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Theorem ballotlem2 33476
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 33473 . . . . 5 𝑂 ∈ V
5 ssrab2 4077 . . . . 5 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ⊆ 𝑂
64, 5elpwi2 5346 . . . 4 {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} ∈ 𝒫 𝑂
7 fveq2 6889 . . . . . 6 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → (♯‘𝑥) = (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}))
87oveq1d 7421 . . . . 5 (𝑥 = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐} → ((♯‘𝑥) / (♯‘𝑂)) = ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)))
9 ballotth.p . . . . 5 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
10 ovex 7439 . . . . 5 ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) ∈ V
118, 9, 10fvmpt 6996 . . . 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 12875 . . . . . . . . . . . . . 14 2 ∈ (ℤ‘1)
15 fzss1 13537 . . . . . . . . . . . . . 14 (2 ∈ (ℤ‘1) → (2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁)))
1614, 15ax-mp 5 . . . . . . . . . . . . 13 (2...(𝑀 + 𝑁)) ⊆ (1...(𝑀 + 𝑁))
1716sspwi 4614 . . . . . . . . . . . 12 𝒫 (2...(𝑀 + 𝑁)) ⊆ 𝒫 (1...(𝑀 + 𝑁))
1817sseli 3978 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → 𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)))
19 1lt2 12380 . . . . . . . . . . . . . . . 16 1 < 2
20 1re 11211 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ
21 2re 12283 . . . . . . . . . . . . . . . . 17 2 ∈ ℝ
2220, 21ltnlei 11332 . . . . . . . . . . . . . . . 16 (1 < 2 ↔ ¬ 2 ≤ 1)
2319, 22mpbi 229 . . . . . . . . . . . . . . 15 ¬ 2 ≤ 1
24 elfzle1 13501 . . . . . . . . . . . . . . 15 (1 ∈ (2...(𝑀 + 𝑁)) → 2 ≤ 1)
2523, 24mto 196 . . . . . . . . . . . . . 14 ¬ 1 ∈ (2...(𝑀 + 𝑁))
26 elelpwi 4612 . . . . . . . . . . . . . 14 ((1 ∈ 𝑐𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) → 1 ∈ (2...(𝑀 + 𝑁)))
2725, 26mto 196 . . . . . . . . . . . . 13 ¬ (1 ∈ 𝑐𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
28 ancom 462 . . . . . . . . . . . . 13 ((1 ∈ 𝑐𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁))) ↔ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐))
2927, 28mtbi 322 . . . . . . . . . . . 12 ¬ (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ 1 ∈ 𝑐)
3029imnani 402 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → ¬ 1 ∈ 𝑐)
3118, 30jca 513 . . . . . . . . . 10 (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) → (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐))
32 ssin 4230 . . . . . . . . . . . 12 ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) ↔ 𝑐 ⊆ ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}))
33 1le2 12418 . . . . . . . . . . . . . . . . . . . . 21 1 ≤ 2
34 1p1e2 12334 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) = 2
35 nnge1 12237 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑀 ∈ ℕ → 1 ≤ 𝑀)
361, 35ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 1 ≤ 𝑀
37 nnge1 12237 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → 1 ≤ 𝑁)
382, 37ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 1 ≤ 𝑁
391nnrei 12218 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑀 ∈ ℝ
402nnrei 12218 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑁 ∈ ℝ
4120, 20, 39, 40le2addi 11774 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ≤ 𝑀 ∧ 1 ≤ 𝑁) → (1 + 1) ≤ (𝑀 + 𝑁))
4236, 38, 41mp2an 691 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) ≤ (𝑀 + 𝑁)
4334, 42eqbrtrri 5171 . . . . . . . . . . . . . . . . . . . . 21 2 ≤ (𝑀 + 𝑁)
4439, 40readdcli 11226 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 + 𝑁) ∈ ℝ
4520, 21, 44letri 11340 . . . . . . . . . . . . . . . . . . . . 21 ((1 ≤ 2 ∧ 2 ≤ (𝑀 + 𝑁)) → 1 ≤ (𝑀 + 𝑁))
4633, 43, 45mp2an 691 . . . . . . . . . . . . . . . . . . . 20 1 ≤ (𝑀 + 𝑁)
47 1z 12589 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℤ
48 nnaddcl 12232 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
491, 2, 48mp2an 691 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 + 𝑁) ∈ ℕ
5049nnzi 12583 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 + 𝑁) ∈ ℤ
51 eluz 12833 . . . . . . . . . . . . . . . . . . . . 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 230 . . . . . . . . . . . . . . . . . . 19 (𝑀 + 𝑁) ∈ (ℤ‘1)
54 elfzp12 13577 . . . . . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...(𝑀 + 𝑁)) → (𝑖 = 1 ∨ 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁))))
5756orcanai 1002 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ ((1 + 1)...(𝑀 + 𝑁)))
5834oveq1i 7416 . . . . . . . . . . . . . . . 16 ((1 + 1)...(𝑀 + 𝑁)) = (2...(𝑀 + 𝑁))
5957, 58eleqtrdi 2844 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1) → 𝑖 ∈ (2...(𝑀 + 𝑁)))
6059ss2abi 4063 . . . . . . . . . . . . . 14 {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} ⊆ {𝑖𝑖 ∈ (2...(𝑀 + 𝑁))}
61 inab 4299 . . . . . . . . . . . . . . 15 ({𝑖𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)}
62 abid2 2872 . . . . . . . . . . . . . . . 16 {𝑖𝑖 ∈ (1...(𝑀 + 𝑁))} = (1...(𝑀 + 𝑁))
6362ineq1i 4208 . . . . . . . . . . . . . . 15 ({𝑖𝑖 ∈ (1...(𝑀 + 𝑁))} ∩ {𝑖 ∣ ¬ 𝑖 = 1}) = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
6461, 63eqtr3i 2763 . . . . . . . . . . . . . 14 {𝑖 ∣ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ ¬ 𝑖 = 1)} = ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1})
65 abid2 2872 . . . . . . . . . . . . . 14 {𝑖𝑖 ∈ (2...(𝑀 + 𝑁))} = (2...(𝑀 + 𝑁))
6660, 64, 653sstr3i 4024 . . . . . . . . . . . . 13 ((1...(𝑀 + 𝑁)) ∩ {𝑖 ∣ ¬ 𝑖 = 1}) ⊆ (2...(𝑀 + 𝑁))
67 sstr 3990 . . . . . . . . . . . . 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 216 . . . . . . . . . . 11 ((𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}) → 𝑐 ⊆ (2...(𝑀 + 𝑁)))
70 velpw 4607 . . . . . . . . . . . 12 (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (1...(𝑀 + 𝑁)))
71 ssab 4058 . . . . . . . . . . . . 13 (𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1} ↔ ∀𝑖(𝑖𝑐 → ¬ 𝑖 = 1))
72 df-ex 1783 . . . . . . . . . . . . . . . 16 (∃𝑖(𝑖 = 1 ∧ 𝑖𝑐) ↔ ¬ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖𝑐))
7372bicomi 223 . . . . . . . . . . . . . . 15 (¬ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖𝑐) ↔ ∃𝑖(𝑖 = 1 ∧ 𝑖𝑐))
7473con1bii 357 . . . . . . . . . . . . . 14 (¬ ∃𝑖(𝑖 = 1 ∧ 𝑖𝑐) ↔ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖𝑐))
75 dfclel 2812 . . . . . . . . . . . . . . 15 (1 ∈ 𝑐 ↔ ∃𝑖(𝑖 = 1 ∧ 𝑖𝑐))
7675notbii 320 . . . . . . . . . . . . . 14 (¬ 1 ∈ 𝑐 ↔ ¬ ∃𝑖(𝑖 = 1 ∧ 𝑖𝑐))
77 imnang 1845 . . . . . . . . . . . . . . 15 (∀𝑖(𝑖𝑐 → ¬ 𝑖 = 1) ↔ ∀𝑖 ¬ (𝑖𝑐𝑖 = 1))
78 ancom 462 . . . . . . . . . . . . . . . . 17 ((𝑖 = 1 ∧ 𝑖𝑐) ↔ (𝑖𝑐𝑖 = 1))
7978notbii 320 . . . . . . . . . . . . . . . 16 (¬ (𝑖 = 1 ∧ 𝑖𝑐) ↔ ¬ (𝑖𝑐𝑖 = 1))
8079albii 1822 . . . . . . . . . . . . . . 15 (∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖𝑐) ↔ ∀𝑖 ¬ (𝑖𝑐𝑖 = 1))
8177, 80bitr4i 278 . . . . . . . . . . . . . 14 (∀𝑖(𝑖𝑐 → ¬ 𝑖 = 1) ↔ ∀𝑖 ¬ (𝑖 = 1 ∧ 𝑖𝑐))
8274, 76, 813bitr4ri 304 . . . . . . . . . . . . 13 (∀𝑖(𝑖𝑐 → ¬ 𝑖 = 1) ↔ ¬ 1 ∈ 𝑐)
8371, 82bitr2i 276 . . . . . . . . . . . 12 (¬ 1 ∈ 𝑐𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1})
8470, 83anbi12i 628 . . . . . . . . . . 11 ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ↔ (𝑐 ⊆ (1...(𝑀 + 𝑁)) ∧ 𝑐 ⊆ {𝑖 ∣ ¬ 𝑖 = 1}))
85 velpw 4607 . . . . . . . . . . 11 (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ↔ 𝑐 ⊆ (2...(𝑀 + 𝑁)))
8669, 84, 853imtr4i 292 . . . . . . . . . 10 ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) → 𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)))
8731, 86impbii 208 . . . . . . . . 9 (𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐))
8887anbi1i 625 . . . . . . . 8 ((𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ ¬ 1 ∈ 𝑐) ∧ (♯‘𝑐) = 𝑀))
893reqabi 3455 . . . . . . . . 9 (𝑐𝑂 ↔ (𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀))
9089anbi1i 625 . . . . . . . 8 ((𝑐𝑂 ∧ ¬ 1 ∈ 𝑐) ↔ ((𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ∧ ¬ 1 ∈ 𝑐))
9113, 88, 903bitr4i 303 . . . . . . 7 ((𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∧ (♯‘𝑐) = 𝑀) ↔ (𝑐𝑂 ∧ ¬ 1 ∈ 𝑐))
9291rabbia2 3436 . . . . . 6 {𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} = {𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}
9392fveq2i 6892 . . . . 5 (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) = (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐})
94 fzfi 13934 . . . . . . 7 (2...(𝑀 + 𝑁)) ∈ Fin
951nnzi 12583 . . . . . . 7 𝑀 ∈ ℤ
96 hashbc 14409 . . . . . . 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 12591 . . . . . . . . . . . 12 2 ∈ ℤ
9998eluz1i 12827 . . . . . . . . . . 11 ((𝑀 + 𝑁) ∈ (ℤ‘2) ↔ ((𝑀 + 𝑁) ∈ ℤ ∧ 2 ≤ (𝑀 + 𝑁)))
10050, 43, 99mpbir2an 710 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ (ℤ‘2)
101 hashfz 14384 . . . . . . . . . 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 12219 . . . . . . . . . . 11 𝑀 ∈ ℂ
1042nncni 12219 . . . . . . . . . . 11 𝑁 ∈ ℂ
105103, 104addcli 11217 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ ℂ
106 2cn 12284 . . . . . . . . . 10 2 ∈ ℂ
107 ax-1cn 11165 . . . . . . . . . 10 1 ∈ ℂ
108 subadd23 11469 . . . . . . . . . 10 (((𝑀 + 𝑁) ∈ ℂ ∧ 2 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2)))
109105, 106, 107, 108mp3an 1462 . . . . . . . . 9 (((𝑀 + 𝑁) − 2) + 1) = ((𝑀 + 𝑁) + (1 − 2))
110106, 107negsubdi2i 11543 . . . . . . . . . . 11 -(2 − 1) = (1 − 2)
111 2m1e1 12335 . . . . . . . . . . . 12 (2 − 1) = 1
112111negeqi 11450 . . . . . . . . . . 11 -(2 − 1) = -1
113110, 112eqtr3i 2763 . . . . . . . . . 10 (1 − 2) = -1
114113oveq2i 7417 . . . . . . . . 9 ((𝑀 + 𝑁) + (1 − 2)) = ((𝑀 + 𝑁) + -1)
115102, 109, 1143eqtri 2765 . . . . . . . 8 (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) + -1)
116105, 107negsubi 11535 . . . . . . . 8 ((𝑀 + 𝑁) + -1) = ((𝑀 + 𝑁) − 1)
117115, 116eqtri 2761 . . . . . . 7 (♯‘(2...(𝑀 + 𝑁))) = ((𝑀 + 𝑁) − 1)
118117oveq1i 7416 . . . . . 6 ((♯‘(2...(𝑀 + 𝑁)))C𝑀) = (((𝑀 + 𝑁) − 1)C𝑀)
11997, 118eqtr3i 2763 . . . . 5 (♯‘{𝑐 ∈ 𝒫 (2...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}) = (((𝑀 + 𝑁) − 1)C𝑀)
12093, 119eqtr3i 2763 . . . 4 (♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (((𝑀 + 𝑁) − 1)C𝑀)
1211, 2, 3ballotlem1 33474 . . . 4 (♯‘𝑂) = ((𝑀 + 𝑁)C𝑀)
122120, 121oveq12i 7418 . . 3 ((♯‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) / (♯‘𝑂)) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
12312, 122eqtri 2761 . 2 (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = ((((𝑀 + 𝑁) − 1)C𝑀) / ((𝑀 + 𝑁)C𝑀))
124 0le1 11734 . . . . 5 0 ≤ 1
125 0re 11213 . . . . . 6 0 ∈ ℝ
126125, 20, 39letri 11340 . . . . 5 ((0 ≤ 1 ∧ 1 ≤ 𝑀) → 0 ≤ 𝑀)
127124, 36, 126mp2an 691 . . . 4 0 ≤ 𝑀
1282nngt0i 12248 . . . . . 6 0 < 𝑁
12940, 128elrpii 12974 . . . . 5 𝑁 ∈ ℝ+
130 ltaddrp 13008 . . . . 5 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → 𝑀 < (𝑀 + 𝑁))
13139, 129, 130mp2an 691 . . . 4 𝑀 < (𝑀 + 𝑁)
132 0z 12566 . . . . 5 0 ∈ ℤ
133 elfzm11 13569 . . . . 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 1342 . . 3 𝑀 ∈ (0...((𝑀 + 𝑁) − 1))
136 bcm1n 31994 . . 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 11464 . . . 4 ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
139103, 104, 138mp2an 691 . . 3 ((𝑀 + 𝑁) − 𝑀) = 𝑁
140139oveq1i 7416 . 2 (((𝑀 + 𝑁) − 𝑀) / (𝑀 + 𝑁)) = (𝑁 / (𝑀 + 𝑁))
141123, 137, 1403eqtri 2765 1 (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3a 1088  wal 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2710  {crab 3433  Vcvv 3475  cin 3947  wss 3948  𝒫 cpw 4602   class class class wbr 5148  cmpt 5231  cfv 6541  (class class class)co 7406  Fincfn 8936  cc 11105  cr 11106  0cc0 11107  1c1 11108   + caddc 11110   < clt 11245  cle 11246  cmin 11441  -cneg 11442   / cdiv 11868  cn 12209  2c2 12264  cz 12555  cuz 12819  +crp 12971  ...cfz 13481  Ccbc 14259  chash 14287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-oadd 8467  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-dju 9893  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-n0 12470  df-z 12556  df-uz 12820  df-rp 12972  df-fz 13482  df-seq 13964  df-fac 14231  df-bc 14260  df-hash 14288
This theorem is referenced by:  ballotth  33525
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