Step |
Hyp |
Ref |
Expression |
1 |
|
poimir.0 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
2 |
|
poimirlem23.1 |
⊢ ( 𝜑 → 𝑇 : ( 1 ... 𝑁 ) ⟶ ( 0 ..^ 𝐾 ) ) |
3 |
|
poimirlem23.2 |
⊢ ( 𝜑 → 𝑈 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) |
4 |
|
poimirlem23.3 |
⊢ ( 𝜑 → 𝑉 ∈ ( 0 ... 𝑁 ) ) |
5 |
|
ovex |
⊢ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∈ V |
6 |
5
|
csbex |
⊢ ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∈ V |
7 |
6
|
rgenw |
⊢ ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∈ V |
8 |
|
eqid |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
9 |
|
fveq1 |
⊢ ( 𝑝 = ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) → ( 𝑝 ‘ 𝑁 ) = ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) ) |
10 |
9
|
neeq1d |
⊢ ( 𝑝 = ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) → ( ( 𝑝 ‘ 𝑁 ) ≠ 0 ↔ ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) ≠ 0 ) ) |
11 |
|
df-ne |
⊢ ( ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) ≠ 0 ↔ ¬ ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) |
12 |
10 11
|
bitrdi |
⊢ ( 𝑝 = ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) → ( ( 𝑝 ‘ 𝑁 ) ≠ 0 ↔ ¬ ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) ) |
13 |
8 12
|
rexrnmptw |
⊢ ( ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∈ V → ( ∃ 𝑝 ∈ ran ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ( 𝑝 ‘ 𝑁 ) ≠ 0 ↔ ∃ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ¬ ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) ) |
14 |
7 13
|
ax-mp |
⊢ ( ∃ 𝑝 ∈ ran ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ( 𝑝 ‘ 𝑁 ) ≠ 0 ↔ ∃ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ¬ ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) |
15 |
|
rexnal |
⊢ ( ∃ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ¬ ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ¬ ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) |
16 |
14 15
|
bitri |
⊢ ( ∃ 𝑝 ∈ ran ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ( 𝑝 ‘ 𝑁 ) ≠ 0 ↔ ¬ ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) |
17 |
1
|
nnzd |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
18 |
|
elfzelz |
⊢ ( 𝑉 ∈ ( 0 ... 𝑁 ) → 𝑉 ∈ ℤ ) |
19 |
4 18
|
syl |
⊢ ( 𝜑 → 𝑉 ∈ ℤ ) |
20 |
|
zlem1lt |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑉 ∈ ℤ ) → ( 𝑁 ≤ 𝑉 ↔ ( 𝑁 − 1 ) < 𝑉 ) ) |
21 |
17 19 20
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ≤ 𝑉 ↔ ( 𝑁 − 1 ) < 𝑉 ) ) |
22 |
|
elfzle2 |
⊢ ( 𝑉 ∈ ( 0 ... 𝑁 ) → 𝑉 ≤ 𝑁 ) |
23 |
4 22
|
syl |
⊢ ( 𝜑 → 𝑉 ≤ 𝑁 ) |
24 |
19
|
zred |
⊢ ( 𝜑 → 𝑉 ∈ ℝ ) |
25 |
1
|
nnred |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
26 |
24 25
|
letri3d |
⊢ ( 𝜑 → ( 𝑉 = 𝑁 ↔ ( 𝑉 ≤ 𝑁 ∧ 𝑁 ≤ 𝑉 ) ) ) |
27 |
26
|
biimprd |
⊢ ( 𝜑 → ( ( 𝑉 ≤ 𝑁 ∧ 𝑁 ≤ 𝑉 ) → 𝑉 = 𝑁 ) ) |
28 |
23 27
|
mpand |
⊢ ( 𝜑 → ( 𝑁 ≤ 𝑉 → 𝑉 = 𝑁 ) ) |
29 |
21 28
|
sylbird |
⊢ ( 𝜑 → ( ( 𝑁 − 1 ) < 𝑉 → 𝑉 = 𝑁 ) ) |
30 |
29
|
necon3ad |
⊢ ( 𝜑 → ( 𝑉 ≠ 𝑁 → ¬ ( 𝑁 − 1 ) < 𝑉 ) ) |
31 |
|
nnm1nn0 |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 − 1 ) ∈ ℕ0 ) |
32 |
1 31
|
syl |
⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ℕ0 ) |
33 |
|
nn0fz0 |
⊢ ( ( 𝑁 − 1 ) ∈ ℕ0 ↔ ( 𝑁 − 1 ) ∈ ( 0 ... ( 𝑁 − 1 ) ) ) |
34 |
32 33
|
sylib |
⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ( 0 ... ( 𝑁 − 1 ) ) ) |
35 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑁 − 1 ) < 𝑉 ) → ( 𝑁 − 1 ) ∈ ( 0 ... ( 𝑁 − 1 ) ) ) |
36 |
|
iffalse |
⊢ ( ¬ ( 𝑁 − 1 ) < 𝑉 → if ( ( 𝑁 − 1 ) < 𝑉 , ( 𝑁 − 1 ) , ( ( 𝑁 − 1 ) + 1 ) ) = ( ( 𝑁 − 1 ) + 1 ) ) |
37 |
1
|
nncnd |
⊢ ( 𝜑 → 𝑁 ∈ ℂ ) |
38 |
|
npcan1 |
⊢ ( 𝑁 ∈ ℂ → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 ) |
39 |
37 38
|
syl |
⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 ) |
40 |
36 39
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑁 − 1 ) < 𝑉 ) → if ( ( 𝑁 − 1 ) < 𝑉 , ( 𝑁 − 1 ) , ( ( 𝑁 − 1 ) + 1 ) ) = 𝑁 ) |
41 |
40
|
csbeq1d |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑁 − 1 ) < 𝑉 ) → ⦋ if ( ( 𝑁 − 1 ) < 𝑉 , ( 𝑁 − 1 ) , ( ( 𝑁 − 1 ) + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ 𝑁 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
42 |
|
oveq2 |
⊢ ( 𝑗 = 𝑁 → ( 1 ... 𝑗 ) = ( 1 ... 𝑁 ) ) |
43 |
42
|
imaeq2d |
⊢ ( 𝑗 = 𝑁 → ( 𝑈 “ ( 1 ... 𝑗 ) ) = ( 𝑈 “ ( 1 ... 𝑁 ) ) ) |
44 |
43
|
xpeq1d |
⊢ ( 𝑗 = 𝑁 → ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( 𝑈 “ ( 1 ... 𝑁 ) ) × { 1 } ) ) |
45 |
|
oveq1 |
⊢ ( 𝑗 = 𝑁 → ( 𝑗 + 1 ) = ( 𝑁 + 1 ) ) |
46 |
45
|
oveq1d |
⊢ ( 𝑗 = 𝑁 → ( ( 𝑗 + 1 ) ... 𝑁 ) = ( ( 𝑁 + 1 ) ... 𝑁 ) ) |
47 |
46
|
imaeq2d |
⊢ ( 𝑗 = 𝑁 → ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( 𝑈 “ ( ( 𝑁 + 1 ) ... 𝑁 ) ) ) |
48 |
47
|
xpeq1d |
⊢ ( 𝑗 = 𝑁 → ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( 𝑈 “ ( ( 𝑁 + 1 ) ... 𝑁 ) ) × { 0 } ) ) |
49 |
44 48
|
uneq12d |
⊢ ( 𝑗 = 𝑁 → ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( 𝑈 “ ( 1 ... 𝑁 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑁 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) |
50 |
|
f1ofo |
⊢ ( 𝑈 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → 𝑈 : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) ) |
51 |
|
foima |
⊢ ( 𝑈 : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) → ( 𝑈 “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) ) |
52 |
3 50 51
|
3syl |
⊢ ( 𝜑 → ( 𝑈 “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) ) |
53 |
52
|
xpeq1d |
⊢ ( 𝜑 → ( ( 𝑈 “ ( 1 ... 𝑁 ) ) × { 1 } ) = ( ( 1 ... 𝑁 ) × { 1 } ) ) |
54 |
25
|
ltp1d |
⊢ ( 𝜑 → 𝑁 < ( 𝑁 + 1 ) ) |
55 |
17
|
peano2zd |
⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ℤ ) |
56 |
|
fzn |
⊢ ( ( ( 𝑁 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 < ( 𝑁 + 1 ) ↔ ( ( 𝑁 + 1 ) ... 𝑁 ) = ∅ ) ) |
57 |
55 17 56
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 < ( 𝑁 + 1 ) ↔ ( ( 𝑁 + 1 ) ... 𝑁 ) = ∅ ) ) |
58 |
54 57
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑁 + 1 ) ... 𝑁 ) = ∅ ) |
59 |
58
|
imaeq2d |
⊢ ( 𝜑 → ( 𝑈 “ ( ( 𝑁 + 1 ) ... 𝑁 ) ) = ( 𝑈 “ ∅ ) ) |
60 |
59
|
xpeq1d |
⊢ ( 𝜑 → ( ( 𝑈 “ ( ( 𝑁 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( 𝑈 “ ∅ ) × { 0 } ) ) |
61 |
|
ima0 |
⊢ ( 𝑈 “ ∅ ) = ∅ |
62 |
61
|
xpeq1i |
⊢ ( ( 𝑈 “ ∅ ) × { 0 } ) = ( ∅ × { 0 } ) |
63 |
|
0xp |
⊢ ( ∅ × { 0 } ) = ∅ |
64 |
62 63
|
eqtri |
⊢ ( ( 𝑈 “ ∅ ) × { 0 } ) = ∅ |
65 |
60 64
|
eqtrdi |
⊢ ( 𝜑 → ( ( 𝑈 “ ( ( 𝑁 + 1 ) ... 𝑁 ) ) × { 0 } ) = ∅ ) |
66 |
53 65
|
uneq12d |
⊢ ( 𝜑 → ( ( ( 𝑈 “ ( 1 ... 𝑁 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑁 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( 1 ... 𝑁 ) × { 1 } ) ∪ ∅ ) ) |
67 |
|
un0 |
⊢ ( ( ( 1 ... 𝑁 ) × { 1 } ) ∪ ∅ ) = ( ( 1 ... 𝑁 ) × { 1 } ) |
68 |
66 67
|
eqtrdi |
⊢ ( 𝜑 → ( ( ( 𝑈 “ ( 1 ... 𝑁 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑁 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( 1 ... 𝑁 ) × { 1 } ) ) |
69 |
49 68
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ 𝑗 = 𝑁 ) → ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( 1 ... 𝑁 ) × { 1 } ) ) |
70 |
69
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 = 𝑁 ) → ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( 𝑇 ∘f + ( ( 1 ... 𝑁 ) × { 1 } ) ) ) |
71 |
1 70
|
csbied |
⊢ ( 𝜑 → ⦋ 𝑁 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( 𝑇 ∘f + ( ( 1 ... 𝑁 ) × { 1 } ) ) ) |
72 |
71
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑁 − 1 ) < 𝑉 ) → ⦋ 𝑁 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( 𝑇 ∘f + ( ( 1 ... 𝑁 ) × { 1 } ) ) ) |
73 |
41 72
|
eqtrd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑁 − 1 ) < 𝑉 ) → ⦋ if ( ( 𝑁 − 1 ) < 𝑉 , ( 𝑁 − 1 ) , ( ( 𝑁 − 1 ) + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( 𝑇 ∘f + ( ( 1 ... 𝑁 ) × { 1 } ) ) ) |
74 |
73
|
fveq1d |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑁 − 1 ) < 𝑉 ) → ( ⦋ if ( ( 𝑁 − 1 ) < 𝑉 , ( 𝑁 − 1 ) , ( ( 𝑁 − 1 ) + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = ( ( 𝑇 ∘f + ( ( 1 ... 𝑁 ) × { 1 } ) ) ‘ 𝑁 ) ) |
75 |
|
elfzonn0 |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝐾 ) → 𝑗 ∈ ℕ0 ) |
76 |
|
nn0p1nn |
⊢ ( 𝑗 ∈ ℕ0 → ( 𝑗 + 1 ) ∈ ℕ ) |
77 |
75 76
|
syl |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝐾 ) → ( 𝑗 + 1 ) ∈ ℕ ) |
78 |
|
elsni |
⊢ ( 𝑦 ∈ { 1 } → 𝑦 = 1 ) |
79 |
78
|
oveq2d |
⊢ ( 𝑦 ∈ { 1 } → ( 𝑗 + 𝑦 ) = ( 𝑗 + 1 ) ) |
80 |
79
|
eleq1d |
⊢ ( 𝑦 ∈ { 1 } → ( ( 𝑗 + 𝑦 ) ∈ ℕ ↔ ( 𝑗 + 1 ) ∈ ℕ ) ) |
81 |
77 80
|
syl5ibrcom |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝐾 ) → ( 𝑦 ∈ { 1 } → ( 𝑗 + 𝑦 ) ∈ ℕ ) ) |
82 |
81
|
imp |
⊢ ( ( 𝑗 ∈ ( 0 ..^ 𝐾 ) ∧ 𝑦 ∈ { 1 } ) → ( 𝑗 + 𝑦 ) ∈ ℕ ) |
83 |
82
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ( 0 ..^ 𝐾 ) ∧ 𝑦 ∈ { 1 } ) ) → ( 𝑗 + 𝑦 ) ∈ ℕ ) |
84 |
|
1ex |
⊢ 1 ∈ V |
85 |
84
|
fconst |
⊢ ( ( 1 ... 𝑁 ) × { 1 } ) : ( 1 ... 𝑁 ) ⟶ { 1 } |
86 |
85
|
a1i |
⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) × { 1 } ) : ( 1 ... 𝑁 ) ⟶ { 1 } ) |
87 |
|
ovexd |
⊢ ( 𝜑 → ( 1 ... 𝑁 ) ∈ V ) |
88 |
|
inidm |
⊢ ( ( 1 ... 𝑁 ) ∩ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) |
89 |
83 2 86 87 87 88
|
off |
⊢ ( 𝜑 → ( 𝑇 ∘f + ( ( 1 ... 𝑁 ) × { 1 } ) ) : ( 1 ... 𝑁 ) ⟶ ℕ ) |
90 |
|
elfz1end |
⊢ ( 𝑁 ∈ ℕ ↔ 𝑁 ∈ ( 1 ... 𝑁 ) ) |
91 |
1 90
|
sylib |
⊢ ( 𝜑 → 𝑁 ∈ ( 1 ... 𝑁 ) ) |
92 |
89 91
|
ffvelrnd |
⊢ ( 𝜑 → ( ( 𝑇 ∘f + ( ( 1 ... 𝑁 ) × { 1 } ) ) ‘ 𝑁 ) ∈ ℕ ) |
93 |
92
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑁 − 1 ) < 𝑉 ) → ( ( 𝑇 ∘f + ( ( 1 ... 𝑁 ) × { 1 } ) ) ‘ 𝑁 ) ∈ ℕ ) |
94 |
74 93
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑁 − 1 ) < 𝑉 ) → ( ⦋ if ( ( 𝑁 − 1 ) < 𝑉 , ( 𝑁 − 1 ) , ( ( 𝑁 − 1 ) + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) ∈ ℕ ) |
95 |
94
|
nnne0d |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑁 − 1 ) < 𝑉 ) → ( ⦋ if ( ( 𝑁 − 1 ) < 𝑉 , ( 𝑁 − 1 ) , ( ( 𝑁 − 1 ) + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) ≠ 0 ) |
96 |
|
breq1 |
⊢ ( 𝑦 = ( 𝑁 − 1 ) → ( 𝑦 < 𝑉 ↔ ( 𝑁 − 1 ) < 𝑉 ) ) |
97 |
|
id |
⊢ ( 𝑦 = ( 𝑁 − 1 ) → 𝑦 = ( 𝑁 − 1 ) ) |
98 |
|
oveq1 |
⊢ ( 𝑦 = ( 𝑁 − 1 ) → ( 𝑦 + 1 ) = ( ( 𝑁 − 1 ) + 1 ) ) |
99 |
96 97 98
|
ifbieq12d |
⊢ ( 𝑦 = ( 𝑁 − 1 ) → if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) = if ( ( 𝑁 − 1 ) < 𝑉 , ( 𝑁 − 1 ) , ( ( 𝑁 − 1 ) + 1 ) ) ) |
100 |
99
|
csbeq1d |
⊢ ( 𝑦 = ( 𝑁 − 1 ) → ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( ( 𝑁 − 1 ) < 𝑉 , ( 𝑁 − 1 ) , ( ( 𝑁 − 1 ) + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
101 |
100
|
fveq1d |
⊢ ( 𝑦 = ( 𝑁 − 1 ) → ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = ( ⦋ if ( ( 𝑁 − 1 ) < 𝑉 , ( 𝑁 − 1 ) , ( ( 𝑁 − 1 ) + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) ) |
102 |
101
|
neeq1d |
⊢ ( 𝑦 = ( 𝑁 − 1 ) → ( ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) ≠ 0 ↔ ( ⦋ if ( ( 𝑁 − 1 ) < 𝑉 , ( 𝑁 − 1 ) , ( ( 𝑁 − 1 ) + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) ≠ 0 ) ) |
103 |
11 102
|
bitr3id |
⊢ ( 𝑦 = ( 𝑁 − 1 ) → ( ¬ ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ( ⦋ if ( ( 𝑁 − 1 ) < 𝑉 , ( 𝑁 − 1 ) , ( ( 𝑁 − 1 ) + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) ≠ 0 ) ) |
104 |
103
|
rspcev |
⊢ ( ( ( 𝑁 − 1 ) ∈ ( 0 ... ( 𝑁 − 1 ) ) ∧ ( ⦋ if ( ( 𝑁 − 1 ) < 𝑉 , ( 𝑁 − 1 ) , ( ( 𝑁 − 1 ) + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) ≠ 0 ) → ∃ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ¬ ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) |
105 |
35 95 104
|
syl2anc |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑁 − 1 ) < 𝑉 ) → ∃ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ¬ ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) |
106 |
105 15
|
sylib |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑁 − 1 ) < 𝑉 ) → ¬ ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) |
107 |
106
|
ex |
⊢ ( 𝜑 → ( ¬ ( 𝑁 − 1 ) < 𝑉 → ¬ ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) ) |
108 |
30 107
|
syld |
⊢ ( 𝜑 → ( 𝑉 ≠ 𝑁 → ¬ ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) ) |
109 |
108
|
necon4ad |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 → 𝑉 = 𝑁 ) ) |
110 |
109
|
pm4.71rd |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ( 𝑉 = 𝑁 ∧ ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) ) ) |
111 |
32
|
nn0zd |
⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ℤ ) |
112 |
|
uzid |
⊢ ( ( 𝑁 − 1 ) ∈ ℤ → ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ ( 𝑁 − 1 ) ) ) |
113 |
|
peano2uz |
⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ ( 𝑁 − 1 ) ) ) |
114 |
111 112 113
|
3syl |
⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ ( 𝑁 − 1 ) ) ) |
115 |
39 114
|
eqeltrrd |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ ( 𝑁 − 1 ) ) ) |
116 |
|
fzss2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ( 𝑁 − 1 ) ) → ( 0 ... ( 𝑁 − 1 ) ) ⊆ ( 0 ... 𝑁 ) ) |
117 |
115 116
|
syl |
⊢ ( 𝜑 → ( 0 ... ( 𝑁 − 1 ) ) ⊆ ( 0 ... 𝑁 ) ) |
118 |
117
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑗 ∈ ( 0 ... 𝑁 ) ) |
119 |
91
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝑁 ∈ ( 1 ... 𝑁 ) ) |
120 |
2
|
ffnd |
⊢ ( 𝜑 → 𝑇 Fn ( 1 ... 𝑁 ) ) |
121 |
120
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝑇 Fn ( 1 ... 𝑁 ) ) |
122 |
84
|
fconst |
⊢ ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) : ( 𝑈 “ ( 1 ... 𝑗 ) ) ⟶ { 1 } |
123 |
|
c0ex |
⊢ 0 ∈ V |
124 |
123
|
fconst |
⊢ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) : ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ⟶ { 0 } |
125 |
122 124
|
pm3.2i |
⊢ ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) : ( 𝑈 “ ( 1 ... 𝑗 ) ) ⟶ { 1 } ∧ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) : ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ⟶ { 0 } ) |
126 |
|
dff1o3 |
⊢ ( 𝑈 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( 𝑈 : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) ∧ Fun ◡ 𝑈 ) ) |
127 |
126
|
simprbi |
⊢ ( 𝑈 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → Fun ◡ 𝑈 ) |
128 |
|
imain |
⊢ ( Fun ◡ 𝑈 → ( 𝑈 “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∩ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) ) |
129 |
3 127 128
|
3syl |
⊢ ( 𝜑 → ( 𝑈 “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∩ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) ) |
130 |
|
elfzelz |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → 𝑗 ∈ ℤ ) |
131 |
130
|
zred |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → 𝑗 ∈ ℝ ) |
132 |
131
|
ltp1d |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → 𝑗 < ( 𝑗 + 1 ) ) |
133 |
|
fzdisj |
⊢ ( 𝑗 < ( 𝑗 + 1 ) → ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ∅ ) |
134 |
132 133
|
syl |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ∅ ) |
135 |
134
|
imaeq2d |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( 𝑈 “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ( 𝑈 “ ∅ ) ) |
136 |
135 61
|
eqtrdi |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( 𝑈 “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ∅ ) |
137 |
129 136
|
sylan9req |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∩ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ∅ ) |
138 |
|
fun |
⊢ ( ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) : ( 𝑈 “ ( 1 ... 𝑗 ) ) ⟶ { 1 } ∧ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) : ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ⟶ { 0 } ) ∧ ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∩ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ∅ ) → ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∪ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) ⟶ ( { 1 } ∪ { 0 } ) ) |
139 |
125 137 138
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∪ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) ⟶ ( { 1 } ∪ { 0 } ) ) |
140 |
|
elfznn0 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → 𝑗 ∈ ℕ0 ) |
141 |
140 76
|
syl |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( 𝑗 + 1 ) ∈ ℕ ) |
142 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
143 |
141 142
|
eleqtrdi |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( 𝑗 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
144 |
|
elfzuz3 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
145 |
|
fzsplit2 |
⊢ ( ( ( 𝑗 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) |
146 |
143 144 145
|
syl2anc |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( 1 ... 𝑁 ) = ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) |
147 |
146
|
imaeq2d |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( 𝑈 “ ( 1 ... 𝑁 ) ) = ( 𝑈 “ ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) ) |
148 |
|
imaundi |
⊢ ( 𝑈 “ ( ( 1 ... 𝑗 ) ∪ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∪ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) |
149 |
147 148
|
eqtr2di |
⊢ ( 𝑗 ∈ ( 0 ... 𝑁 ) → ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∪ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ( 𝑈 “ ( 1 ... 𝑁 ) ) ) |
150 |
149 52
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∪ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) ) |
151 |
150
|
feq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∪ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) ⟶ ( { 1 } ∪ { 0 } ) ↔ ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( { 1 } ∪ { 0 } ) ) ) |
152 |
139 151
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( { 1 } ∪ { 0 } ) ) |
153 |
152
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ) |
154 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 1 ... 𝑁 ) ∈ V ) |
155 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑁 ∈ ( 1 ... 𝑁 ) ) → ( 𝑇 ‘ 𝑁 ) = ( 𝑇 ‘ 𝑁 ) ) |
156 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑁 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ) |
157 |
121 153 154 154 88 155 156
|
ofval |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑁 ∈ ( 1 ... 𝑁 ) ) → ( ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = ( ( 𝑇 ‘ 𝑁 ) + ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ) ) |
158 |
119 157
|
mpdan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = ( ( 𝑇 ‘ 𝑁 ) + ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ) ) |
159 |
158
|
eqeq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ( ( 𝑇 ‘ 𝑁 ) + ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ) = 0 ) ) |
160 |
2 91
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑇 ‘ 𝑁 ) ∈ ( 0 ..^ 𝐾 ) ) |
161 |
|
elfzonn0 |
⊢ ( ( 𝑇 ‘ 𝑁 ) ∈ ( 0 ..^ 𝐾 ) → ( 𝑇 ‘ 𝑁 ) ∈ ℕ0 ) |
162 |
160 161
|
syl |
⊢ ( 𝜑 → ( 𝑇 ‘ 𝑁 ) ∈ ℕ0 ) |
163 |
162
|
nn0red |
⊢ ( 𝜑 → ( 𝑇 ‘ 𝑁 ) ∈ ℝ ) |
164 |
163
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝑇 ‘ 𝑁 ) ∈ ℝ ) |
165 |
162
|
nn0ge0d |
⊢ ( 𝜑 → 0 ≤ ( 𝑇 ‘ 𝑁 ) ) |
166 |
165
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 0 ≤ ( 𝑇 ‘ 𝑁 ) ) |
167 |
|
1re |
⊢ 1 ∈ ℝ |
168 |
|
snssi |
⊢ ( 1 ∈ ℝ → { 1 } ⊆ ℝ ) |
169 |
167 168
|
ax-mp |
⊢ { 1 } ⊆ ℝ |
170 |
|
0re |
⊢ 0 ∈ ℝ |
171 |
|
snssi |
⊢ ( 0 ∈ ℝ → { 0 } ⊆ ℝ ) |
172 |
170 171
|
ax-mp |
⊢ { 0 } ⊆ ℝ |
173 |
169 172
|
unssi |
⊢ ( { 1 } ∪ { 0 } ) ⊆ ℝ |
174 |
152 119
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ∈ ( { 1 } ∪ { 0 } ) ) |
175 |
173 174
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ∈ ℝ ) |
176 |
|
elun |
⊢ ( ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ∈ ( { 1 } ∪ { 0 } ) ↔ ( ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ∈ { 1 } ∨ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ∈ { 0 } ) ) |
177 |
|
0le1 |
⊢ 0 ≤ 1 |
178 |
|
elsni |
⊢ ( ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ∈ { 1 } → ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 1 ) |
179 |
177 178
|
breqtrrid |
⊢ ( ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ∈ { 1 } → 0 ≤ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ) |
180 |
|
0le0 |
⊢ 0 ≤ 0 |
181 |
|
elsni |
⊢ ( ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ∈ { 0 } → ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) |
182 |
180 181
|
breqtrrid |
⊢ ( ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ∈ { 0 } → 0 ≤ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ) |
183 |
179 182
|
jaoi |
⊢ ( ( ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ∈ { 1 } ∨ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ∈ { 0 } ) → 0 ≤ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ) |
184 |
176 183
|
sylbi |
⊢ ( ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ∈ ( { 1 } ∪ { 0 } ) → 0 ≤ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ) |
185 |
174 184
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 0 ≤ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ) |
186 |
|
add20 |
⊢ ( ( ( ( 𝑇 ‘ 𝑁 ) ∈ ℝ ∧ 0 ≤ ( 𝑇 ‘ 𝑁 ) ) ∧ ( ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ∈ ℝ ∧ 0 ≤ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ) ) → ( ( ( 𝑇 ‘ 𝑁 ) + ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ) = 0 ↔ ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) ) |
187 |
164 166 175 185 186
|
syl22anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ( 𝑇 ‘ 𝑁 ) + ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ) = 0 ↔ ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) ) |
188 |
159 187
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) ) |
189 |
118 188
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) ) |
190 |
189
|
ralbidva |
⊢ ( 𝜑 → ( ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) ) |
191 |
190
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑉 = 𝑁 ) → ( ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) ) |
192 |
|
breq2 |
⊢ ( 𝑉 = 𝑁 → ( 𝑦 < 𝑉 ↔ 𝑦 < 𝑁 ) ) |
193 |
192
|
ifbid |
⊢ ( 𝑉 = 𝑁 → if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) = if ( 𝑦 < 𝑁 , 𝑦 , ( 𝑦 + 1 ) ) ) |
194 |
|
elfzelz |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℤ ) |
195 |
194
|
zred |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℝ ) |
196 |
195
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑦 ∈ ℝ ) |
197 |
32
|
nn0red |
⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ℝ ) |
198 |
197
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑁 − 1 ) ∈ ℝ ) |
199 |
25
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ℝ ) |
200 |
|
elfzle2 |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ≤ ( 𝑁 − 1 ) ) |
201 |
200
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑦 ≤ ( 𝑁 − 1 ) ) |
202 |
25
|
ltm1d |
⊢ ( 𝜑 → ( 𝑁 − 1 ) < 𝑁 ) |
203 |
202
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑁 − 1 ) < 𝑁 ) |
204 |
196 198 199 201 203
|
lelttrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑦 < 𝑁 ) |
205 |
204
|
iftrued |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → if ( 𝑦 < 𝑁 , 𝑦 , ( 𝑦 + 1 ) ) = 𝑦 ) |
206 |
193 205
|
sylan9eqr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑉 = 𝑁 ) → if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) = 𝑦 ) |
207 |
206
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑉 = 𝑁 ) ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) = 𝑦 ) |
208 |
207
|
csbeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑉 = 𝑁 ) ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ 𝑦 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
209 |
208
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑉 = 𝑁 ) ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = ( ⦋ 𝑦 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) ) |
210 |
209
|
eqeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑉 = 𝑁 ) ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ( ⦋ 𝑦 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) ) |
211 |
210
|
ralbidva |
⊢ ( ( 𝜑 ∧ 𝑉 = 𝑁 ) → ( ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ 𝑦 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) ) |
212 |
|
nfv |
⊢ Ⅎ 𝑦 ( ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 |
213 |
|
nfcsb1v |
⊢ Ⅎ 𝑗 ⦋ 𝑦 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) |
214 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑁 |
215 |
213 214
|
nffv |
⊢ Ⅎ 𝑗 ( ⦋ 𝑦 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) |
216 |
215
|
nfeq1 |
⊢ Ⅎ 𝑗 ( ⦋ 𝑦 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 |
217 |
|
csbeq1a |
⊢ ( 𝑗 = 𝑦 → ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ 𝑦 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
218 |
217
|
fveq1d |
⊢ ( 𝑗 = 𝑦 → ( ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = ( ⦋ 𝑦 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) ) |
219 |
218
|
eqeq1d |
⊢ ( 𝑗 = 𝑦 → ( ( ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ( ⦋ 𝑦 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) ) |
220 |
212 216 219
|
cbvralw |
⊢ ( ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ 𝑦 / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) |
221 |
211 220
|
bitr4di |
⊢ ( ( 𝜑 ∧ 𝑉 = 𝑁 ) → ( ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) ) |
222 |
|
ne0i |
⊢ ( ( 𝑁 − 1 ) ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 0 ... ( 𝑁 − 1 ) ) ≠ ∅ ) |
223 |
|
r19.3rzv |
⊢ ( ( 0 ... ( 𝑁 − 1 ) ) ≠ ∅ → ( ( 𝑇 ‘ 𝑁 ) = 0 ↔ ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( 𝑇 ‘ 𝑁 ) = 0 ) ) |
224 |
34 222 223
|
3syl |
⊢ ( 𝜑 → ( ( 𝑇 ‘ 𝑁 ) = 0 ↔ ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( 𝑇 ‘ 𝑁 ) = 0 ) ) |
225 |
|
elfzelz |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑗 ∈ ℤ ) |
226 |
225
|
zred |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑗 ∈ ℝ ) |
227 |
226
|
ltp1d |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑗 < ( 𝑗 + 1 ) ) |
228 |
227 133
|
syl |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ∅ ) |
229 |
228
|
imaeq2d |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑈 “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ( 𝑈 “ ∅ ) ) |
230 |
229 61
|
eqtrdi |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑈 “ ( ( 1 ... 𝑗 ) ∩ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ∅ ) |
231 |
129 230
|
sylan9req |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∩ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ∅ ) |
232 |
231
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∩ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ∅ ) |
233 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑈 ‘ 𝑁 ) = 𝑁 ) |
234 |
|
f1ofn |
⊢ ( 𝑈 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → 𝑈 Fn ( 1 ... 𝑁 ) ) |
235 |
3 234
|
syl |
⊢ ( 𝜑 → 𝑈 Fn ( 1 ... 𝑁 ) ) |
236 |
235
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑈 Fn ( 1 ... 𝑁 ) ) |
237 |
|
elfznn0 |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑗 ∈ ℕ0 ) |
238 |
237 76
|
syl |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑗 + 1 ) ∈ ℕ ) |
239 |
238 142
|
eleqtrdi |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑗 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
240 |
|
fzss1 |
⊢ ( ( 𝑗 + 1 ) ∈ ( ℤ≥ ‘ 1 ) → ( ( 𝑗 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ) |
241 |
239 240
|
syl |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑗 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ) |
242 |
241
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑗 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ) |
243 |
39
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 ) |
244 |
|
elfzuz3 |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ 𝑗 ) ) |
245 |
|
eluzp1p1 |
⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ 𝑗 ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) |
246 |
244 245
|
syl |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) |
247 |
246
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) |
248 |
243 247
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) ) |
249 |
|
eluzfz2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ( 𝑗 + 1 ) ) → 𝑁 ∈ ( ( 𝑗 + 1 ) ... 𝑁 ) ) |
250 |
248 249
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ( 𝑗 + 1 ) ... 𝑁 ) ) |
251 |
|
fnfvima |
⊢ ( ( 𝑈 Fn ( 1 ... 𝑁 ) ∧ ( ( 𝑗 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ∧ 𝑁 ∈ ( ( 𝑗 + 1 ) ... 𝑁 ) ) → ( 𝑈 ‘ 𝑁 ) ∈ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) |
252 |
236 242 250 251
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑈 ‘ 𝑁 ) ∈ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) |
253 |
252
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑈 ‘ 𝑁 ) ∈ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) |
254 |
233 253
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) |
255 |
|
fnconstg |
⊢ ( 1 ∈ V → ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) Fn ( 𝑈 “ ( 1 ... 𝑗 ) ) ) |
256 |
84 255
|
ax-mp |
⊢ ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) Fn ( 𝑈 “ ( 1 ... 𝑗 ) ) |
257 |
|
fnconstg |
⊢ ( 0 ∈ V → ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) |
258 |
123 257
|
ax-mp |
⊢ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) |
259 |
|
fvun2 |
⊢ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) Fn ( 𝑈 “ ( 1 ... 𝑗 ) ) ∧ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ∧ ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∩ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ∅ ∧ 𝑁 ∈ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) ) → ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = ( ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ 𝑁 ) ) |
260 |
256 258 259
|
mp3an12 |
⊢ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) ∩ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) = ∅ ∧ 𝑁 ∈ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) → ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = ( ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ 𝑁 ) ) |
261 |
232 254 260
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = ( ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ 𝑁 ) ) |
262 |
123
|
fvconst2 |
⊢ ( 𝑁 ∈ ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) → ( ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ 𝑁 ) = 0 ) |
263 |
254 262
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ 𝑁 ) = 0 ) |
264 |
261 263
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ∧ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) |
265 |
264
|
ralrimiva |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) → ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) |
266 |
265
|
ex |
⊢ ( 𝜑 → ( ( 𝑈 ‘ 𝑁 ) = 𝑁 → ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) |
267 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 ) → ( 𝑁 − 1 ) ∈ ( 0 ... ( 𝑁 − 1 ) ) ) |
268 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
269 |
|
imain |
⊢ ( Fun ◡ 𝑈 → ( 𝑈 “ ( ( 1 ... ( 𝑁 − 1 ) ) ∩ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ∩ ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) ) |
270 |
3 127 269
|
3syl |
⊢ ( 𝜑 → ( 𝑈 “ ( ( 1 ... ( 𝑁 − 1 ) ) ∩ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ∩ ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) ) |
271 |
202 39
|
breqtrrd |
⊢ ( 𝜑 → ( 𝑁 − 1 ) < ( ( 𝑁 − 1 ) + 1 ) ) |
272 |
|
fzdisj |
⊢ ( ( 𝑁 − 1 ) < ( ( 𝑁 − 1 ) + 1 ) → ( ( 1 ... ( 𝑁 − 1 ) ) ∩ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) = ∅ ) |
273 |
271 272
|
syl |
⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∩ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) = ∅ ) |
274 |
273
|
imaeq2d |
⊢ ( 𝜑 → ( 𝑈 “ ( ( 1 ... ( 𝑁 − 1 ) ) ∩ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) = ( 𝑈 “ ∅ ) ) |
275 |
274 61
|
eqtrdi |
⊢ ( 𝜑 → ( 𝑈 “ ( ( 1 ... ( 𝑁 − 1 ) ) ∩ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ) |
276 |
270 275
|
eqtr3d |
⊢ ( 𝜑 → ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ∩ ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ) |
277 |
276
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 ) → ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ∩ ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ) |
278 |
91
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 ) → 𝑁 ∈ ( 1 ... 𝑁 ) ) |
279 |
|
elimasni |
⊢ ( 𝑁 ∈ ( 𝑈 “ { 𝑁 } ) → 𝑁 𝑈 𝑁 ) |
280 |
|
fnbrfvb |
⊢ ( ( 𝑈 Fn ( 1 ... 𝑁 ) ∧ 𝑁 ∈ ( 1 ... 𝑁 ) ) → ( ( 𝑈 ‘ 𝑁 ) = 𝑁 ↔ 𝑁 𝑈 𝑁 ) ) |
281 |
235 91 280
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑈 ‘ 𝑁 ) = 𝑁 ↔ 𝑁 𝑈 𝑁 ) ) |
282 |
279 281
|
syl5ibr |
⊢ ( 𝜑 → ( 𝑁 ∈ ( 𝑈 “ { 𝑁 } ) → ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ) |
283 |
282
|
necon3ad |
⊢ ( 𝜑 → ( ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 → ¬ 𝑁 ∈ ( 𝑈 “ { 𝑁 } ) ) ) |
284 |
283
|
imp |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 ) → ¬ 𝑁 ∈ ( 𝑈 “ { 𝑁 } ) ) |
285 |
278 284
|
eldifd |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 ) → 𝑁 ∈ ( ( 1 ... 𝑁 ) ∖ ( 𝑈 “ { 𝑁 } ) ) ) |
286 |
|
imadif |
⊢ ( Fun ◡ 𝑈 → ( 𝑈 “ ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) = ( ( 𝑈 “ ( 1 ... 𝑁 ) ) ∖ ( 𝑈 “ { 𝑁 } ) ) ) |
287 |
3 127 286
|
3syl |
⊢ ( 𝜑 → ( 𝑈 “ ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) = ( ( 𝑈 “ ( 1 ... 𝑁 ) ) ∖ ( 𝑈 “ { 𝑁 } ) ) ) |
288 |
|
difun2 |
⊢ ( ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ∖ { 𝑁 } ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } ) |
289 |
|
elun |
⊢ ( 𝑗 ∈ ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ↔ ( 𝑗 ∈ ( 1 ... ( 𝑁 − 1 ) ) ∨ 𝑗 ∈ { 𝑁 } ) ) |
290 |
|
velsn |
⊢ ( 𝑗 ∈ { 𝑁 } ↔ 𝑗 = 𝑁 ) |
291 |
290
|
orbi2i |
⊢ ( ( 𝑗 ∈ ( 1 ... ( 𝑁 − 1 ) ) ∨ 𝑗 ∈ { 𝑁 } ) ↔ ( 𝑗 ∈ ( 1 ... ( 𝑁 − 1 ) ) ∨ 𝑗 = 𝑁 ) ) |
292 |
289 291
|
bitri |
⊢ ( 𝑗 ∈ ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ↔ ( 𝑗 ∈ ( 1 ... ( 𝑁 − 1 ) ) ∨ 𝑗 = 𝑁 ) ) |
293 |
1 142
|
eleqtrdi |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 1 ) ) |
294 |
|
fzm1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 1 ) → ( 𝑗 ∈ ( 1 ... 𝑁 ) ↔ ( 𝑗 ∈ ( 1 ... ( 𝑁 − 1 ) ) ∨ 𝑗 = 𝑁 ) ) ) |
295 |
293 294
|
syl |
⊢ ( 𝜑 → ( 𝑗 ∈ ( 1 ... 𝑁 ) ↔ ( 𝑗 ∈ ( 1 ... ( 𝑁 − 1 ) ) ∨ 𝑗 = 𝑁 ) ) ) |
296 |
292 295
|
bitr4id |
⊢ ( 𝜑 → ( 𝑗 ∈ ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ↔ 𝑗 ∈ ( 1 ... 𝑁 ) ) ) |
297 |
296
|
eqrdv |
⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) = ( 1 ... 𝑁 ) ) |
298 |
297
|
difeq1d |
⊢ ( 𝜑 → ( ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ∖ { 𝑁 } ) = ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) |
299 |
197 25
|
ltnled |
⊢ ( 𝜑 → ( ( 𝑁 − 1 ) < 𝑁 ↔ ¬ 𝑁 ≤ ( 𝑁 − 1 ) ) ) |
300 |
202 299
|
mpbid |
⊢ ( 𝜑 → ¬ 𝑁 ≤ ( 𝑁 − 1 ) ) |
301 |
|
elfzle2 |
⊢ ( 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) → 𝑁 ≤ ( 𝑁 − 1 ) ) |
302 |
300 301
|
nsyl |
⊢ ( 𝜑 → ¬ 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) |
303 |
|
difsn |
⊢ ( ¬ 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) → ( ( 1 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } ) = ( 1 ... ( 𝑁 − 1 ) ) ) |
304 |
302 303
|
syl |
⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } ) = ( 1 ... ( 𝑁 − 1 ) ) ) |
305 |
288 298 304
|
3eqtr3a |
⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) = ( 1 ... ( 𝑁 − 1 ) ) ) |
306 |
305
|
imaeq2d |
⊢ ( 𝜑 → ( 𝑈 “ ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) = ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ) |
307 |
52
|
difeq1d |
⊢ ( 𝜑 → ( ( 𝑈 “ ( 1 ... 𝑁 ) ) ∖ ( 𝑈 “ { 𝑁 } ) ) = ( ( 1 ... 𝑁 ) ∖ ( 𝑈 “ { 𝑁 } ) ) ) |
308 |
287 306 307
|
3eqtr3rd |
⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) ∖ ( 𝑈 “ { 𝑁 } ) ) = ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ) |
309 |
308
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 ) → ( ( 1 ... 𝑁 ) ∖ ( 𝑈 “ { 𝑁 } ) ) = ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ) |
310 |
285 309
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 ) → 𝑁 ∈ ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ) |
311 |
|
fnconstg |
⊢ ( 1 ∈ V → ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) Fn ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ) |
312 |
84 311
|
ax-mp |
⊢ ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) Fn ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) |
313 |
|
fnconstg |
⊢ ( 0 ∈ V → ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) |
314 |
123 313
|
ax-mp |
⊢ ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) |
315 |
|
fvun1 |
⊢ ( ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) Fn ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ∧ ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ∧ ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ∩ ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ∧ 𝑁 ∈ ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ) ) → ( ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ‘ 𝑁 ) ) |
316 |
312 314 315
|
mp3an12 |
⊢ ( ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ∩ ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ∧ 𝑁 ∈ ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ) → ( ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ‘ 𝑁 ) ) |
317 |
277 310 316
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 ) → ( ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ‘ 𝑁 ) ) |
318 |
84
|
fvconst2 |
⊢ ( 𝑁 ∈ ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) → ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ‘ 𝑁 ) = 1 ) |
319 |
310 318
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 ) → ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ‘ 𝑁 ) = 1 ) |
320 |
317 319
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 ) → ( ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 1 ) |
321 |
320
|
neeq1d |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 ) → ( ( ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ≠ 0 ↔ 1 ≠ 0 ) ) |
322 |
268 321
|
mpbiri |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 ) → ( ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ≠ 0 ) |
323 |
|
df-ne |
⊢ ( ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ≠ 0 ↔ ¬ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) |
324 |
|
oveq2 |
⊢ ( 𝑗 = ( 𝑁 − 1 ) → ( 1 ... 𝑗 ) = ( 1 ... ( 𝑁 − 1 ) ) ) |
325 |
324
|
imaeq2d |
⊢ ( 𝑗 = ( 𝑁 − 1 ) → ( 𝑈 “ ( 1 ... 𝑗 ) ) = ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) ) |
326 |
325
|
xpeq1d |
⊢ ( 𝑗 = ( 𝑁 − 1 ) → ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ) |
327 |
|
oveq1 |
⊢ ( 𝑗 = ( 𝑁 − 1 ) → ( 𝑗 + 1 ) = ( ( 𝑁 − 1 ) + 1 ) ) |
328 |
327
|
oveq1d |
⊢ ( 𝑗 = ( 𝑁 − 1 ) → ( ( 𝑗 + 1 ) ... 𝑁 ) = ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) |
329 |
328
|
imaeq2d |
⊢ ( 𝑗 = ( 𝑁 − 1 ) → ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) |
330 |
329
|
xpeq1d |
⊢ ( 𝑗 = ( 𝑁 − 1 ) → ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) |
331 |
326 330
|
uneq12d |
⊢ ( 𝑗 = ( 𝑁 − 1 ) → ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) |
332 |
331
|
fveq1d |
⊢ ( 𝑗 = ( 𝑁 − 1 ) → ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = ( ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ) |
333 |
332
|
neeq1d |
⊢ ( 𝑗 = ( 𝑁 − 1 ) → ( ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ≠ 0 ↔ ( ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ≠ 0 ) ) |
334 |
323 333
|
bitr3id |
⊢ ( 𝑗 = ( 𝑁 − 1 ) → ( ¬ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ↔ ( ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ≠ 0 ) ) |
335 |
334
|
rspcev |
⊢ ( ( ( 𝑁 − 1 ) ∈ ( 0 ... ( 𝑁 − 1 ) ) ∧ ( ( ( ( 𝑈 “ ( 1 ... ( 𝑁 − 1 ) ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) ≠ 0 ) → ∃ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ¬ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) |
336 |
267 322 335
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 ) → ∃ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ¬ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) |
337 |
336
|
ex |
⊢ ( 𝜑 → ( ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 → ∃ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ¬ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) |
338 |
|
rexnal |
⊢ ( ∃ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ¬ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ↔ ¬ ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) |
339 |
337 338
|
syl6ib |
⊢ ( 𝜑 → ( ( 𝑈 ‘ 𝑁 ) ≠ 𝑁 → ¬ ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) |
340 |
339
|
necon4ad |
⊢ ( 𝜑 → ( ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 → ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ) |
341 |
266 340
|
impbid |
⊢ ( 𝜑 → ( ( 𝑈 ‘ 𝑁 ) = 𝑁 ↔ ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) |
342 |
224 341
|
anbi12d |
⊢ ( 𝜑 → ( ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ↔ ( ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( 𝑇 ‘ 𝑁 ) = 0 ∧ ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) ) |
343 |
|
r19.26 |
⊢ ( ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ↔ ( ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( 𝑇 ‘ 𝑁 ) = 0 ∧ ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) |
344 |
342 343
|
bitr4di |
⊢ ( 𝜑 → ( ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ↔ ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) ) |
345 |
344
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑉 = 𝑁 ) → ( ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ↔ ∀ 𝑗 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑁 ) = 0 ) ) ) |
346 |
191 221 345
|
3bitr4d |
⊢ ( ( 𝜑 ∧ 𝑉 = 𝑁 ) → ( ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ) ) |
347 |
346
|
pm5.32da |
⊢ ( 𝜑 → ( ( 𝑉 = 𝑁 ∧ ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ) ↔ ( 𝑉 = 𝑁 ∧ ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ) ) ) |
348 |
110 347
|
bitrd |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ( 𝑉 = 𝑁 ∧ ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ) ) ) |
349 |
348
|
notbid |
⊢ ( 𝜑 → ( ¬ ∀ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ( ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑁 ) = 0 ↔ ¬ ( 𝑉 = 𝑁 ∧ ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ) ) ) |
350 |
16 349
|
syl5bb |
⊢ ( 𝜑 → ( ∃ 𝑝 ∈ ran ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < 𝑉 , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( 𝑇 ∘f + ( ( ( 𝑈 “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( 𝑈 “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ( 𝑝 ‘ 𝑁 ) ≠ 0 ↔ ¬ ( 𝑉 = 𝑁 ∧ ( ( 𝑇 ‘ 𝑁 ) = 0 ∧ ( 𝑈 ‘ 𝑁 ) = 𝑁 ) ) ) ) |