Step |
Hyp |
Ref |
Expression |
1 |
|
ballotth.m |
⊢ 𝑀 ∈ ℕ |
2 |
|
ballotth.n |
⊢ 𝑁 ∈ ℕ |
3 |
|
ballotth.o |
⊢ 𝑂 = { 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∣ ( ♯ ‘ 𝑐 ) = 𝑀 } |
4 |
|
ballotth.p |
⊢ 𝑃 = ( 𝑥 ∈ 𝒫 𝑂 ↦ ( ( ♯ ‘ 𝑥 ) / ( ♯ ‘ 𝑂 ) ) ) |
5 |
1 2 3
|
ballotlemoex |
⊢ 𝑂 ∈ V |
6 |
|
ssrab2 |
⊢ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ⊆ 𝑂 |
7 |
5 6
|
elpwi2 |
⊢ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ∈ 𝒫 𝑂 |
8 |
|
fveq2 |
⊢ ( 𝑥 = { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } → ( ♯ ‘ 𝑥 ) = ( ♯ ‘ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ) ) |
9 |
8
|
oveq1d |
⊢ ( 𝑥 = { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } → ( ( ♯ ‘ 𝑥 ) / ( ♯ ‘ 𝑂 ) ) = ( ( ♯ ‘ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ) / ( ♯ ‘ 𝑂 ) ) ) |
10 |
|
ovex |
⊢ ( ( ♯ ‘ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ) / ( ♯ ‘ 𝑂 ) ) ∈ V |
11 |
9 4 10
|
fvmpt |
⊢ ( { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ∈ 𝒫 𝑂 → ( 𝑃 ‘ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ) = ( ( ♯ ‘ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ) / ( ♯ ‘ 𝑂 ) ) ) |
12 |
7 11
|
ax-mp |
⊢ ( 𝑃 ‘ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ) = ( ( ♯ ‘ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ) / ( ♯ ‘ 𝑂 ) ) |
13 |
|
an32 |
⊢ ( ( ( 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ¬ 1 ∈ 𝑐 ) ∧ ( ♯ ‘ 𝑐 ) = 𝑀 ) ↔ ( ( 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ♯ ‘ 𝑐 ) = 𝑀 ) ∧ ¬ 1 ∈ 𝑐 ) ) |
14 |
|
2eluzge1 |
⊢ 2 ∈ ( ℤ≥ ‘ 1 ) |
15 |
|
fzss1 |
⊢ ( 2 ∈ ( ℤ≥ ‘ 1 ) → ( 2 ... ( 𝑀 + 𝑁 ) ) ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) ) |
16 |
14 15
|
ax-mp |
⊢ ( 2 ... ( 𝑀 + 𝑁 ) ) ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) |
17 |
16
|
sspwi |
⊢ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ⊆ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) |
18 |
17
|
sseli |
⊢ ( 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) → 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ) |
19 |
|
1lt2 |
⊢ 1 < 2 |
20 |
|
1re |
⊢ 1 ∈ ℝ |
21 |
|
2re |
⊢ 2 ∈ ℝ |
22 |
20 21
|
ltnlei |
⊢ ( 1 < 2 ↔ ¬ 2 ≤ 1 ) |
23 |
19 22
|
mpbi |
⊢ ¬ 2 ≤ 1 |
24 |
|
elfzle1 |
⊢ ( 1 ∈ ( 2 ... ( 𝑀 + 𝑁 ) ) → 2 ≤ 1 ) |
25 |
23 24
|
mto |
⊢ ¬ 1 ∈ ( 2 ... ( 𝑀 + 𝑁 ) ) |
26 |
|
elelpwi |
⊢ ( ( 1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ) → 1 ∈ ( 2 ... ( 𝑀 + 𝑁 ) ) ) |
27 |
25 26
|
mto |
⊢ ¬ ( 1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ) |
28 |
|
ancom |
⊢ ( ( 1 ∈ 𝑐 ∧ 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ) ↔ ( 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ∧ 1 ∈ 𝑐 ) ) |
29 |
27 28
|
mtbi |
⊢ ¬ ( 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ∧ 1 ∈ 𝑐 ) |
30 |
29
|
imnani |
⊢ ( 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) → ¬ 1 ∈ 𝑐 ) |
31 |
18 30
|
jca |
⊢ ( 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) → ( 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ¬ 1 ∈ 𝑐 ) ) |
32 |
|
ssin |
⊢ ( ( 𝑐 ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝑐 ⊆ { 𝑖 ∣ ¬ 𝑖 = 1 } ) ↔ 𝑐 ⊆ ( ( 1 ... ( 𝑀 + 𝑁 ) ) ∩ { 𝑖 ∣ ¬ 𝑖 = 1 } ) ) |
33 |
|
1le2 |
⊢ 1 ≤ 2 |
34 |
|
1p1e2 |
⊢ ( 1 + 1 ) = 2 |
35 |
|
nnge1 |
⊢ ( 𝑀 ∈ ℕ → 1 ≤ 𝑀 ) |
36 |
1 35
|
ax-mp |
⊢ 1 ≤ 𝑀 |
37 |
|
nnge1 |
⊢ ( 𝑁 ∈ ℕ → 1 ≤ 𝑁 ) |
38 |
2 37
|
ax-mp |
⊢ 1 ≤ 𝑁 |
39 |
1
|
nnrei |
⊢ 𝑀 ∈ ℝ |
40 |
2
|
nnrei |
⊢ 𝑁 ∈ ℝ |
41 |
20 20 39 40
|
le2addi |
⊢ ( ( 1 ≤ 𝑀 ∧ 1 ≤ 𝑁 ) → ( 1 + 1 ) ≤ ( 𝑀 + 𝑁 ) ) |
42 |
36 38 41
|
mp2an |
⊢ ( 1 + 1 ) ≤ ( 𝑀 + 𝑁 ) |
43 |
34 42
|
eqbrtrri |
⊢ 2 ≤ ( 𝑀 + 𝑁 ) |
44 |
39 40
|
readdcli |
⊢ ( 𝑀 + 𝑁 ) ∈ ℝ |
45 |
20 21 44
|
letri |
⊢ ( ( 1 ≤ 2 ∧ 2 ≤ ( 𝑀 + 𝑁 ) ) → 1 ≤ ( 𝑀 + 𝑁 ) ) |
46 |
33 43 45
|
mp2an |
⊢ 1 ≤ ( 𝑀 + 𝑁 ) |
47 |
|
1z |
⊢ 1 ∈ ℤ |
48 |
|
nnaddcl |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝑀 + 𝑁 ) ∈ ℕ ) |
49 |
1 2 48
|
mp2an |
⊢ ( 𝑀 + 𝑁 ) ∈ ℕ |
50 |
49
|
nnzi |
⊢ ( 𝑀 + 𝑁 ) ∈ ℤ |
51 |
|
eluz |
⊢ ( ( 1 ∈ ℤ ∧ ( 𝑀 + 𝑁 ) ∈ ℤ ) → ( ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 1 ) ↔ 1 ≤ ( 𝑀 + 𝑁 ) ) ) |
52 |
47 50 51
|
mp2an |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 1 ) ↔ 1 ≤ ( 𝑀 + 𝑁 ) ) |
53 |
46 52
|
mpbir |
⊢ ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 1 ) |
54 |
|
elfzp12 |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 1 ) → ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ↔ ( 𝑖 = 1 ∨ 𝑖 ∈ ( ( 1 + 1 ) ... ( 𝑀 + 𝑁 ) ) ) ) ) |
55 |
53 54
|
ax-mp |
⊢ ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ↔ ( 𝑖 = 1 ∨ 𝑖 ∈ ( ( 1 + 1 ) ... ( 𝑀 + 𝑁 ) ) ) ) |
56 |
55
|
biimpi |
⊢ ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) → ( 𝑖 = 1 ∨ 𝑖 ∈ ( ( 1 + 1 ) ... ( 𝑀 + 𝑁 ) ) ) ) |
57 |
56
|
orcanai |
⊢ ( ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ¬ 𝑖 = 1 ) → 𝑖 ∈ ( ( 1 + 1 ) ... ( 𝑀 + 𝑁 ) ) ) |
58 |
34
|
oveq1i |
⊢ ( ( 1 + 1 ) ... ( 𝑀 + 𝑁 ) ) = ( 2 ... ( 𝑀 + 𝑁 ) ) |
59 |
57 58
|
eleqtrdi |
⊢ ( ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ¬ 𝑖 = 1 ) → 𝑖 ∈ ( 2 ... ( 𝑀 + 𝑁 ) ) ) |
60 |
59
|
ss2abi |
⊢ { 𝑖 ∣ ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ¬ 𝑖 = 1 ) } ⊆ { 𝑖 ∣ 𝑖 ∈ ( 2 ... ( 𝑀 + 𝑁 ) ) } |
61 |
|
inab |
⊢ ( { 𝑖 ∣ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) } ∩ { 𝑖 ∣ ¬ 𝑖 = 1 } ) = { 𝑖 ∣ ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ¬ 𝑖 = 1 ) } |
62 |
|
abid2 |
⊢ { 𝑖 ∣ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) } = ( 1 ... ( 𝑀 + 𝑁 ) ) |
63 |
62
|
ineq1i |
⊢ ( { 𝑖 ∣ 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) } ∩ { 𝑖 ∣ ¬ 𝑖 = 1 } ) = ( ( 1 ... ( 𝑀 + 𝑁 ) ) ∩ { 𝑖 ∣ ¬ 𝑖 = 1 } ) |
64 |
61 63
|
eqtr3i |
⊢ { 𝑖 ∣ ( 𝑖 ∈ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ¬ 𝑖 = 1 ) } = ( ( 1 ... ( 𝑀 + 𝑁 ) ) ∩ { 𝑖 ∣ ¬ 𝑖 = 1 } ) |
65 |
|
abid2 |
⊢ { 𝑖 ∣ 𝑖 ∈ ( 2 ... ( 𝑀 + 𝑁 ) ) } = ( 2 ... ( 𝑀 + 𝑁 ) ) |
66 |
60 64 65
|
3sstr3i |
⊢ ( ( 1 ... ( 𝑀 + 𝑁 ) ) ∩ { 𝑖 ∣ ¬ 𝑖 = 1 } ) ⊆ ( 2 ... ( 𝑀 + 𝑁 ) ) |
67 |
|
sstr |
⊢ ( ( 𝑐 ⊆ ( ( 1 ... ( 𝑀 + 𝑁 ) ) ∩ { 𝑖 ∣ ¬ 𝑖 = 1 } ) ∧ ( ( 1 ... ( 𝑀 + 𝑁 ) ) ∩ { 𝑖 ∣ ¬ 𝑖 = 1 } ) ⊆ ( 2 ... ( 𝑀 + 𝑁 ) ) ) → 𝑐 ⊆ ( 2 ... ( 𝑀 + 𝑁 ) ) ) |
68 |
66 67
|
mpan2 |
⊢ ( 𝑐 ⊆ ( ( 1 ... ( 𝑀 + 𝑁 ) ) ∩ { 𝑖 ∣ ¬ 𝑖 = 1 } ) → 𝑐 ⊆ ( 2 ... ( 𝑀 + 𝑁 ) ) ) |
69 |
32 68
|
sylbi |
⊢ ( ( 𝑐 ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝑐 ⊆ { 𝑖 ∣ ¬ 𝑖 = 1 } ) → 𝑐 ⊆ ( 2 ... ( 𝑀 + 𝑁 ) ) ) |
70 |
|
velpw |
⊢ ( 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ↔ 𝑐 ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) ) |
71 |
|
ssab |
⊢ ( 𝑐 ⊆ { 𝑖 ∣ ¬ 𝑖 = 1 } ↔ ∀ 𝑖 ( 𝑖 ∈ 𝑐 → ¬ 𝑖 = 1 ) ) |
72 |
|
df-ex |
⊢ ( ∃ 𝑖 ( 𝑖 = 1 ∧ 𝑖 ∈ 𝑐 ) ↔ ¬ ∀ 𝑖 ¬ ( 𝑖 = 1 ∧ 𝑖 ∈ 𝑐 ) ) |
73 |
72
|
bicomi |
⊢ ( ¬ ∀ 𝑖 ¬ ( 𝑖 = 1 ∧ 𝑖 ∈ 𝑐 ) ↔ ∃ 𝑖 ( 𝑖 = 1 ∧ 𝑖 ∈ 𝑐 ) ) |
74 |
73
|
con1bii |
⊢ ( ¬ ∃ 𝑖 ( 𝑖 = 1 ∧ 𝑖 ∈ 𝑐 ) ↔ ∀ 𝑖 ¬ ( 𝑖 = 1 ∧ 𝑖 ∈ 𝑐 ) ) |
75 |
|
dfclel |
⊢ ( 1 ∈ 𝑐 ↔ ∃ 𝑖 ( 𝑖 = 1 ∧ 𝑖 ∈ 𝑐 ) ) |
76 |
75
|
notbii |
⊢ ( ¬ 1 ∈ 𝑐 ↔ ¬ ∃ 𝑖 ( 𝑖 = 1 ∧ 𝑖 ∈ 𝑐 ) ) |
77 |
|
imnang |
⊢ ( ∀ 𝑖 ( 𝑖 ∈ 𝑐 → ¬ 𝑖 = 1 ) ↔ ∀ 𝑖 ¬ ( 𝑖 ∈ 𝑐 ∧ 𝑖 = 1 ) ) |
78 |
|
ancom |
⊢ ( ( 𝑖 = 1 ∧ 𝑖 ∈ 𝑐 ) ↔ ( 𝑖 ∈ 𝑐 ∧ 𝑖 = 1 ) ) |
79 |
78
|
notbii |
⊢ ( ¬ ( 𝑖 = 1 ∧ 𝑖 ∈ 𝑐 ) ↔ ¬ ( 𝑖 ∈ 𝑐 ∧ 𝑖 = 1 ) ) |
80 |
79
|
albii |
⊢ ( ∀ 𝑖 ¬ ( 𝑖 = 1 ∧ 𝑖 ∈ 𝑐 ) ↔ ∀ 𝑖 ¬ ( 𝑖 ∈ 𝑐 ∧ 𝑖 = 1 ) ) |
81 |
77 80
|
bitr4i |
⊢ ( ∀ 𝑖 ( 𝑖 ∈ 𝑐 → ¬ 𝑖 = 1 ) ↔ ∀ 𝑖 ¬ ( 𝑖 = 1 ∧ 𝑖 ∈ 𝑐 ) ) |
82 |
74 76 81
|
3bitr4ri |
⊢ ( ∀ 𝑖 ( 𝑖 ∈ 𝑐 → ¬ 𝑖 = 1 ) ↔ ¬ 1 ∈ 𝑐 ) |
83 |
71 82
|
bitr2i |
⊢ ( ¬ 1 ∈ 𝑐 ↔ 𝑐 ⊆ { 𝑖 ∣ ¬ 𝑖 = 1 } ) |
84 |
70 83
|
anbi12i |
⊢ ( ( 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ¬ 1 ∈ 𝑐 ) ↔ ( 𝑐 ⊆ ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ 𝑐 ⊆ { 𝑖 ∣ ¬ 𝑖 = 1 } ) ) |
85 |
|
velpw |
⊢ ( 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ↔ 𝑐 ⊆ ( 2 ... ( 𝑀 + 𝑁 ) ) ) |
86 |
69 84 85
|
3imtr4i |
⊢ ( ( 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ¬ 1 ∈ 𝑐 ) → 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ) |
87 |
31 86
|
impbii |
⊢ ( 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ↔ ( 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ¬ 1 ∈ 𝑐 ) ) |
88 |
87
|
anbi1i |
⊢ ( ( 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ∧ ( ♯ ‘ 𝑐 ) = 𝑀 ) ↔ ( ( 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ¬ 1 ∈ 𝑐 ) ∧ ( ♯ ‘ 𝑐 ) = 𝑀 ) ) |
89 |
3
|
rabeq2i |
⊢ ( 𝑐 ∈ 𝑂 ↔ ( 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ♯ ‘ 𝑐 ) = 𝑀 ) ) |
90 |
89
|
anbi1i |
⊢ ( ( 𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐 ) ↔ ( ( 𝑐 ∈ 𝒫 ( 1 ... ( 𝑀 + 𝑁 ) ) ∧ ( ♯ ‘ 𝑐 ) = 𝑀 ) ∧ ¬ 1 ∈ 𝑐 ) ) |
91 |
13 88 90
|
3bitr4i |
⊢ ( ( 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ∧ ( ♯ ‘ 𝑐 ) = 𝑀 ) ↔ ( 𝑐 ∈ 𝑂 ∧ ¬ 1 ∈ 𝑐 ) ) |
92 |
91
|
rabbia2 |
⊢ { 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ∣ ( ♯ ‘ 𝑐 ) = 𝑀 } = { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } |
93 |
92
|
fveq2i |
⊢ ( ♯ ‘ { 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ∣ ( ♯ ‘ 𝑐 ) = 𝑀 } ) = ( ♯ ‘ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ) |
94 |
|
fzfi |
⊢ ( 2 ... ( 𝑀 + 𝑁 ) ) ∈ Fin |
95 |
1
|
nnzi |
⊢ 𝑀 ∈ ℤ |
96 |
|
hashbc |
⊢ ( ( ( 2 ... ( 𝑀 + 𝑁 ) ) ∈ Fin ∧ 𝑀 ∈ ℤ ) → ( ( ♯ ‘ ( 2 ... ( 𝑀 + 𝑁 ) ) ) C 𝑀 ) = ( ♯ ‘ { 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ∣ ( ♯ ‘ 𝑐 ) = 𝑀 } ) ) |
97 |
94 95 96
|
mp2an |
⊢ ( ( ♯ ‘ ( 2 ... ( 𝑀 + 𝑁 ) ) ) C 𝑀 ) = ( ♯ ‘ { 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ∣ ( ♯ ‘ 𝑐 ) = 𝑀 } ) |
98 |
|
2z |
⊢ 2 ∈ ℤ |
99 |
98
|
eluz1i |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 2 ) ↔ ( ( 𝑀 + 𝑁 ) ∈ ℤ ∧ 2 ≤ ( 𝑀 + 𝑁 ) ) ) |
100 |
50 43 99
|
mpbir2an |
⊢ ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 2 ) |
101 |
|
hashfz |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 2 ) → ( ♯ ‘ ( 2 ... ( 𝑀 + 𝑁 ) ) ) = ( ( ( 𝑀 + 𝑁 ) − 2 ) + 1 ) ) |
102 |
100 101
|
ax-mp |
⊢ ( ♯ ‘ ( 2 ... ( 𝑀 + 𝑁 ) ) ) = ( ( ( 𝑀 + 𝑁 ) − 2 ) + 1 ) |
103 |
1
|
nncni |
⊢ 𝑀 ∈ ℂ |
104 |
2
|
nncni |
⊢ 𝑁 ∈ ℂ |
105 |
103 104
|
addcli |
⊢ ( 𝑀 + 𝑁 ) ∈ ℂ |
106 |
|
2cn |
⊢ 2 ∈ ℂ |
107 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
108 |
|
subadd23 |
⊢ ( ( ( 𝑀 + 𝑁 ) ∈ ℂ ∧ 2 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( ( 𝑀 + 𝑁 ) − 2 ) + 1 ) = ( ( 𝑀 + 𝑁 ) + ( 1 − 2 ) ) ) |
109 |
105 106 107 108
|
mp3an |
⊢ ( ( ( 𝑀 + 𝑁 ) − 2 ) + 1 ) = ( ( 𝑀 + 𝑁 ) + ( 1 − 2 ) ) |
110 |
106 107
|
negsubdi2i |
⊢ - ( 2 − 1 ) = ( 1 − 2 ) |
111 |
|
2m1e1 |
⊢ ( 2 − 1 ) = 1 |
112 |
111
|
negeqi |
⊢ - ( 2 − 1 ) = - 1 |
113 |
110 112
|
eqtr3i |
⊢ ( 1 − 2 ) = - 1 |
114 |
113
|
oveq2i |
⊢ ( ( 𝑀 + 𝑁 ) + ( 1 − 2 ) ) = ( ( 𝑀 + 𝑁 ) + - 1 ) |
115 |
102 109 114
|
3eqtri |
⊢ ( ♯ ‘ ( 2 ... ( 𝑀 + 𝑁 ) ) ) = ( ( 𝑀 + 𝑁 ) + - 1 ) |
116 |
105 107
|
negsubi |
⊢ ( ( 𝑀 + 𝑁 ) + - 1 ) = ( ( 𝑀 + 𝑁 ) − 1 ) |
117 |
115 116
|
eqtri |
⊢ ( ♯ ‘ ( 2 ... ( 𝑀 + 𝑁 ) ) ) = ( ( 𝑀 + 𝑁 ) − 1 ) |
118 |
117
|
oveq1i |
⊢ ( ( ♯ ‘ ( 2 ... ( 𝑀 + 𝑁 ) ) ) C 𝑀 ) = ( ( ( 𝑀 + 𝑁 ) − 1 ) C 𝑀 ) |
119 |
97 118
|
eqtr3i |
⊢ ( ♯ ‘ { 𝑐 ∈ 𝒫 ( 2 ... ( 𝑀 + 𝑁 ) ) ∣ ( ♯ ‘ 𝑐 ) = 𝑀 } ) = ( ( ( 𝑀 + 𝑁 ) − 1 ) C 𝑀 ) |
120 |
93 119
|
eqtr3i |
⊢ ( ♯ ‘ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ) = ( ( ( 𝑀 + 𝑁 ) − 1 ) C 𝑀 ) |
121 |
1 2 3
|
ballotlem1 |
⊢ ( ♯ ‘ 𝑂 ) = ( ( 𝑀 + 𝑁 ) C 𝑀 ) |
122 |
120 121
|
oveq12i |
⊢ ( ( ♯ ‘ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ) / ( ♯ ‘ 𝑂 ) ) = ( ( ( ( 𝑀 + 𝑁 ) − 1 ) C 𝑀 ) / ( ( 𝑀 + 𝑁 ) C 𝑀 ) ) |
123 |
12 122
|
eqtri |
⊢ ( 𝑃 ‘ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ) = ( ( ( ( 𝑀 + 𝑁 ) − 1 ) C 𝑀 ) / ( ( 𝑀 + 𝑁 ) C 𝑀 ) ) |
124 |
|
0le1 |
⊢ 0 ≤ 1 |
125 |
|
0re |
⊢ 0 ∈ ℝ |
126 |
125 20 39
|
letri |
⊢ ( ( 0 ≤ 1 ∧ 1 ≤ 𝑀 ) → 0 ≤ 𝑀 ) |
127 |
124 36 126
|
mp2an |
⊢ 0 ≤ 𝑀 |
128 |
2
|
nngt0i |
⊢ 0 < 𝑁 |
129 |
40 128
|
elrpii |
⊢ 𝑁 ∈ ℝ+ |
130 |
|
ltaddrp |
⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ+ ) → 𝑀 < ( 𝑀 + 𝑁 ) ) |
131 |
39 129 130
|
mp2an |
⊢ 𝑀 < ( 𝑀 + 𝑁 ) |
132 |
|
0z |
⊢ 0 ∈ ℤ |
133 |
|
elfzm11 |
⊢ ( ( 0 ∈ ℤ ∧ ( 𝑀 + 𝑁 ) ∈ ℤ ) → ( 𝑀 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 1 ) ) ↔ ( 𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < ( 𝑀 + 𝑁 ) ) ) ) |
134 |
132 50 133
|
mp2an |
⊢ ( 𝑀 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 1 ) ) ↔ ( 𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ∧ 𝑀 < ( 𝑀 + 𝑁 ) ) ) |
135 |
95 127 131 134
|
mpbir3an |
⊢ 𝑀 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 1 ) ) |
136 |
|
bcm1n |
⊢ ( ( 𝑀 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 1 ) ) ∧ ( 𝑀 + 𝑁 ) ∈ ℕ ) → ( ( ( ( 𝑀 + 𝑁 ) − 1 ) C 𝑀 ) / ( ( 𝑀 + 𝑁 ) C 𝑀 ) ) = ( ( ( 𝑀 + 𝑁 ) − 𝑀 ) / ( 𝑀 + 𝑁 ) ) ) |
137 |
135 49 136
|
mp2an |
⊢ ( ( ( ( 𝑀 + 𝑁 ) − 1 ) C 𝑀 ) / ( ( 𝑀 + 𝑁 ) C 𝑀 ) ) = ( ( ( 𝑀 + 𝑁 ) − 𝑀 ) / ( 𝑀 + 𝑁 ) ) |
138 |
|
pncan2 |
⊢ ( ( 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ ) → ( ( 𝑀 + 𝑁 ) − 𝑀 ) = 𝑁 ) |
139 |
103 104 138
|
mp2an |
⊢ ( ( 𝑀 + 𝑁 ) − 𝑀 ) = 𝑁 |
140 |
139
|
oveq1i |
⊢ ( ( ( 𝑀 + 𝑁 ) − 𝑀 ) / ( 𝑀 + 𝑁 ) ) = ( 𝑁 / ( 𝑀 + 𝑁 ) ) |
141 |
123 137 140
|
3eqtri |
⊢ ( 𝑃 ‘ { 𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐 } ) = ( 𝑁 / ( 𝑀 + 𝑁 ) ) |