 0)
59 divsubdir 10921 . . . 4 (((♯‘𝑂) ∈ ℂ ∧ (♯‘(𝑂𝐸)) ∈ ℂ ∧ ((♯‘𝑂) ∈ ℂ ∧ (♯‘𝑂) ≠ 0)) → (((♯‘𝑂) − (♯‘(𝑂𝐸))) / (♯‘𝑂)) = (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂𝐸)) / (♯‘𝑂))))
6028, 37, 58, 59mp3an 1572 . . 3 (((♯‘𝑂) − (♯‘(𝑂𝐸))) / (♯‘𝑂)) = (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂𝐸)) / (♯‘𝑂)))
6128, 57dividi 10958 . . . 4 ((♯‘𝑂) / (♯‘𝑂)) = 1
6261oveq1i 6801 . . 3 (((♯‘𝑂) / (♯‘𝑂)) − ((♯‘(𝑂𝐸)) / (♯‘𝑂))) = (1 − ((♯‘(𝑂𝐸)) / (♯‘𝑂)))
6341, 60, 623eqtri 2797 . 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 30931 . . . . . 6 (♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
7069oveq1i 6801 . . . . 5 ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
7170oveq1i 6801 . . . 4 (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
72 rabxm 4105 . . . . . . 7 (𝑂𝐸) = ({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
7372fveq2i 6333 . . . . . 6 (♯‘(𝑂𝐸)) = (♯‘({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
74 ssrab2 3836 . . . . . . . . . 10 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂𝐸)
7574, 32sstri 3761 . . . . . . . . 9 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝑂
7675, 9sstri 3761 . . . . . . . 8 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
77 ssfi 8334 . . . . . . . 8 ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))) → {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∈ Fin)
786, 76, 77mp2an 672 . . . . . . 7 {𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∈ Fin
79 ssrab2 3836 . . . . . . . . . 10 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂𝐸)
8079, 32sstri 3761 . . . . . . . . 9 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
8180, 9sstri 3761 . . . . . . . 8 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))
82 ssfi 8334 . . . . . . . 8 ((𝒫 (1...(𝑀 + 𝑁)) ∈ Fin ∧ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ 𝒫 (1...(𝑀 + 𝑁))) → {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin)
836, 81, 82mp2an 672 . . . . . . 7 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin
84 rabnc 4106 . . . . . . 7 ({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∩ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) = ∅
85 hashun 13366 . . . . . . 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 1572 . . . . . 6 (♯‘({𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐} ∪ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
8773, 86eqtri 2793 . . . . 5 (♯‘(𝑂𝐸)) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
8887oveq1i 6801 . . . 4 ((♯‘(𝑂𝐸)) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
89 ssrab2 3836 . . . . . . . . 9 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
9011elexi 3365 . . . . . . . . . 10 𝑂 ∈ V
9190elpw2 4959 . . . . . . . . 9 ({𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 ↔ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂)
9289, 91mpbir 221 . . . . . . . 8 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
93 fveq2 6330 . . . . . . . . . 10 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}))
9493oveq1d 6806 . . . . . . . . 9 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
95 ovex 6821 . . . . . . . . 9 ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
9694, 18, 95fvmpt 6422 . . . . . . . 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 30883 . . . . . . 7 (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))
99 nfrab1 3271 . . . . . . . . . . . 12 𝑐{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}
100 nfrab1 3271 . . . . . . . . . . . 12 𝑐{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
10199, 100dfss2f 3743 . . . . . . . . . . 11 ({𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ ∀𝑐(𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
10242, 43, 7, 18, 64, 1ballotlem4 30893 . . . . . . . . . . . . . 14 (𝑐𝑂 → (¬ 1 ∈ 𝑐 → ¬ 𝑐𝐸))
103102imdistani 558 . . . . . . . . . . . . 13 ((𝑐𝑂 ∧ ¬ 1 ∈ 𝑐) → (𝑐𝑂 ∧ ¬ 𝑐𝐸))
104 rabid 3264 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐𝑂 ∧ ¬ 1 ∈ 𝑐))
105 eldif 3733 . . . . . . . . . . . . 13 (𝑐 ∈ (𝑂𝐸) ↔ (𝑐𝑂 ∧ ¬ 𝑐𝐸))
106103, 104, 1053imtr4i 281 . . . . . . . . . . . 12 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ (𝑂𝐸))
107104simprbi 484 . . . . . . . . . . . 12 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → ¬ 1 ∈ 𝑐)
108 rabid 3264 . . . . . . . . . . . 12 (𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝑐))
109106, 107, 108sylanbrc 572 . . . . . . . . . . 11 (𝑐 ∈ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → 𝑐 ∈ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
110101, 109mpgbir 1874 . . . . . . . . . 10 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
111 rabss2 3834 . . . . . . . . . . 11 ((𝑂𝐸) ⊆ 𝑂 → {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐})
11232, 111ax-mp 5 . . . . . . . . . 10 {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}
113110, 112eqssi 3768 . . . . . . . . 9 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} = {𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}
114113fveq2i 6333 . . . . . . . 8 (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})
115114oveq1i 6801 . . . . . . 7 ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
11697, 98, 1153eqtr3i 2801 . . . . . 6 (𝑁 / (𝑀 + 𝑁)) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
117116oveq2i 6802 . . . . 5 (2 · (𝑁 / (𝑀 + 𝑁))) = (2 · ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
118 2cn 11291 . . . . . 6 2 ∈ ℂ
119 hashcl 13342 . . . . . . . 8 ({𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐} ∈ Fin → (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ0)
12083, 119ax-mp 5 . . . . . . 7 (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℕ0
121120nn0cni 11504 . . . . . 6 (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) ∈ ℂ
122118, 121, 28, 57divassi 10981 . . . . 5 ((2 · (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (2 · ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
1231212timesi 11347 . . . . . 6 (2 · (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) = ((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}))
124123oveq1i 6801 . . . . 5 ((2 · (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂)) = (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
125117, 122, 1243eqtr2i 2799 . . . 4 (2 · (𝑁 / (𝑀 + 𝑁))) = (((♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐}) + (♯‘{𝑐 ∈ (𝑂𝐸) ∣ ¬ 1 ∈ 𝑐})) / (♯‘𝑂))
12671, 88, 1253eqtr4ri 2804 . . 3 (2 · (𝑁 / (𝑀 + 𝑁))) = ((♯‘(𝑂𝐸)) / (♯‘𝑂))
127126oveq2i 6802 . 2 (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = (1 − ((♯‘(𝑂𝐸)) / (♯‘𝑂)))
12847nncni 11230 . . . 4 (𝑀 + 𝑁) ∈ ℂ
12943nncni 11230 . . . . 5 𝑁 ∈ ℂ
130118, 129mulcli 10245 . . . 4 (2 · 𝑁) ∈ ℂ
13147nnne0i 11255 . . . . 5 (𝑀 + 𝑁) ≠ 0
132128, 131pm3.2i 456 . . . 4 ((𝑀 + 𝑁) ∈ ℂ ∧ (𝑀 + 𝑁) ≠ 0)
133 divsubdir 10921 . . . 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 1572 . . 3 (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁)))
1351292timesi 11347 . . . . . 6 (2 · 𝑁) = (𝑁 + 𝑁)
136135oveq2i 6802 . . . . 5 ((𝑀 + 𝑁) − (2 · 𝑁)) = ((𝑀 + 𝑁) − (𝑁 + 𝑁))
13742nncni 11230 . . . . . . 7 𝑀 ∈ ℂ
138137, 129, 129, 129addsub4i 10577 . . . . . 6 ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = ((𝑀𝑁) + (𝑁𝑁))
139129subidi 10552 . . . . . . 7 (𝑁𝑁) = 0
140139oveq2i 6802 . . . . . 6 ((𝑀𝑁) + (𝑁𝑁)) = ((𝑀𝑁) + 0)
141137, 129subcli 10557 . . . . . . 7 (𝑀𝑁) ∈ ℂ
142141addid1i 10423 . . . . . 6 ((𝑀𝑁) + 0) = (𝑀𝑁)
143138, 140, 1423eqtri 2797 . . . . 5 ((𝑀 + 𝑁) − (𝑁 + 𝑁)) = (𝑀𝑁)
144136, 143eqtri 2793 . . . 4 ((𝑀 + 𝑁) − (2 · 𝑁)) = (𝑀𝑁)
145144oveq1i 6801 . . 3 (((𝑀 + 𝑁) − (2 · 𝑁)) / (𝑀 + 𝑁)) = ((𝑀𝑁) / (𝑀 + 𝑁))
146128, 131dividi 10958 . . . 4 ((𝑀 + 𝑁) / (𝑀 + 𝑁)) = 1
147118, 129, 128, 131divassi 10981 . . . 4 ((2 · 𝑁) / (𝑀 + 𝑁)) = (2 · (𝑁 / (𝑀 + 𝑁)))
148146, 147oveq12i 6803 . . 3 (((𝑀 + 𝑁) / (𝑀 + 𝑁)) − ((2 · 𝑁) / (𝑀 + 𝑁))) = (1 − (2 · (𝑁 / (𝑀 + 𝑁))))
149134, 145, 1483eqtr3ri 2802 . 2 (1 − (2 · (𝑁 / (𝑀 + 𝑁)))) = ((𝑀𝑁) / (𝑀 + 𝑁))
15063, 127, 1493eqtr2i 2799 1 (𝑃𝐸) = ((𝑀𝑁) / (𝑀 + 𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  {crab 3065  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063  ifcif 4225  𝒫 cpw 4297   class class class wbr 4786  cmpt 4863  cima 5252  cfv 6029  (class class class)co 6791  Fincfn 8107  infcinf 8501  cc 10134  cr 10135  0cc0 10136  1c1 10137   + caddc 10139   · cmul 10141   < clt 10274  cle 10275  cmin 10466   / cdiv 10884  cn 11220  2c2 11270  0cn0 11492  cz 11577  ...cfz 12526  Ccbc 13286  chash 13314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-2o 7712  df-oadd 7715  df-er 7894  df-map 8009  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-sup 8502  df-inf 8503  df-card 8963  df-cda 9190  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-div 10885  df-nn 11221  df-2 11279  df-n0 11493  df-z 11578  df-uz 11887  df-rp 12029  df-fz 12527  df-seq 13002  df-fac 13258  df-bc 13287  df-hash 13315
This theorem is referenced by: (None)
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