Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ballotlem2 Structured version   Visualization version   GIF version

Theorem ballotlem2 31922
 Description: The probability that the first vote picked in a count is a B. (Contributed by Thierry Arnoux, 23-Nov-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)))
Assertion
Ref Expression
ballotlem2 (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))
Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐,𝑥
Allowed substitution hints:   𝑃(𝑥,𝑐)   𝑀(𝑥)   𝑁(𝑥)

Proof of Theorem ballotlem2
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 ballotth.m . . . . . 6 𝑀 ∈ ℕ
2 ballotth.n . . . . . 6 𝑁 ∈ ℕ
3 ballotth.o . . . . . 6 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
41, 2, 3ballotlemoex 31919 . . . . 5 𝑂 ∈ V
5 ssrab2 4009 . . . . 5 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
64, 5elpwi2 5217 . . . 4 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
7 fveq2 6655 . . . . . 6 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}))
87oveq1d 7160 . . . . 5 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
9 ballotth.p . . . . 5 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
10 ovex 7178 . . . . 5 ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
118, 9, 10fvmpt 6755 . . . 4 ({𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂 → (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
126, 11ax-mp 5 . . 3 (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂))
13 an32 645 . . . . . . . 8 (((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ∧ (♯‘𝑐) = 𝑀) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
14 2eluzge1 12302 . . . . . . . . . . . . . 14 2 ∈ (ℤ‘1)
15 fzss1 12961 . . . . . . . . . . . . . 14 (2 ∈ (ℤ‘1) → (2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁)))
1614, 15ax-mp 5 . . . . . . . . . . . . 13 (2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁))
1716sspwi 4514 . . . . . . . . . . . 12 𝒫 (2...(𝑀 + 𝑁)) ⊆ 𝒫 (1...(𝑀 + 𝑁))
1817sseli 3913 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
19 1lt2 11814 . . . . . . . . . . . . . . . 16 1 < 2
20 1re 10648 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ
21 2re 11717 . . . . . . . . . . . . . . . . 17 2 ∈ ℝ
2220, 21ltnlei 10768 . . . . . . . . . . . . . . . 16 (1 < 2 ↔ ¬ 2 ≤ 1)
2319, 22mpbi 233 . . . . . . . . . . . . . . 15 ¬ 2 ≤ 1
24 elfzle1 12925 . . . . . . . . . . . . . . 15 (1 ∈ (2...(𝑀 + 𝑁)) → 2 ≤ 1)
2523, 24mto 200 . . . . . . . . . . . . . 14 ¬ 1 ∈ (2...(𝑀 + 𝑁))
26 elelpwi 4512 . . . . . . . . . . . . . 14 ((1 ∈ 𝑐𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) → 1 ∈ (2...(𝑀 + 𝑁)))
2725, 26mto 200 . . . . . . . . . . . . 13 ¬ (1 ∈ 𝑐𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
28 ancom 464 . . . . . . . . . . . . 13 ((1 ∈ 𝑐𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) ↔ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐))
2927, 28mtbi 325 . . . . . . . . . . . 12 ¬ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐)
3029imnani 404 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → ¬ 1 ∈ 𝑐)
3118, 30jca 515 . . . . . . . . . 10 (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐))
32 ssin 4160 . . . . . . . . . . . 12 ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) ↔ 𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}))
33 1le2 11852 . . . . . . . . . . . . . . . . . . . . 21 1 ≤ 2
34 1p1e2 11768 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) = 2
35 nnge1 11671 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑀 ∈ ℕ → 1 ≤ 𝑀)
361, 35ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 1 ≤ 𝑀
37 nnge1 11671 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → 1 ≤ 𝑁)
382, 37ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 1 ≤ 𝑁
391nnrei 11652 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑀 ∈ ℝ
402nnrei 11652 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑁 ∈ ℝ
4120, 20, 39, 40le2addi 11210 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ≤ 𝑀 ∧ 1 ≤ 𝑁) → (1 + 1) ≤ (𝑀 + 𝑁))
4236, 38, 41mp2an 691 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) ≤ (𝑀 + 𝑁)
4334, 42eqbrtrri 5057 . . . . . . . . . . . . . . . . . . . . 21 2 ≤ (𝑀 + 𝑁)
4439, 40readdcli 10663 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 + 𝑁) ∈ ℝ
4520, 21, 44letri 10776 . . . . . . . . . . . . . . . . . . . . 21 ((1 ≤ 2 ∧ 2 ≤ (𝑀 + 𝑁)) → 1 ≤ (𝑀 + 𝑁))
4633, 43, 45mp2an 691 . . . . . . . . . . . . . . . . . . . 20 1 ≤ (𝑀 + 𝑁)
47 1z 12020 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℤ
48 nnaddcl 11666 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
491, 2, 48mp2an 691 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 + 𝑁) ∈ ℕ
5049nnzi 12014 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 + 𝑁) ∈ ℤ
51 eluz 12265 . . . . . . . . . . . . . . . . . . . . 21 ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝑀 + 𝑁) ∈ (ℤ‘1) ↔ 1 ≤ (𝑀 + 𝑁)))
5247, 50, 51mp2an 691 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 + 𝑁) ∈ (ℤ‘1) ↔ 1 ≤ (𝑀 + 𝑁))
5346, 52mpbir 234 . . . . . . . . . . . . . . . . . . 19 (𝑀 + 𝑁) ∈ (ℤ‘1)
54 elfzp12 13001 . . . . . . . . . . . . . . . . . . 19 ((𝑀 + 𝑁) ∈ (ℤ‘1) → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↔ (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁)))))
5553, 54ax-mp 5 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (1...(𝑀 + 𝑁)) ↔ (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁))))
5655biimpi 219 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...(𝑀 + 𝑁)) → (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁))))
5756orcanai 1000 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁)))
5834oveq1i 7155 . . . . . . . . . . . . . . . 16 ((1 + 1)...(𝑀 + 𝑁)) = (2...(𝑀 + 𝑁))
5957, 58eleqtrdi 2900 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ (2...(𝑀 + 𝑁)))
6059ss2abi 3996 . . . . . . . . . . . . . 14 {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} ⊆ {𝑖𝑖 ∈ (2...(𝑀 + 𝑁))}
61 inab 4226 . . . . . . . . . . . . . . 15 ({𝑖𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)}
62 abid2 2932 . . . . . . . . . . . . . . . 16 {𝑖𝑖 ∈ (1...(𝑀 + 𝑁))} = (1...(𝑀 + 𝑁))
6362ineq1i 4138 . . . . . . . . . . . . . . 15 ({𝑖𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
6461, 63eqtr3i 2823 . . . . . . . . . . . . . 14 {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
65 abid2 2932 . . . . . . . . . . . . . 14 {𝑖𝑖 ∈ (2...(𝑀 + 𝑁))} = (2...(𝑀 + 𝑁))
6660, 64, 653sstr3i 3959 . . . . . . . . . . . . 13 ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))
67 sstr 3925 . . . . . . . . . . . . 13 ((𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ∧ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
6866, 67mpan2 690 . . . . . . . . . . . 12 (𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
6932, 68sylbi 220 . . . . . . . . . . 11 ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
70 velpw 4505 . . . . . . . . . . . 12 (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (1...(𝑀 + 𝑁)))
71 ssab 3991 . . . . . . . . . . . . 13 (𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1} ↔ ∀𝑖(𝑖𝑐 → ¬ 𝑖 = 1))
72 df-ex 1782 . . . . . . . . . . . . . . . 16 (∃𝑖(𝑖 = 1 ∧ 𝑖𝑐) ↔ ¬ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖𝑐))
7372bicomi 227 . . . . . . . . . . . . . . 15 (¬ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖𝑐) ↔ ∃𝑖(𝑖 = 1 ∧ 𝑖𝑐))
7473con1bii 360 . . . . . . . . . . . . . 14 (¬ ∃𝑖(𝑖 = 1 ∧ 𝑖𝑐) ↔ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖𝑐))
75 dfclel 2871 . . . . . . . . . . . . . . 15 (1 ∈ 𝑐 ↔ ∃𝑖(𝑖 = 1 ∧ 𝑖𝑐))
7675notbii 323 . . . . . . . . . . . . . 14 (¬ 1 ∈ 𝑐 ↔ ¬ ∃𝑖(𝑖 = 1 ∧ 𝑖𝑐))
77 imnang 1843 . . . . . . . . . . . . . . 15 (∀𝑖(𝑖𝑐 → ¬ 𝑖 = 1) ↔ ∀𝑖 ¬ (𝑖𝑐𝑖 = 1))
78 ancom 464 . . . . . . . . . . . . . . . . 17 ((𝑖 = 1 ∧ 𝑖𝑐) ↔ (𝑖𝑐𝑖 = 1))
7978notbii 323 . . . . . . . . . . . . . . . 16 (¬ (𝑖 = 1 ∧ 𝑖𝑐) ↔ ¬ (𝑖𝑐𝑖 = 1))
8079albii 1821 . . . . . . . . . . . . . . 15 (∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖𝑐) ↔ ∀𝑖 ¬ (𝑖𝑐𝑖 = 1))
8177, 80bitr4i 281 . . . . . . . . . . . . . 14 (∀𝑖(𝑖𝑐 → ¬ 𝑖 = 1) ↔ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖𝑐))
8274, 76, 813bitr4ri 307 . . . . . . . . . . . . 13 (∀𝑖(𝑖𝑐 → ¬ 𝑖 = 1) ↔ ¬ 1 ∈ 𝑐)
8371, 82bitr2i 279 . . . . . . . . . . . 12 (¬ 1 ∈ 𝑐𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1})
8470, 83anbi12i 629 . . . . . . . . . . 11 ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ↔ (𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}))
85 velpw 4505 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (2...(𝑀 + 𝑁)))
8669, 84, 853imtr4i 295 . . . . . . . . . 10 ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) → 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
8731, 86impbii 212 . . . . . . . . 9 (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐))
8887anbi1i 626 . . . . . . . 8 ((𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ∧ (♯‘𝑐) = 𝑀))
893rabeq2i 3436 . . . . . . . . 9 (𝑐𝑂 ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀))
9089anbi1i 626 . . . . . . . 8 ((𝑐𝑂 ∧ ¬ 1 ∈ 𝑐) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
9113, 88, 903bitr4i 306 . . . . . . 7 ((𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ↔ (𝑐𝑂 ∧ ¬ 1 ∈ 𝑐))
9291rabbia2 3425 . . . . . 6 {𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}
9392fveq2i 6658 . . . . 5 (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) = (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐})
94 fzfi 13355 . . . . . . 7 (2...(𝑀 + 𝑁)) ∈ Fin
951nnzi 12014 . . . . . . 7 𝑀 ∈ ℤ
96 hashbc 13827 . . . . . . 7 (((2...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑀 ∈ ℤ) → ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}))
9794, 95, 96mp2an 691 . . . . . 6 ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀})
98 2z 12022 . . . . . . . . . . . 12 2 ∈ ℤ
9998eluz1i 12259 . . . . . . . . . . 11 ((𝑀 + 𝑁) ∈ (ℤ‘2) ↔ ((𝑀 + 𝑁) ∈ ℤ ∧ 2 ≤ (𝑀 + 𝑁)))
10050, 43, 99mpbir2an 710 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ (ℤ‘2)
101 hashfz 13804 . . . . . . . . . 10 ((𝑀 + 𝑁) ∈ (ℤ‘2) → (♯‘(2...(𝑀 + 𝑁))) = (((𝑀 + 𝑁) − 2) + 1))
102100, 101ax-mp 5 . . . . . . . . 9 (♯‘(2...(𝑀 + 𝑁))) = (((𝑀 + 𝑁) − 2) + 1)
1031nncni 11653 . . . . . . . . . . 11 𝑀 ∈ ℂ
1042nncni 11653 . . . . . . . . . . 11 𝑁 ∈ ℂ
105103, 104addcli 10654 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ ℂ
106 2cn 11718 . . . . . . . . . 10 2 ∈ ℂ
107 ax-1cn 10602 . . . . . . . . . 10 1 ∈ ℂ
108 subadd23 10905 . . . . . . . . . 10 (((𝑀 + 𝑁) ∈ ℂ ∧ 2 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2)))
109105, 106, 107, 108mp3an 1458 . . . . . . . . 9 (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2))
110106, 107negsubdi2i 10979 . . . . . . . . . . 11 -(2 − 1) = (1 − 2)
111 2m1e1 11769 . . . . . . . . . . . 12 (2 − 1) = 1
112111negeqi 10886 . . . . . . . . . . 11 -(2 − 1) = -1
113110, 112eqtr3i 2823 . . . . . . . . . 10 (1 − 2) = -1
114113oveq2i 7156 . . . . . . . . 9 ((𝑀 + 𝑁) + (1 − 2)) = ((𝑀 + 𝑁) + -1)
115102, 109, 1143eqtri 2825 . . . . . . . 8 (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) + -1)
116105, 107negsubi 10971 . . . . . . . 8 ((𝑀 + 𝑁) + -1) = ((𝑀 + 𝑁) − 1)
117115, 116eqtri 2821 . . . . . . 7 (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) − 1)
118117oveq1i 7155 . . . . . 6 ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (((𝑀 + 𝑁) − 1)C𝑀)
11997, 118eqtr3i 2823 . . . . 5 (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) = (((𝑀 + 𝑁) − 1)C𝑀)
12093, 119eqtr3i 2823 . . . 4 (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (((𝑀 + 𝑁) − 1)C𝑀)
1211, 2, 3ballotlem1 31920 . . . 4 (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
122120, 121oveq12i 7157 . . 3 ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
12312, 122eqtri 2821 . 2 (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
124 0le1 11170 . . . . 5 0 ≤ 1
125 0re 10650 . . . . . 6 0 ∈ ℝ
126125, 20, 39letri 10776 . . . . 5 ((0 ≤ 1 ∧ 1 ≤ 𝑀) → 0 ≤ 𝑀)
127124, 36, 126mp2an 691 . . . 4 0 ≤ 𝑀
1282nngt0i 11682 . . . . . 6 0 < 𝑁
12940, 128elrpii 12400 . . . . 5 𝑁 ∈ ℝ+
130 ltaddrp 12434 . . . . 5 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → 𝑀 < (𝑀 + 𝑁))
13139, 129, 130mp2an 691 . . . 4 𝑀 < (𝑀 + 𝑁)
132 0z 12000 . . . . 5 0 ∈ ℤ
133 elfzm11 12993 . . . . 5 ((0 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀𝑀 < (𝑀 + 𝑁))))
134132, 50, 133mp2an 691 . . . 4 (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀𝑀 < (𝑀 + 𝑁)))
13595, 127, 131, 134mpbir3an 1338 . . 3 𝑀 ∈ (0...((𝑀 + 𝑁) − 1))
136 bcm1n 30588 . . 3 ((𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ∧ (𝑀 + 𝑁) ∈ ℕ) → ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀)) = (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)))
137135, 49, 136mp2an 691 . 2 ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀)) = (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁))
138 pncan2 10900 . . . 4 ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
139103, 104, 138mp2an 691 . . 3 ((𝑀 + 𝑁) − 𝑀) = 𝑁
140139oveq1i 7155 . 2 (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)) = (𝑁 / (𝑀 + 𝑁))
141123, 137, 1403eqtri 2825 1 (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  {crab 3110  Vcvv 3442   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4500   class class class wbr 5034   ↦ cmpt 5114  ‘cfv 6332  (class class class)co 7145  Fincfn 8510  ℂcc 10542  ℝcr 10543  0cc0 10544  1c1 10545   + caddc 10547   < clt 10682   ≤ cle 10683   − cmin 10877  -cneg 10878   / cdiv 11304  ℕcn 11643  2c2 11698  ℤcz 11989  ℤ≥cuz 12251  ℝ+crp 12397  ...cfz 12905  Ccbc 13678  ♯chash 13706 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-2o 8104  df-oadd 8107  df-er 8290  df-map 8409  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-dju 9332  df-card 9370  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-div 11305  df-nn 11644  df-2 11706  df-n0 11904  df-z 11990  df-uz 12252  df-rp 12398  df-fz 12906  df-seq 13385  df-fac 13650  df-bc 13679  df-hash 13707 This theorem is referenced by:  ballotth  31971
 Copyright terms: Public domain W3C validator