Step |
Hyp |
Ref |
Expression |
1 |
|
metakunt24.1 |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
2 |
|
metakunt24.2 |
⊢ ( 𝜑 → 𝐼 ∈ ℕ ) |
3 |
|
metakunt24.3 |
⊢ ( 𝜑 → 𝐼 ≤ 𝑀 ) |
4 |
|
indir |
⊢ ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∩ { 𝑀 } ) = ( ( ( 1 ... ( 𝐼 − 1 ) ) ∩ { 𝑀 } ) ∪ ( ( 𝐼 ... ( 𝑀 − 1 ) ) ∩ { 𝑀 } ) ) |
5 |
4
|
a1i |
⊢ ( 𝜑 → ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∩ { 𝑀 } ) = ( ( ( 1 ... ( 𝐼 − 1 ) ) ∩ { 𝑀 } ) ∪ ( ( 𝐼 ... ( 𝑀 − 1 ) ) ∩ { 𝑀 } ) ) ) |
6 |
1 2 3
|
metakunt18 |
⊢ ( 𝜑 → ( ( ( ( 1 ... ( 𝐼 − 1 ) ) ∩ ( 𝐼 ... ( 𝑀 − 1 ) ) ) = ∅ ∧ ( ( 1 ... ( 𝐼 − 1 ) ) ∩ { 𝑀 } ) = ∅ ∧ ( ( 𝐼 ... ( 𝑀 − 1 ) ) ∩ { 𝑀 } ) = ∅ ) ∧ ( ( ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ∩ ( 1 ... ( 𝑀 − 𝐼 ) ) ) = ∅ ∧ ( ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ∩ { 𝑀 } ) = ∅ ∧ ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∩ { 𝑀 } ) = ∅ ) ) ) |
7 |
6
|
simpld |
⊢ ( 𝜑 → ( ( ( 1 ... ( 𝐼 − 1 ) ) ∩ ( 𝐼 ... ( 𝑀 − 1 ) ) ) = ∅ ∧ ( ( 1 ... ( 𝐼 − 1 ) ) ∩ { 𝑀 } ) = ∅ ∧ ( ( 𝐼 ... ( 𝑀 − 1 ) ) ∩ { 𝑀 } ) = ∅ ) ) |
8 |
7
|
simp2d |
⊢ ( 𝜑 → ( ( 1 ... ( 𝐼 − 1 ) ) ∩ { 𝑀 } ) = ∅ ) |
9 |
7
|
simp3d |
⊢ ( 𝜑 → ( ( 𝐼 ... ( 𝑀 − 1 ) ) ∩ { 𝑀 } ) = ∅ ) |
10 |
8 9
|
uneq12d |
⊢ ( 𝜑 → ( ( ( 1 ... ( 𝐼 − 1 ) ) ∩ { 𝑀 } ) ∪ ( ( 𝐼 ... ( 𝑀 − 1 ) ) ∩ { 𝑀 } ) ) = ( ∅ ∪ ∅ ) ) |
11 |
|
unidm |
⊢ ( ∅ ∪ ∅ ) = ∅ |
12 |
11
|
a1i |
⊢ ( 𝜑 → ( ∅ ∪ ∅ ) = ∅ ) |
13 |
10 12
|
eqtrd |
⊢ ( 𝜑 → ( ( ( 1 ... ( 𝐼 − 1 ) ) ∩ { 𝑀 } ) ∪ ( ( 𝐼 ... ( 𝑀 − 1 ) ) ∩ { 𝑀 } ) ) = ∅ ) |
14 |
5 13
|
eqtrd |
⊢ ( 𝜑 → ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∩ { 𝑀 } ) = ∅ ) |
15 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
16 |
1
|
nnzd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
17 |
1
|
nnge1d |
⊢ ( 𝜑 → 1 ≤ 𝑀 ) |
18 |
1
|
nnred |
⊢ ( 𝜑 → 𝑀 ∈ ℝ ) |
19 |
18
|
leidd |
⊢ ( 𝜑 → 𝑀 ≤ 𝑀 ) |
20 |
15 16 16 17 19
|
elfzd |
⊢ ( 𝜑 → 𝑀 ∈ ( 1 ... 𝑀 ) ) |
21 |
20
|
fzsplitnd |
⊢ ( 𝜑 → ( 1 ... 𝑀 ) = ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ( 𝑀 ... 𝑀 ) ) ) |
22 |
|
oveq1 |
⊢ ( 𝐼 = 𝑀 → ( 𝐼 − 1 ) = ( 𝑀 − 1 ) ) |
23 |
22
|
oveq2d |
⊢ ( 𝐼 = 𝑀 → ( 1 ... ( 𝐼 − 1 ) ) = ( 1 ... ( 𝑀 − 1 ) ) ) |
24 |
|
oveq1 |
⊢ ( 𝐼 = 𝑀 → ( 𝐼 ... ( 𝑀 − 1 ) ) = ( 𝑀 ... ( 𝑀 − 1 ) ) ) |
25 |
23 24
|
uneq12d |
⊢ ( 𝐼 = 𝑀 → ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) = ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ( 𝑀 ... ( 𝑀 − 1 ) ) ) ) |
26 |
25
|
uneq1d |
⊢ ( 𝐼 = 𝑀 → ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) = ( ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ( 𝑀 ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) ) |
27 |
26
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) = ( ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ( 𝑀 ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) ) |
28 |
18
|
ltm1d |
⊢ ( 𝜑 → ( 𝑀 − 1 ) < 𝑀 ) |
29 |
16 15
|
zsubcld |
⊢ ( 𝜑 → ( 𝑀 − 1 ) ∈ ℤ ) |
30 |
|
fzn |
⊢ ( ( 𝑀 ∈ ℤ ∧ ( 𝑀 − 1 ) ∈ ℤ ) → ( ( 𝑀 − 1 ) < 𝑀 ↔ ( 𝑀 ... ( 𝑀 − 1 ) ) = ∅ ) ) |
31 |
16 29 30
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑀 − 1 ) < 𝑀 ↔ ( 𝑀 ... ( 𝑀 − 1 ) ) = ∅ ) ) |
32 |
28 31
|
mpbid |
⊢ ( 𝜑 → ( 𝑀 ... ( 𝑀 − 1 ) ) = ∅ ) |
33 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( 𝑀 ... ( 𝑀 − 1 ) ) = ∅ ) |
34 |
33
|
uneq2d |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ( 𝑀 ... ( 𝑀 − 1 ) ) ) = ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ∅ ) ) |
35 |
|
un0 |
⊢ ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ∅ ) = ( 1 ... ( 𝑀 − 1 ) ) |
36 |
35
|
a1i |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ∅ ) = ( 1 ... ( 𝑀 − 1 ) ) ) |
37 |
34 36
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ( 𝑀 ... ( 𝑀 − 1 ) ) ) = ( 1 ... ( 𝑀 − 1 ) ) ) |
38 |
37
|
uneq1d |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ( 𝑀 ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) = ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ( 𝑀 ... 𝑀 ) ) ) |
39 |
27 38
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) = ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ( 𝑀 ... 𝑀 ) ) ) |
40 |
39
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ( 𝑀 ... 𝑀 ) ) = ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) ) |
41 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → 1 ∈ ℤ ) |
42 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → 𝑀 ∈ ℤ ) |
43 |
42 41
|
zsubcld |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → ( 𝑀 − 1 ) ∈ ℤ ) |
44 |
2
|
nnzd |
⊢ ( 𝜑 → 𝐼 ∈ ℤ ) |
45 |
44
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → 𝐼 ∈ ℤ ) |
46 |
2
|
nnge1d |
⊢ ( 𝜑 → 1 ≤ 𝐼 ) |
47 |
46
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → 1 ≤ 𝐼 ) |
48 |
|
eqid |
⊢ 𝑀 = 𝑀 |
49 |
|
eqeq1 |
⊢ ( 𝑀 = 𝐼 → ( 𝑀 = 𝑀 ↔ 𝐼 = 𝑀 ) ) |
50 |
48 49
|
mpbii |
⊢ ( 𝑀 = 𝐼 → 𝐼 = 𝑀 ) |
51 |
50
|
necon3bi |
⊢ ( ¬ 𝐼 = 𝑀 → 𝑀 ≠ 𝐼 ) |
52 |
51
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → 𝑀 ≠ 𝐼 ) |
53 |
2
|
nnred |
⊢ ( 𝜑 → 𝐼 ∈ ℝ ) |
54 |
53 18 3
|
leltned |
⊢ ( 𝜑 → ( 𝐼 < 𝑀 ↔ 𝑀 ≠ 𝐼 ) ) |
55 |
54
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → ( 𝐼 < 𝑀 ↔ 𝑀 ≠ 𝐼 ) ) |
56 |
52 55
|
mpbird |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → 𝐼 < 𝑀 ) |
57 |
|
zltlem1 |
⊢ ( ( 𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( 𝐼 < 𝑀 ↔ 𝐼 ≤ ( 𝑀 − 1 ) ) ) |
58 |
44 16 57
|
syl2anc |
⊢ ( 𝜑 → ( 𝐼 < 𝑀 ↔ 𝐼 ≤ ( 𝑀 − 1 ) ) ) |
59 |
58
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → ( 𝐼 < 𝑀 ↔ 𝐼 ≤ ( 𝑀 − 1 ) ) ) |
60 |
56 59
|
mpbid |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → 𝐼 ≤ ( 𝑀 − 1 ) ) |
61 |
41 43 45 47 60
|
fzsplitnr |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → ( 1 ... ( 𝑀 − 1 ) ) = ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ) |
62 |
61
|
uneq1d |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ( 𝑀 ... 𝑀 ) ) = ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) ) |
63 |
40 62
|
pm2.61dan |
⊢ ( 𝜑 → ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ( 𝑀 ... 𝑀 ) ) = ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) ) |
64 |
|
fzsn |
⊢ ( 𝑀 ∈ ℤ → ( 𝑀 ... 𝑀 ) = { 𝑀 } ) |
65 |
16 64
|
syl |
⊢ ( 𝜑 → ( 𝑀 ... 𝑀 ) = { 𝑀 } ) |
66 |
65
|
uneq2d |
⊢ ( 𝜑 → ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) = ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∪ { 𝑀 } ) ) |
67 |
21 63 66
|
3eqtrd |
⊢ ( 𝜑 → ( 1 ... 𝑀 ) = ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∪ { 𝑀 } ) ) |
68 |
|
uncom |
⊢ ( ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ∪ ( 1 ... ( 𝑀 − 𝐼 ) ) ) = ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) |
69 |
68
|
a1i |
⊢ ( 𝜑 → ( ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ∪ ( 1 ... ( 𝑀 − 𝐼 ) ) ) = ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ) |
70 |
69
|
uneq1d |
⊢ ( 𝜑 → ( ( ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ∪ ( 1 ... ( 𝑀 − 𝐼 ) ) ) ∪ { 𝑀 } ) = ( ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ∪ { 𝑀 } ) ) |
71 |
65
|
uneq2d |
⊢ ( 𝜑 → ( ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) = ( ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ∪ { 𝑀 } ) ) |
72 |
71
|
eqcomd |
⊢ ( 𝜑 → ( ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ∪ { 𝑀 } ) = ( ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) ) |
73 |
|
fz10 |
⊢ ( 1 ... 0 ) = ∅ |
74 |
73
|
uneq1i |
⊢ ( ( 1 ... 0 ) ∪ ( 1 ... ( 𝑀 − 1 ) ) ) = ( ∅ ∪ ( 1 ... ( 𝑀 − 1 ) ) ) |
75 |
74
|
a1i |
⊢ ( 𝜑 → ( ( 1 ... 0 ) ∪ ( 1 ... ( 𝑀 − 1 ) ) ) = ( ∅ ∪ ( 1 ... ( 𝑀 − 1 ) ) ) ) |
76 |
75
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( 1 ... 0 ) ∪ ( 1 ... ( 𝑀 − 1 ) ) ) = ( ∅ ∪ ( 1 ... ( 𝑀 − 1 ) ) ) ) |
77 |
|
uncom |
⊢ ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ∅ ) = ( ∅ ∪ ( 1 ... ( 𝑀 − 1 ) ) ) |
78 |
77
|
eqeq1i |
⊢ ( ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ∅ ) = ( 1 ... ( 𝑀 − 1 ) ) ↔ ( ∅ ∪ ( 1 ... ( 𝑀 − 1 ) ) ) = ( 1 ... ( 𝑀 − 1 ) ) ) |
79 |
78
|
imbi2i |
⊢ ( ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ∅ ) = ( 1 ... ( 𝑀 − 1 ) ) ) ↔ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ∅ ∪ ( 1 ... ( 𝑀 − 1 ) ) ) = ( 1 ... ( 𝑀 − 1 ) ) ) ) |
80 |
36 79
|
mpbi |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ∅ ∪ ( 1 ... ( 𝑀 − 1 ) ) ) = ( 1 ... ( 𝑀 − 1 ) ) ) |
81 |
76 80
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( 1 ... 0 ) ∪ ( 1 ... ( 𝑀 − 1 ) ) ) = ( 1 ... ( 𝑀 − 1 ) ) ) |
82 |
81
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( 1 ... ( 𝑀 − 1 ) ) = ( ( 1 ... 0 ) ∪ ( 1 ... ( 𝑀 − 1 ) ) ) ) |
83 |
|
oveq2 |
⊢ ( 𝐼 = 𝑀 → ( 𝑀 − 𝐼 ) = ( 𝑀 − 𝑀 ) ) |
84 |
83
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( 𝑀 − 𝐼 ) = ( 𝑀 − 𝑀 ) ) |
85 |
18
|
recnd |
⊢ ( 𝜑 → 𝑀 ∈ ℂ ) |
86 |
85
|
subidd |
⊢ ( 𝜑 → ( 𝑀 − 𝑀 ) = 0 ) |
87 |
86
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( 𝑀 − 𝑀 ) = 0 ) |
88 |
84 87
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( 𝑀 − 𝐼 ) = 0 ) |
89 |
88
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( 1 ... ( 𝑀 − 𝐼 ) ) = ( 1 ... 0 ) ) |
90 |
83
|
oveq1d |
⊢ ( 𝐼 = 𝑀 → ( ( 𝑀 − 𝐼 ) + 1 ) = ( ( 𝑀 − 𝑀 ) + 1 ) ) |
91 |
90
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( 𝑀 − 𝐼 ) + 1 ) = ( ( 𝑀 − 𝑀 ) + 1 ) ) |
92 |
87
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( 𝑀 − 𝑀 ) + 1 ) = ( 0 + 1 ) ) |
93 |
91 92
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( 𝑀 − 𝐼 ) + 1 ) = ( 0 + 1 ) ) |
94 |
|
1e0p1 |
⊢ 1 = ( 0 + 1 ) |
95 |
93 94
|
eqtr4di |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( 𝑀 − 𝐼 ) + 1 ) = 1 ) |
96 |
95
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) = ( 1 ... ( 𝑀 − 1 ) ) ) |
97 |
89 96
|
uneq12d |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) = ( ( 1 ... 0 ) ∪ ( 1 ... ( 𝑀 − 1 ) ) ) ) |
98 |
97
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( ( 1 ... 0 ) ∪ ( 1 ... ( 𝑀 − 1 ) ) ) = ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ) |
99 |
82 98
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐼 = 𝑀 ) → ( 1 ... ( 𝑀 − 1 ) ) = ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ) |
100 |
42 45
|
zsubcld |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → ( 𝑀 − 𝐼 ) ∈ ℤ ) |
101 |
53
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → 𝐼 ∈ ℝ ) |
102 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → 𝑀 ∈ ℝ ) |
103 |
|
1red |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → 1 ∈ ℝ ) |
104 |
101 102 103 60
|
lesubd |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → 1 ≤ ( 𝑀 − 𝐼 ) ) |
105 |
103 101 102 47
|
lesub2dd |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → ( 𝑀 − 𝐼 ) ≤ ( 𝑀 − 1 ) ) |
106 |
41 43 100 104 105
|
elfzd |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → ( 𝑀 − 𝐼 ) ∈ ( 1 ... ( 𝑀 − 1 ) ) ) |
107 |
|
fzsplit |
⊢ ( ( 𝑀 − 𝐼 ) ∈ ( 1 ... ( 𝑀 − 1 ) ) → ( 1 ... ( 𝑀 − 1 ) ) = ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ) |
108 |
106 107
|
syl |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝑀 ) → ( 1 ... ( 𝑀 − 1 ) ) = ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ) |
109 |
99 108
|
pm2.61dan |
⊢ ( 𝜑 → ( 1 ... ( 𝑀 − 1 ) ) = ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ) |
110 |
109
|
uneq1d |
⊢ ( 𝜑 → ( ( 1 ... ( 𝑀 − 1 ) ) ∪ ( 𝑀 ... 𝑀 ) ) = ( ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) ) |
111 |
21 110
|
eqtrd |
⊢ ( 𝜑 → ( 1 ... 𝑀 ) = ( ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) ) |
112 |
111
|
eqcomd |
⊢ ( 𝜑 → ( ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ∪ ( 𝑀 ... 𝑀 ) ) = ( 1 ... 𝑀 ) ) |
113 |
72 112
|
eqtrd |
⊢ ( 𝜑 → ( ( ( 1 ... ( 𝑀 − 𝐼 ) ) ∪ ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ) ∪ { 𝑀 } ) = ( 1 ... 𝑀 ) ) |
114 |
70 113
|
eqtrd |
⊢ ( 𝜑 → ( ( ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ∪ ( 1 ... ( 𝑀 − 𝐼 ) ) ) ∪ { 𝑀 } ) = ( 1 ... 𝑀 ) ) |
115 |
114
|
eqcomd |
⊢ ( 𝜑 → ( 1 ... 𝑀 ) = ( ( ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ∪ ( 1 ... ( 𝑀 − 𝐼 ) ) ) ∪ { 𝑀 } ) ) |
116 |
14 67 115
|
3jca |
⊢ ( 𝜑 → ( ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∩ { 𝑀 } ) = ∅ ∧ ( 1 ... 𝑀 ) = ( ( ( 1 ... ( 𝐼 − 1 ) ) ∪ ( 𝐼 ... ( 𝑀 − 1 ) ) ) ∪ { 𝑀 } ) ∧ ( 1 ... 𝑀 ) = ( ( ( ( ( 𝑀 − 𝐼 ) + 1 ) ... ( 𝑀 − 1 ) ) ∪ ( 1 ... ( 𝑀 − 𝐼 ) ) ) ∪ { 𝑀 } ) ) ) |