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Theorem ballotlem2 32167
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 32164 . . . . 5 𝑂 ∈ V
5 ssrab2 3993 . . . . 5 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
64, 5elpwi2 5239 . . . 4 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
7 fveq2 6717 . . . . . 6 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}))
87oveq1d 7228 . . . . 5 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
9 ballotth.p . . . . 5 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
10 ovex 7246 . . . . 5 ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
118, 9, 10fvmpt 6818 . . . 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 646 . . . . . . . 8 (((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ∧ (♯‘𝑐) = 𝑀) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
14 2eluzge1 12490 . . . . . . . . . . . . . 14 2 ∈ (ℤ‘1)
15 fzss1 13151 . . . . . . . . . . . . . 14 (2 ∈ (ℤ‘1) → (2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁)))
1614, 15ax-mp 5 . . . . . . . . . . . . 13 (2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁))
1716sspwi 4527 . . . . . . . . . . . 12 𝒫 (2...(𝑀 + 𝑁)) ⊆ 𝒫 (1...(𝑀 + 𝑁))
1817sseli 3896 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
19 1lt2 12001 . . . . . . . . . . . . . . . 16 1 < 2
20 1re 10833 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ
21 2re 11904 . . . . . . . . . . . . . . . . 17 2 ∈ ℝ
2220, 21ltnlei 10953 . . . . . . . . . . . . . . . 16 (1 < 2 ↔ ¬ 2 ≤ 1)
2319, 22mpbi 233 . . . . . . . . . . . . . . 15 ¬ 2 ≤ 1
24 elfzle1 13115 . . . . . . . . . . . . . . 15 (1 ∈ (2...(𝑀 + 𝑁)) → 2 ≤ 1)
2523, 24mto 200 . . . . . . . . . . . . . 14 ¬ 1 ∈ (2...(𝑀 + 𝑁))
26 elelpwi 4525 . . . . . . . . . . . . . 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 4145 . . . . . . . . . . . 12 ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) ↔ 𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}))
33 1le2 12039 . . . . . . . . . . . . . . . . . . . . 21 1 ≤ 2
34 1p1e2 11955 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) = 2
35 nnge1 11858 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑀 ∈ ℕ → 1 ≤ 𝑀)
361, 35ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 1 ≤ 𝑀
37 nnge1 11858 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → 1 ≤ 𝑁)
382, 37ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 1 ≤ 𝑁
391nnrei 11839 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑀 ∈ ℝ
402nnrei 11839 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑁 ∈ ℝ
4120, 20, 39, 40le2addi 11395 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ≤ 𝑀 ∧ 1 ≤ 𝑁) → (1 + 1) ≤ (𝑀 + 𝑁))
4236, 38, 41mp2an 692 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) ≤ (𝑀 + 𝑁)
4334, 42eqbrtrri 5076 . . . . . . . . . . . . . . . . . . . . 21 2 ≤ (𝑀 + 𝑁)
4439, 40readdcli 10848 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 + 𝑁) ∈ ℝ
4520, 21, 44letri 10961 . . . . . . . . . . . . . . . . . . . . 21 ((1 ≤ 2 ∧ 2 ≤ (𝑀 + 𝑁)) → 1 ≤ (𝑀 + 𝑁))
4633, 43, 45mp2an 692 . . . . . . . . . . . . . . . . . . . 20 1 ≤ (𝑀 + 𝑁)
47 1z 12207 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℤ
48 nnaddcl 11853 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
491, 2, 48mp2an 692 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 + 𝑁) ∈ ℕ
5049nnzi 12201 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 + 𝑁) ∈ ℤ
51 eluz 12452 . . . . . . . . . . . . . . . . . . . . 21 ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝑀 + 𝑁) ∈ (ℤ‘1) ↔ 1 ≤ (𝑀 + 𝑁)))
5247, 50, 51mp2an 692 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 + 𝑁) ∈ (ℤ‘1) ↔ 1 ≤ (𝑀 + 𝑁))
5346, 52mpbir 234 . . . . . . . . . . . . . . . . . . 19 (𝑀 + 𝑁) ∈ (ℤ‘1)
54 elfzp12 13191 . . . . . . . . . . . . . . . . . . 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 1003 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁)))
5834oveq1i 7223 . . . . . . . . . . . . . . . 16 ((1 + 1)...(𝑀 + 𝑁)) = (2...(𝑀 + 𝑁))
5957, 58eleqtrdi 2848 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ (2...(𝑀 + 𝑁)))
6059ss2abi 3980 . . . . . . . . . . . . . 14 {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} ⊆ {𝑖𝑖 ∈ (2...(𝑀 + 𝑁))}
61 inab 4214 . . . . . . . . . . . . . . 15 ({𝑖𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)}
62 abid2 2879 . . . . . . . . . . . . . . . 16 {𝑖𝑖 ∈ (1...(𝑀 + 𝑁))} = (1...(𝑀 + 𝑁))
6362ineq1i 4123 . . . . . . . . . . . . . . 15 ({𝑖𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
6461, 63eqtr3i 2767 . . . . . . . . . . . . . 14 {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
65 abid2 2879 . . . . . . . . . . . . . 14 {𝑖𝑖 ∈ (2...(𝑀 + 𝑁))} = (2...(𝑀 + 𝑁))
6660, 64, 653sstr3i 3943 . . . . . . . . . . . . 13 ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))
67 sstr 3909 . . . . . . . . . . . . 13 ((𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ∧ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
6866, 67mpan2 691 . . . . . . . . . . . 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 4518 . . . . . . . . . . . 12 (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (1...(𝑀 + 𝑁)))
71 ssab 3975 . . . . . . . . . . . . 13 (𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1} ↔ ∀𝑖(𝑖𝑐 → ¬ 𝑖 = 1))
72 df-ex 1788 . . . . . . . . . . . . . . . 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 2817 . . . . . . . . . . . . . . 15 (1 ∈ 𝑐 ↔ ∃𝑖(𝑖 = 1 ∧ 𝑖𝑐))
7675notbii 323 . . . . . . . . . . . . . 14 (¬ 1 ∈ 𝑐 ↔ ¬ ∃𝑖(𝑖 = 1 ∧ 𝑖𝑐))
77 imnang 1849 . . . . . . . . . . . . . . 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 1827 . . . . . . . . . . . . . . 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 630 . . . . . . . . . . 11 ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ↔ (𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}))
85 velpw 4518 . . . . . . . . . . 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 627 . . . . . . . 8 ((𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ∧ (♯‘𝑐) = 𝑀))
893rabeq2i 3398 . . . . . . . . 9 (𝑐𝑂 ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀))
9089anbi1i 627 . . . . . . . 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 3387 . . . . . 6 {𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}
9392fveq2i 6720 . . . . 5 (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) = (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐})
94 fzfi 13545 . . . . . . 7 (2...(𝑀 + 𝑁)) ∈ Fin
951nnzi 12201 . . . . . . 7 𝑀 ∈ ℤ
96 hashbc 14017 . . . . . . 7 (((2...(𝑀 + 𝑁)) ∈ Fin ∧ 𝑀 ∈ ℤ) → ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}))
9794, 95, 96mp2an 692 . . . . . 6 ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀})
98 2z 12209 . . . . . . . . . . . 12 2 ∈ ℤ
9998eluz1i 12446 . . . . . . . . . . 11 ((𝑀 + 𝑁) ∈ (ℤ‘2) ↔ ((𝑀 + 𝑁) ∈ ℤ ∧ 2 ≤ (𝑀 + 𝑁)))
10050, 43, 99mpbir2an 711 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ (ℤ‘2)
101 hashfz 13994 . . . . . . . . . 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 11840 . . . . . . . . . . 11 𝑀 ∈ ℂ
1042nncni 11840 . . . . . . . . . . 11 𝑁 ∈ ℂ
105103, 104addcli 10839 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ ℂ
106 2cn 11905 . . . . . . . . . 10 2 ∈ ℂ
107 ax-1cn 10787 . . . . . . . . . 10 1 ∈ ℂ
108 subadd23 11090 . . . . . . . . . 10 (((𝑀 + 𝑁) ∈ ℂ ∧ 2 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2)))
109105, 106, 107, 108mp3an 1463 . . . . . . . . 9 (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2))
110106, 107negsubdi2i 11164 . . . . . . . . . . 11 -(2 − 1) = (1 − 2)
111 2m1e1 11956 . . . . . . . . . . . 12 (2 − 1) = 1
112111negeqi 11071 . . . . . . . . . . 11 -(2 − 1) = -1
113110, 112eqtr3i 2767 . . . . . . . . . 10 (1 − 2) = -1
114113oveq2i 7224 . . . . . . . . 9 ((𝑀 + 𝑁) + (1 − 2)) = ((𝑀 + 𝑁) + -1)
115102, 109, 1143eqtri 2769 . . . . . . . 8 (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) + -1)
116105, 107negsubi 11156 . . . . . . . 8 ((𝑀 + 𝑁) + -1) = ((𝑀 + 𝑁) − 1)
117115, 116eqtri 2765 . . . . . . 7 (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) − 1)
118117oveq1i 7223 . . . . . 6 ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (((𝑀 + 𝑁) − 1)C𝑀)
11997, 118eqtr3i 2767 . . . . 5 (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) = (((𝑀 + 𝑁) − 1)C𝑀)
12093, 119eqtr3i 2767 . . . 4 (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (((𝑀 + 𝑁) − 1)C𝑀)
1211, 2, 3ballotlem1 32165 . . . 4 (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
122120, 121oveq12i 7225 . . 3 ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
12312, 122eqtri 2765 . 2 (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
124 0le1 11355 . . . . 5 0 ≤ 1
125 0re 10835 . . . . . 6 0 ∈ ℝ
126125, 20, 39letri 10961 . . . . 5 ((0 ≤ 1 ∧ 1 ≤ 𝑀) → 0 ≤ 𝑀)
127124, 36, 126mp2an 692 . . . 4 0 ≤ 𝑀
1282nngt0i 11869 . . . . . 6 0 < 𝑁
12940, 128elrpii 12589 . . . . 5 𝑁 ∈ ℝ+
130 ltaddrp 12623 . . . . 5 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → 𝑀 < (𝑀 + 𝑁))
13139, 129, 130mp2an 692 . . . 4 𝑀 < (𝑀 + 𝑁)
132 0z 12187 . . . . 5 0 ∈ ℤ
133 elfzm11 13183 . . . . 5 ((0 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀𝑀 < (𝑀 + 𝑁))))
134132, 50, 133mp2an 692 . . . 4 (𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ↔ (𝑀 ∈ ℤ ∧ 0 ≤ 𝑀𝑀 < (𝑀 + 𝑁)))
13595, 127, 131, 134mpbir3an 1343 . . 3 𝑀 ∈ (0...((𝑀 + 𝑁) − 1))
136 bcm1n 30836 . . 3 ((𝑀 ∈ (0...((𝑀 + 𝑁) − 1)) ∧ (𝑀 + 𝑁) ∈ ℕ) → ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀)) = (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)))
137135, 49, 136mp2an 692 . 2 ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀)) = (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁))
138 pncan2 11085 . . . 4 ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
139103, 104, 138mp2an 692 . . 3 ((𝑀 + 𝑁) − 𝑀) = 𝑁
140139oveq1i 7223 . 2 (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)) = (𝑁 / (𝑀 + 𝑁))
141123, 137, 1403eqtri 2769 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 847  w3a 1089  wal 1541   = wceq 1543  wex 1787  wcel 2110  {cab 2714  {crab 3065  Vcvv 3408  cin 3865  wss 3866  𝒫 cpw 4513   class class class wbr 5053  cmpt 5135  cfv 6380  (class class class)co 7213  Fincfn 8626  cc 10727  cr 10728  0cc0 10729  1c1 10730   + caddc 10732   < clt 10867  cle 10868  cmin 11062  -cneg 11063   / cdiv 11489  cn 11830  2c2 11885  cz 12176  cuz 12438  +crp 12586  ...cfz 13095  Ccbc 13868  chash 13896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-dju 9517  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-n0 12091  df-z 12177  df-uz 12439  df-rp 12587  df-fz 13096  df-seq 13575  df-fac 13840  df-bc 13869  df-hash 13897
This theorem is referenced by:  ballotth  32216
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