Step |
Hyp |
Ref |
Expression |
1 |
|
poimir.0 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
2 |
|
poimirlem22.s |
⊢ 𝑆 = { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ∣ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑡 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) } |
3 |
|
poimirlem22.1 |
⊢ ( 𝜑 → 𝐹 : ( 0 ... ( 𝑁 − 1 ) ) ⟶ ( ( 0 ... 𝐾 ) ↑m ( 1 ... 𝑁 ) ) ) |
4 |
|
poimirlem22.2 |
⊢ ( 𝜑 → 𝑇 ∈ 𝑆 ) |
5 |
|
poimirlem22.3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ∃ 𝑝 ∈ ran 𝐹 ( 𝑝 ‘ 𝑛 ) ≠ 0 ) |
6 |
|
poimirlem21.4 |
⊢ ( 𝜑 → ( 2nd ‘ 𝑇 ) = 𝑁 ) |
7 |
|
fveq2 |
⊢ ( 𝑡 = 𝑇 → ( 2nd ‘ 𝑡 ) = ( 2nd ‘ 𝑇 ) ) |
8 |
7
|
breq2d |
⊢ ( 𝑡 = 𝑇 → ( 𝑦 < ( 2nd ‘ 𝑡 ) ↔ 𝑦 < ( 2nd ‘ 𝑇 ) ) ) |
9 |
8
|
ifbid |
⊢ ( 𝑡 = 𝑇 → if ( 𝑦 < ( 2nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) = if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) ) |
10 |
|
2fveq3 |
⊢ ( 𝑡 = 𝑇 → ( 1st ‘ ( 1st ‘ 𝑡 ) ) = ( 1st ‘ ( 1st ‘ 𝑇 ) ) ) |
11 |
|
2fveq3 |
⊢ ( 𝑡 = 𝑇 → ( 2nd ‘ ( 1st ‘ 𝑡 ) ) = ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ) |
12 |
11
|
imaeq1d |
⊢ ( 𝑡 = 𝑇 → ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) ) |
13 |
12
|
xpeq1d |
⊢ ( 𝑡 = 𝑇 → ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ) |
14 |
11
|
imaeq1d |
⊢ ( 𝑡 = 𝑇 → ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) |
15 |
14
|
xpeq1d |
⊢ ( 𝑡 = 𝑇 → ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) |
16 |
13 15
|
uneq12d |
⊢ ( 𝑡 = 𝑇 → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) |
17 |
10 16
|
oveq12d |
⊢ ( 𝑡 = 𝑇 → ( ( 1st ‘ ( 1st ‘ 𝑡 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
18 |
9 17
|
csbeq12dv |
⊢ ( 𝑡 = 𝑇 → ⦋ if ( 𝑦 < ( 2nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑡 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
19 |
18
|
mpteq2dv |
⊢ ( 𝑡 = 𝑇 → ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑡 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) |
20 |
19
|
eqeq2d |
⊢ ( 𝑡 = 𝑇 → ( 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑡 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ↔ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) ) |
21 |
20 2
|
elrab2 |
⊢ ( 𝑇 ∈ 𝑆 ↔ ( 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ∧ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) ) |
22 |
21
|
simprbi |
⊢ ( 𝑇 ∈ 𝑆 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) |
23 |
4 22
|
syl |
⊢ ( 𝜑 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) |
24 |
|
elrabi |
⊢ ( 𝑇 ∈ { 𝑡 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ∣ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑡 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) } → 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ) |
25 |
24 2
|
eleq2s |
⊢ ( 𝑇 ∈ 𝑆 → 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ) |
26 |
4 25
|
syl |
⊢ ( 𝜑 → 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ) |
27 |
|
xp1st |
⊢ ( 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) → ( 1st ‘ 𝑇 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) |
28 |
26 27
|
syl |
⊢ ( 𝜑 → ( 1st ‘ 𝑇 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) |
29 |
|
xp1st |
⊢ ( ( 1st ‘ 𝑇 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) → ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) ) |
30 |
28 29
|
syl |
⊢ ( 𝜑 → ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) ) |
31 |
|
elmapfn |
⊢ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) → ( 1st ‘ ( 1st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ) |
32 |
30 31
|
syl |
⊢ ( 𝜑 → ( 1st ‘ ( 1st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ) |
33 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1st ‘ ( 1st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ) |
34 |
|
1ex |
⊢ 1 ∈ V |
35 |
|
fnconstg |
⊢ ( 1 ∈ V → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ) |
36 |
34 35
|
ax-mp |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) |
37 |
|
c0ex |
⊢ 0 ∈ V |
38 |
|
fnconstg |
⊢ ( 0 ∈ V → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) |
39 |
37 38
|
ax-mp |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) |
40 |
36 39
|
pm3.2i |
⊢ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) |
41 |
|
xp2nd |
⊢ ( ( 1st ‘ 𝑇 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) → ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) |
42 |
28 41
|
syl |
⊢ ( 𝜑 → ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) |
43 |
|
fvex |
⊢ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∈ V |
44 |
|
f1oeq1 |
⊢ ( 𝑓 = ( 2nd ‘ ( 1st ‘ 𝑇 ) ) → ( 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) ) |
45 |
43 44
|
elab |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ↔ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) |
46 |
42 45
|
sylib |
⊢ ( 𝜑 → ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) |
47 |
|
dff1o3 |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) ∧ Fun ◡ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ) ) |
48 |
47
|
simprbi |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → Fun ◡ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ) |
49 |
46 48
|
syl |
⊢ ( 𝜑 → Fun ◡ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ) |
50 |
|
imain |
⊢ ( Fun ◡ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ) |
51 |
49 50
|
syl |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ) |
52 |
|
elfznn0 |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℕ0 ) |
53 |
52
|
nn0red |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℝ ) |
54 |
53
|
ltp1d |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 < ( 𝑦 + 1 ) ) |
55 |
|
fzdisj |
⊢ ( 𝑦 < ( 𝑦 + 1 ) → ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ∅ ) |
56 |
54 55
|
syl |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ∅ ) |
57 |
56
|
imaeq2d |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ∅ ) ) |
58 |
|
ima0 |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ∅ ) = ∅ |
59 |
57 58
|
eqtrdi |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ ) |
60 |
51 59
|
sylan9req |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ ) |
61 |
|
fnun |
⊢ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ) |
62 |
40 60 61
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ) |
63 |
|
imaundi |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) |
64 |
|
nn0p1nn |
⊢ ( 𝑦 ∈ ℕ0 → ( 𝑦 + 1 ) ∈ ℕ ) |
65 |
52 64
|
syl |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℕ ) |
66 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
67 |
65 66
|
eleqtrdi |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
68 |
67
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
69 |
1
|
nncnd |
⊢ ( 𝜑 → 𝑁 ∈ ℂ ) |
70 |
|
npcan1 |
⊢ ( 𝑁 ∈ ℂ → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 ) |
71 |
69 70
|
syl |
⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 ) |
72 |
71
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 ) |
73 |
|
elfzuz3 |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ 𝑦 ) ) |
74 |
|
peano2uz |
⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ 𝑦 ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ 𝑦 ) ) |
75 |
73 74
|
syl |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ 𝑦 ) ) |
76 |
75
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ 𝑦 ) ) |
77 |
72 76
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑦 ) ) |
78 |
|
fzsplit2 |
⊢ ( ( ( 𝑦 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑦 ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) |
79 |
68 77 78
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) |
80 |
79
|
imaeq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ) |
81 |
|
f1ofo |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) ) |
82 |
|
foima |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) ) |
83 |
46 81 82
|
3syl |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) ) |
84 |
83
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) ) |
85 |
80 84
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) ) |
86 |
63 85
|
eqtr3id |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) ) |
87 |
86
|
fneq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ↔ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ) ) |
88 |
62 87
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ) |
89 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑁 ) ∈ V ) |
90 |
|
inidm |
⊢ ( ( 1 ... 𝑁 ) ∩ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) |
91 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) = ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) ) |
92 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) |
93 |
33 88 89 89 90 91 92
|
offval |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ) |
94 |
|
elmapi |
⊢ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) → ( 1st ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ..^ 𝐾 ) ) |
95 |
30 94
|
syl |
⊢ ( 𝜑 → ( 1st ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ..^ 𝐾 ) ) |
96 |
95
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ( 0 ..^ 𝐾 ) ) |
97 |
|
elfzonn0 |
⊢ ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ( 0 ..^ 𝐾 ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ℕ0 ) |
98 |
96 97
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ℕ0 ) |
99 |
98
|
nn0cnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ℂ ) |
100 |
99
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) ∈ ℂ ) |
101 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
102 |
|
0cn |
⊢ 0 ∈ ℂ |
103 |
101 102
|
ifcli |
⊢ if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ∈ ℂ |
104 |
103
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ∈ ℂ ) |
105 |
|
snssi |
⊢ ( 1 ∈ ℂ → { 1 } ⊆ ℂ ) |
106 |
101 105
|
ax-mp |
⊢ { 1 } ⊆ ℂ |
107 |
|
snssi |
⊢ ( 0 ∈ ℂ → { 0 } ⊆ ℂ ) |
108 |
102 107
|
ax-mp |
⊢ { 0 } ⊆ ℂ |
109 |
106 108
|
unssi |
⊢ ( { 1 } ∪ { 0 } ) ⊆ ℂ |
110 |
34
|
fconst |
⊢ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) : ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⟶ { 1 } |
111 |
37
|
fconst |
⊢ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) : ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ⟶ { 0 } |
112 |
110 111
|
pm3.2i |
⊢ ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) : ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⟶ { 1 } ∧ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) : ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ⟶ { 0 } ) |
113 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) |
114 |
1
|
nnzd |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
115 |
|
1z |
⊢ 1 ∈ ℤ |
116 |
|
peano2z |
⊢ ( 1 ∈ ℤ → ( 1 + 1 ) ∈ ℤ ) |
117 |
115 116
|
ax-mp |
⊢ ( 1 + 1 ) ∈ ℤ |
118 |
114 117
|
jctil |
⊢ ( 𝜑 → ( ( 1 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) |
119 |
|
elfzelz |
⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) → 𝑛 ∈ ℤ ) |
120 |
119 115
|
jctir |
⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) → ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) ) |
121 |
|
fzsubel |
⊢ ( ( ( ( 1 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) ) |
122 |
118 120 121
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) ) |
123 |
113 122
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) |
124 |
101 101
|
pncan3oi |
⊢ ( ( 1 + 1 ) − 1 ) = 1 |
125 |
124
|
oveq1i |
⊢ ( ( ( 1 + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) = ( 1 ... ( 𝑁 − 1 ) ) |
126 |
123 125
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → ( 𝑛 − 1 ) ∈ ( 1 ... ( 𝑁 − 1 ) ) ) |
127 |
126
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ( 𝑛 − 1 ) ∈ ( 1 ... ( 𝑁 − 1 ) ) ) |
128 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) |
129 |
|
peano2zm |
⊢ ( 𝑁 ∈ ℤ → ( 𝑁 − 1 ) ∈ ℤ ) |
130 |
114 129
|
syl |
⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ℤ ) |
131 |
130 115
|
jctil |
⊢ ( 𝜑 → ( 1 ∈ ℤ ∧ ( 𝑁 − 1 ) ∈ ℤ ) ) |
132 |
|
elfzelz |
⊢ ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℤ ) |
133 |
132 115
|
jctir |
⊢ ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) → ( 𝑦 ∈ ℤ ∧ 1 ∈ ℤ ) ) |
134 |
|
fzaddel |
⊢ ( ( ( 1 ∈ ℤ ∧ ( 𝑁 − 1 ) ∈ ℤ ) ∧ ( 𝑦 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↔ ( 𝑦 + 1 ) ∈ ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) ) |
135 |
131 133 134
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ↔ ( 𝑦 + 1 ) ∈ ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) ) |
136 |
128 135
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 + 1 ) ∈ ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) ) |
137 |
71
|
oveq2d |
⊢ ( 𝜑 → ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) = ( ( 1 + 1 ) ... 𝑁 ) ) |
138 |
137
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ( ( 1 + 1 ) ... ( ( 𝑁 − 1 ) + 1 ) ) = ( ( 1 + 1 ) ... 𝑁 ) ) |
139 |
136 138
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ( 𝑦 + 1 ) ∈ ( ( 1 + 1 ) ... 𝑁 ) ) |
140 |
119
|
zcnd |
⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) → 𝑛 ∈ ℂ ) |
141 |
132
|
zcnd |
⊢ ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℂ ) |
142 |
|
subadd2 |
⊢ ( ( 𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( 𝑛 − 1 ) = 𝑦 ↔ ( 𝑦 + 1 ) = 𝑛 ) ) |
143 |
101 142
|
mp3an2 |
⊢ ( ( 𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( 𝑛 − 1 ) = 𝑦 ↔ ( 𝑦 + 1 ) = 𝑛 ) ) |
144 |
|
eqcom |
⊢ ( 𝑦 = ( 𝑛 − 1 ) ↔ ( 𝑛 − 1 ) = 𝑦 ) |
145 |
|
eqcom |
⊢ ( 𝑛 = ( 𝑦 + 1 ) ↔ ( 𝑦 + 1 ) = 𝑛 ) |
146 |
143 144 145
|
3bitr4g |
⊢ ( ( 𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑦 = ( 𝑛 − 1 ) ↔ 𝑛 = ( 𝑦 + 1 ) ) ) |
147 |
140 141 146
|
syl2anr |
⊢ ( ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ∧ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) → ( 𝑦 = ( 𝑛 − 1 ) ↔ 𝑛 = ( 𝑦 + 1 ) ) ) |
148 |
147
|
ralrimiva |
⊢ ( 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) → ∀ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ( 𝑦 = ( 𝑛 − 1 ) ↔ 𝑛 = ( 𝑦 + 1 ) ) ) |
149 |
148
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ∀ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ( 𝑦 = ( 𝑛 − 1 ) ↔ 𝑛 = ( 𝑦 + 1 ) ) ) |
150 |
|
reu6i |
⊢ ( ( ( 𝑦 + 1 ) ∈ ( ( 1 + 1 ) ... 𝑁 ) ∧ ∀ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ( 𝑦 = ( 𝑛 − 1 ) ↔ 𝑛 = ( 𝑦 + 1 ) ) ) → ∃! 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) 𝑦 = ( 𝑛 − 1 ) ) |
151 |
139 149 150
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) → ∃! 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) 𝑦 = ( 𝑛 − 1 ) ) |
152 |
151
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ∃! 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) 𝑦 = ( 𝑛 − 1 ) ) |
153 |
|
eqid |
⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) = ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) |
154 |
153
|
f1ompt |
⊢ ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) : ( ( 1 + 1 ) ... 𝑁 ) –1-1-onto→ ( 1 ... ( 𝑁 − 1 ) ) ↔ ( ∀ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ( 𝑛 − 1 ) ∈ ( 1 ... ( 𝑁 − 1 ) ) ∧ ∀ 𝑦 ∈ ( 1 ... ( 𝑁 − 1 ) ) ∃! 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) 𝑦 = ( 𝑛 − 1 ) ) ) |
155 |
127 152 154
|
sylanbrc |
⊢ ( 𝜑 → ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) : ( ( 1 + 1 ) ... 𝑁 ) –1-1-onto→ ( 1 ... ( 𝑁 − 1 ) ) ) |
156 |
|
f1osng |
⊢ ( ( 1 ∈ V ∧ 𝑁 ∈ ℕ ) → { 〈 1 , 𝑁 〉 } : { 1 } –1-1-onto→ { 𝑁 } ) |
157 |
34 1 156
|
sylancr |
⊢ ( 𝜑 → { 〈 1 , 𝑁 〉 } : { 1 } –1-1-onto→ { 𝑁 } ) |
158 |
1
|
nnred |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
159 |
158
|
ltm1d |
⊢ ( 𝜑 → ( 𝑁 − 1 ) < 𝑁 ) |
160 |
130
|
zred |
⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ℝ ) |
161 |
160 158
|
ltnled |
⊢ ( 𝜑 → ( ( 𝑁 − 1 ) < 𝑁 ↔ ¬ 𝑁 ≤ ( 𝑁 − 1 ) ) ) |
162 |
159 161
|
mpbid |
⊢ ( 𝜑 → ¬ 𝑁 ≤ ( 𝑁 − 1 ) ) |
163 |
|
elfzle2 |
⊢ ( 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) → 𝑁 ≤ ( 𝑁 − 1 ) ) |
164 |
162 163
|
nsyl |
⊢ ( 𝜑 → ¬ 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) |
165 |
|
disjsn |
⊢ ( ( ( 1 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ ↔ ¬ 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) ) |
166 |
164 165
|
sylibr |
⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ ) |
167 |
|
1re |
⊢ 1 ∈ ℝ |
168 |
167
|
ltp1i |
⊢ 1 < ( 1 + 1 ) |
169 |
117
|
zrei |
⊢ ( 1 + 1 ) ∈ ℝ |
170 |
167 169
|
ltnlei |
⊢ ( 1 < ( 1 + 1 ) ↔ ¬ ( 1 + 1 ) ≤ 1 ) |
171 |
168 170
|
mpbi |
⊢ ¬ ( 1 + 1 ) ≤ 1 |
172 |
|
elfzle1 |
⊢ ( 1 ∈ ( ( 1 + 1 ) ... 𝑁 ) → ( 1 + 1 ) ≤ 1 ) |
173 |
171 172
|
mto |
⊢ ¬ 1 ∈ ( ( 1 + 1 ) ... 𝑁 ) |
174 |
|
disjsn |
⊢ ( ( ( ( 1 + 1 ) ... 𝑁 ) ∩ { 1 } ) = ∅ ↔ ¬ 1 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) |
175 |
173 174
|
mpbir |
⊢ ( ( ( 1 + 1 ) ... 𝑁 ) ∩ { 1 } ) = ∅ |
176 |
|
f1oun |
⊢ ( ( ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) : ( ( 1 + 1 ) ... 𝑁 ) –1-1-onto→ ( 1 ... ( 𝑁 − 1 ) ) ∧ { 〈 1 , 𝑁 〉 } : { 1 } –1-1-onto→ { 𝑁 } ) ∧ ( ( ( ( 1 + 1 ) ... 𝑁 ) ∩ { 1 } ) = ∅ ∧ ( ( 1 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ ) ) → ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ∪ { 〈 1 , 𝑁 〉 } ) : ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ) |
177 |
175 176
|
mpanr1 |
⊢ ( ( ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) : ( ( 1 + 1 ) ... 𝑁 ) –1-1-onto→ ( 1 ... ( 𝑁 − 1 ) ) ∧ { 〈 1 , 𝑁 〉 } : { 1 } –1-1-onto→ { 𝑁 } ) ∧ ( ( 1 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ ) → ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ∪ { 〈 1 , 𝑁 〉 } ) : ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ) |
178 |
155 157 166 177
|
syl21anc |
⊢ ( 𝜑 → ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ∪ { 〈 1 , 𝑁 〉 } ) : ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ) |
179 |
|
eleq1 |
⊢ ( 𝑛 = 1 → ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↔ 1 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) ) |
180 |
173 179
|
mtbiri |
⊢ ( 𝑛 = 1 → ¬ 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ) |
181 |
180
|
necon2ai |
⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) → 𝑛 ≠ 1 ) |
182 |
|
ifnefalse |
⊢ ( 𝑛 ≠ 1 → if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) = ( 𝑛 − 1 ) ) |
183 |
181 182
|
syl |
⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) → if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) = ( 𝑛 − 1 ) ) |
184 |
183
|
mpteq2ia |
⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) = ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) |
185 |
184
|
uneq1i |
⊢ ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ∪ { 〈 1 , 𝑁 〉 } ) = ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ∪ { 〈 1 , 𝑁 〉 } ) |
186 |
34
|
a1i |
⊢ ( 𝜑 → 1 ∈ V ) |
187 |
1 66
|
eleqtrdi |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 1 ) ) |
188 |
|
fzpred |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 1 ) → ( 1 ... 𝑁 ) = ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) ) |
189 |
187 188
|
syl |
⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) ) |
190 |
|
uncom |
⊢ ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) = ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) |
191 |
189 190
|
eqtr2di |
⊢ ( 𝜑 → ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) = ( 1 ... 𝑁 ) ) |
192 |
|
iftrue |
⊢ ( 𝑛 = 1 → if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) = 𝑁 ) |
193 |
192
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 = 1 ) → if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) = 𝑁 ) |
194 |
186 1 191 193
|
fmptapd |
⊢ ( 𝜑 → ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ∪ { 〈 1 , 𝑁 〉 } ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) |
195 |
185 194
|
eqtr3id |
⊢ ( 𝜑 → ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ∪ { 〈 1 , 𝑁 〉 } ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) |
196 |
71 187
|
eqeltrd |
⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
197 |
|
uzid |
⊢ ( ( 𝑁 − 1 ) ∈ ℤ → ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ ( 𝑁 − 1 ) ) ) |
198 |
|
peano2uz |
⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ ( 𝑁 − 1 ) ) ) |
199 |
130 197 198
|
3syl |
⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ ( 𝑁 − 1 ) ) ) |
200 |
71 199
|
eqeltrrd |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ ( 𝑁 − 1 ) ) ) |
201 |
|
fzsplit2 |
⊢ ( ( ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) |
202 |
196 200 201
|
syl2anc |
⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) |
203 |
71
|
oveq1d |
⊢ ( 𝜑 → ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) = ( 𝑁 ... 𝑁 ) ) |
204 |
|
fzsn |
⊢ ( 𝑁 ∈ ℤ → ( 𝑁 ... 𝑁 ) = { 𝑁 } ) |
205 |
114 204
|
syl |
⊢ ( 𝜑 → ( 𝑁 ... 𝑁 ) = { 𝑁 } ) |
206 |
203 205
|
eqtrd |
⊢ ( 𝜑 → ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) = { 𝑁 } ) |
207 |
206
|
uneq2d |
⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ) |
208 |
202 207
|
eqtr2d |
⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) = ( 1 ... 𝑁 ) ) |
209 |
195 191 208
|
f1oeq123d |
⊢ ( 𝜑 → ( ( ( 𝑛 ∈ ( ( 1 + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ∪ { 〈 1 , 𝑁 〉 } ) : ( ( ( 1 + 1 ) ... 𝑁 ) ∪ { 1 } ) –1-1-onto→ ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) ) |
210 |
178 209
|
mpbid |
⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) |
211 |
|
f1oco |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) |
212 |
46 210 211
|
syl2anc |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) |
213 |
|
dff1o3 |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) ∧ Fun ◡ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) ) ) |
214 |
213
|
simprbi |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → Fun ◡ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) ) |
215 |
|
imain |
⊢ ( Fun ◡ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) ) |
216 |
212 214 215
|
3syl |
⊢ ( 𝜑 → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) ) |
217 |
65
|
nnred |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℝ ) |
218 |
217
|
ltp1d |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) < ( ( 𝑦 + 1 ) + 1 ) ) |
219 |
|
fzdisj |
⊢ ( ( 𝑦 + 1 ) < ( ( 𝑦 + 1 ) + 1 ) → ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ ) |
220 |
218 219
|
syl |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ∅ ) |
221 |
220
|
imaeq2d |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ∅ ) ) |
222 |
|
ima0 |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ∅ ) = ∅ |
223 |
221 222
|
eqtrdi |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∩ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ) |
224 |
216 223
|
sylan9req |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ) |
225 |
|
fun |
⊢ ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) : ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⟶ { 1 } ∧ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) : ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ⟶ { 0 } ) ∧ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) ⟶ ( { 1 } ∪ { 0 } ) ) |
226 |
112 224 225
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) ⟶ ( { 1 } ∪ { 0 } ) ) |
227 |
|
imaundi |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) |
228 |
65
|
peano2nnd |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ℕ ) |
229 |
228 66
|
eleqtrdi |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
230 |
229
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
231 |
|
eluzp1p1 |
⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ 𝑦 ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ ( 𝑦 + 1 ) ) ) |
232 |
73 231
|
syl |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ ( 𝑦 + 1 ) ) ) |
233 |
232
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ ( 𝑦 + 1 ) ) ) |
234 |
72 233
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ℤ≥ ‘ ( 𝑦 + 1 ) ) ) |
235 |
|
fzsplit2 |
⊢ ( ( ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ ( 𝑦 + 1 ) ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) |
236 |
230 234 235
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) |
237 |
236
|
imaeq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... 𝑁 ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) ) |
238 |
|
f1ofo |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) ) |
239 |
|
foima |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) ) |
240 |
212 238 239
|
3syl |
⊢ ( 𝜑 → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) ) |
241 |
240
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) ) |
242 |
237 241
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 ... ( 𝑦 + 1 ) ) ∪ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) ) |
243 |
227 242
|
eqtr3id |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) ) |
244 |
243
|
feq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) ⟶ ( { 1 } ∪ { 0 } ) ↔ ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( { 1 } ∪ { 0 } ) ) ) |
245 |
226 244
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) : ( 1 ... 𝑁 ) ⟶ ( { 1 } ∪ { 0 } ) ) |
246 |
245
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ∈ ( { 1 } ∪ { 0 } ) ) |
247 |
109 246
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ∈ ℂ ) |
248 |
100 104 247
|
subadd23d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) = ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) ) |
249 |
|
oveq2 |
⊢ ( 1 = if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) → ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) |
250 |
249
|
eqeq1d |
⊢ ( 1 = if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) → ( ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ↔ ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) |
251 |
|
oveq2 |
⊢ ( 0 = if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) → ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 0 ) = ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) |
252 |
251
|
eqeq1d |
⊢ ( 0 = if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) → ( ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 0 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ↔ ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) |
253 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
254 |
|
f1ofn |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 2nd ‘ ( 1st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ) |
255 |
46 254
|
syl |
⊢ ( 𝜑 → ( 2nd ‘ ( 1st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ) |
256 |
255
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 2nd ‘ ( 1st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ) |
257 |
|
imassrn |
⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⊆ ran ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) |
258 |
|
f1of |
⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) ) |
259 |
210 258
|
syl |
⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) ) |
260 |
259
|
frnd |
⊢ ( 𝜑 → ran ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ⊆ ( 1 ... 𝑁 ) ) |
261 |
257 260
|
sstrid |
⊢ ( 𝜑 → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⊆ ( 1 ... 𝑁 ) ) |
262 |
261
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⊆ ( 1 ... 𝑁 ) ) |
263 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) |
264 |
|
eluzfz1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ ( 1 ... 𝑁 ) ) |
265 |
187 264
|
syl |
⊢ ( 𝜑 → 1 ∈ ( 1 ... 𝑁 ) ) |
266 |
263 193 265 1
|
fvmptd |
⊢ ( 𝜑 → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) = 𝑁 ) |
267 |
266
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) = 𝑁 ) |
268 |
|
f1ofn |
⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) Fn ( 1 ... 𝑁 ) ) |
269 |
210 268
|
syl |
⊢ ( 𝜑 → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) Fn ( 1 ... 𝑁 ) ) |
270 |
269
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) Fn ( 1 ... 𝑁 ) ) |
271 |
|
fzss2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ( 𝑦 + 1 ) ) → ( 1 ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... 𝑁 ) ) |
272 |
234 271
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... 𝑁 ) ) |
273 |
|
eluzfz1 |
⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ ( 1 ... ( 𝑦 + 1 ) ) ) |
274 |
67 273
|
syl |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 1 ∈ ( 1 ... ( 𝑦 + 1 ) ) ) |
275 |
274
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 1 ∈ ( 1 ... ( 𝑦 + 1 ) ) ) |
276 |
|
fnfvima |
⊢ ( ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) Fn ( 1 ... 𝑁 ) ∧ ( 1 ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... 𝑁 ) ∧ 1 ∈ ( 1 ... ( 𝑦 + 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) ∈ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) |
277 |
270 272 275 276
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) ∈ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) |
278 |
267 277
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) |
279 |
|
fnfvima |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ∧ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ⊆ ( 1 ... 𝑁 ) ∧ 𝑁 ∈ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) ) |
280 |
256 262 278 279
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) ) |
281 |
|
imaco |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) |
282 |
280 281
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) |
283 |
|
fnconstg |
⊢ ( 1 ∈ V → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) |
284 |
34 283
|
ax-mp |
⊢ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) |
285 |
|
fnconstg |
⊢ ( 0 ∈ V → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) |
286 |
37 285
|
ax-mp |
⊢ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) |
287 |
|
fvun1 |
⊢ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∧ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ∧ ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ∧ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) |
288 |
284 286 287
|
mp3an12 |
⊢ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ∩ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ∅ ∧ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) |
289 |
224 282 288
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) |
290 |
34
|
fvconst2 |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = 1 ) |
291 |
282 290
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = 1 ) |
292 |
289 291
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = 1 ) |
293 |
292
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) − 1 ) = ( 1 − 1 ) ) |
294 |
|
fzss1 |
⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ≥ ‘ 1 ) → ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ) |
295 |
67 294
|
syl |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ) |
296 |
295
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ) |
297 |
|
eluzfz2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ( 𝑦 + 1 ) ) → 𝑁 ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) ) |
298 |
234 297
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) ) |
299 |
|
fnfvima |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ∧ ( ( 𝑦 + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ∧ 𝑁 ∈ ( ( 𝑦 + 1 ) ... 𝑁 ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) |
300 |
256 296 298 299
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) |
301 |
|
fvun2 |
⊢ ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ∧ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ ∧ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) |
302 |
36 39 301
|
mp3an12 |
⊢ ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) = ∅ ∧ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) |
303 |
60 300 302
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) |
304 |
37
|
fvconst2 |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = 0 ) |
305 |
300 304
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = 0 ) |
306 |
303 305
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) = 0 ) |
307 |
253 293 306
|
3eqtr4a |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) − 1 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) |
308 |
|
fveq2 |
⊢ ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) |
309 |
308
|
oveq1d |
⊢ ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) → ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) − 1 ) ) |
310 |
|
fveq2 |
⊢ ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) |
311 |
309 310
|
eqeq12d |
⊢ ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) → ( ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ↔ ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) − 1 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) ) |
312 |
307 311
|
syl5ibrcom |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) → ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) |
313 |
312
|
imp |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) |
314 |
313
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 1 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) |
315 |
247
|
subid1d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 0 ) = ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) |
316 |
315
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 0 ) = ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) |
317 |
|
eldifsn |
⊢ ( 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑛 ≠ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) |
318 |
|
df-ne |
⊢ ( 𝑛 ≠ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ↔ ¬ 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) |
319 |
318
|
anbi2i |
⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ 𝑛 ≠ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ¬ 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) |
320 |
317 319
|
bitri |
⊢ ( 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ↔ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ¬ 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) |
321 |
|
fnconstg |
⊢ ( 0 ∈ V → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) |
322 |
37 321
|
ax-mp |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) |
323 |
36 322
|
pm3.2i |
⊢ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) |
324 |
|
imain |
⊢ ( Fun ◡ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) ) |
325 |
49 324
|
syl |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) ) |
326 |
|
fzdisj |
⊢ ( 𝑦 < ( 𝑦 + 1 ) → ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) = ∅ ) |
327 |
54 326
|
syl |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) = ∅ ) |
328 |
327
|
imaeq2d |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ∅ ) ) |
329 |
328 58
|
eqtrdi |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∩ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ∅ ) |
330 |
325 329
|
sylan9req |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ∅ ) |
331 |
|
fnun |
⊢ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ∅ ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) ) |
332 |
323 330 331
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) ) |
333 |
|
imaundi |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) |
334 |
202 207
|
eqtrd |
⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ) |
335 |
334
|
difeq1d |
⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) = ( ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ∖ { 𝑁 } ) ) |
336 |
|
difun2 |
⊢ ( ( ( 1 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ∖ { 𝑁 } ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } ) |
337 |
335 336
|
eqtrdi |
⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) = ( ( 1 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } ) ) |
338 |
|
difsn |
⊢ ( ¬ 𝑁 ∈ ( 1 ... ( 𝑁 − 1 ) ) → ( ( 1 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } ) = ( 1 ... ( 𝑁 − 1 ) ) ) |
339 |
164 338
|
syl |
⊢ ( 𝜑 → ( ( 1 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } ) = ( 1 ... ( 𝑁 − 1 ) ) ) |
340 |
337 339
|
eqtrd |
⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) = ( 1 ... ( 𝑁 − 1 ) ) ) |
341 |
340
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) = ( 1 ... ( 𝑁 − 1 ) ) ) |
342 |
73
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ 𝑦 ) ) |
343 |
|
fzsplit2 |
⊢ ( ( ( 𝑦 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ∧ ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ 𝑦 ) ) → ( 1 ... ( 𝑁 − 1 ) ) = ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) |
344 |
68 342 343
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... ( 𝑁 − 1 ) ) = ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) |
345 |
341 344
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) = ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) |
346 |
345
|
imaeq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) ) |
347 |
|
imadif |
⊢ ( Fun ◡ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) ∖ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ { 𝑁 } ) ) ) |
348 |
49 347
|
syl |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) ∖ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ { 𝑁 } ) ) ) |
349 |
|
elfz1end |
⊢ ( 𝑁 ∈ ℕ ↔ 𝑁 ∈ ( 1 ... 𝑁 ) ) |
350 |
1 349
|
sylib |
⊢ ( 𝜑 → 𝑁 ∈ ( 1 ... 𝑁 ) ) |
351 |
|
fnsnfv |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ∧ 𝑁 ∈ ( 1 ... 𝑁 ) ) → { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ { 𝑁 } ) ) |
352 |
255 350 351
|
syl2anc |
⊢ ( 𝜑 → { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ { 𝑁 } ) ) |
353 |
352
|
eqcomd |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ { 𝑁 } ) = { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) |
354 |
83 353
|
difeq12d |
⊢ ( 𝜑 → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) ∖ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ { 𝑁 } ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) |
355 |
348 354
|
eqtrd |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) |
356 |
355
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑁 ) ∖ { 𝑁 } ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) |
357 |
346 356
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑦 ) ∪ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) |
358 |
333 357
|
eqtr3id |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) = ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) |
359 |
358
|
fneq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) ↔ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) ) |
360 |
332 359
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) |
361 |
|
disjdifr |
⊢ ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∩ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ∅ |
362 |
|
fnconstg |
⊢ ( 1 ∈ V → ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) Fn { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) |
363 |
34 362
|
ax-mp |
⊢ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) Fn { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } |
364 |
|
fvun1 |
⊢ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∧ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) Fn { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∩ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) |
365 |
363 364
|
mp3an2 |
⊢ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∩ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) |
366 |
|
fnconstg |
⊢ ( 0 ∈ V → ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) Fn { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) |
367 |
37 366
|
ax-mp |
⊢ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) Fn { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } |
368 |
|
fvun1 |
⊢ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∧ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) Fn { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∩ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) |
369 |
367 368
|
mp3an2 |
⊢ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∩ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ‘ 𝑛 ) ) |
370 |
365 369
|
eqtr4d |
⊢ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∧ ( ( ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∩ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ∅ ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) ) |
371 |
361 370
|
mpanr1 |
⊢ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) Fn ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) ) |
372 |
360 371
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∖ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) ) |
373 |
320 372
|
sylan2br |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ ( 𝑛 ∈ ( 1 ... 𝑁 ) ∧ ¬ 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) ) |
374 |
373
|
anassrs |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) ) |
375 |
|
imaundi |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ { 1 } ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ { 1 } ) ) |
376 |
|
imaco |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) |
377 |
|
imaco |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ { 1 } ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) |
378 |
376 377
|
uneq12i |
⊢ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ { 1 } ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) ) |
379 |
375 378
|
eqtri |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ { 1 } ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) ) |
380 |
|
fzpred |
⊢ ( ( 𝑦 + 1 ) ∈ ( ℤ≥ ‘ 1 ) → ( 1 ... ( 𝑦 + 1 ) ) = ( { 1 } ∪ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) |
381 |
67 380
|
syl |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 1 ... ( 𝑦 + 1 ) ) = ( { 1 } ∪ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) |
382 |
|
uncom |
⊢ ( { 1 } ∪ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ { 1 } ) |
383 |
381 382
|
eqtrdi |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 1 ... ( 𝑦 + 1 ) ) = ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ { 1 } ) ) |
384 |
383
|
imaeq2d |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ { 1 } ) ) ) |
385 |
384
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∪ { 1 } ) ) ) |
386 |
|
elfzelz |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ∈ ℤ ) |
387 |
124
|
a1i |
⊢ ( 𝑦 ∈ ℤ → ( ( 1 + 1 ) − 1 ) = 1 ) |
388 |
|
zcn |
⊢ ( 𝑦 ∈ ℤ → 𝑦 ∈ ℂ ) |
389 |
|
pncan1 |
⊢ ( 𝑦 ∈ ℂ → ( ( 𝑦 + 1 ) − 1 ) = 𝑦 ) |
390 |
388 389
|
syl |
⊢ ( 𝑦 ∈ ℤ → ( ( 𝑦 + 1 ) − 1 ) = 𝑦 ) |
391 |
387 390
|
oveq12d |
⊢ ( 𝑦 ∈ ℤ → ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) = ( 1 ... 𝑦 ) ) |
392 |
|
elfzelz |
⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → 𝑗 ∈ ℤ ) |
393 |
392
|
zcnd |
⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → 𝑗 ∈ ℂ ) |
394 |
|
pncan1 |
⊢ ( 𝑗 ∈ ℂ → ( ( 𝑗 + 1 ) − 1 ) = 𝑗 ) |
395 |
393 394
|
syl |
⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → ( ( 𝑗 + 1 ) − 1 ) = 𝑗 ) |
396 |
395
|
eleq1d |
⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → ( ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ↔ 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) ) |
397 |
396
|
ibir |
⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) |
398 |
397
|
adantl |
⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) → ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) |
399 |
|
peano2z |
⊢ ( 𝑦 ∈ ℤ → ( 𝑦 + 1 ) ∈ ℤ ) |
400 |
399 117
|
jctil |
⊢ ( 𝑦 ∈ ℤ → ( ( 1 + 1 ) ∈ ℤ ∧ ( 𝑦 + 1 ) ∈ ℤ ) ) |
401 |
392
|
peano2zd |
⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → ( 𝑗 + 1 ) ∈ ℤ ) |
402 |
401 115
|
jctir |
⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → ( ( 𝑗 + 1 ) ∈ ℤ ∧ 1 ∈ ℤ ) ) |
403 |
|
fzsubel |
⊢ ( ( ( ( 1 + 1 ) ∈ ℤ ∧ ( 𝑦 + 1 ) ∈ ℤ ) ∧ ( ( 𝑗 + 1 ) ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( ( 𝑗 + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↔ ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) ) |
404 |
400 402 403
|
syl2an |
⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) → ( ( 𝑗 + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↔ ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) ) |
405 |
398 404
|
mpbird |
⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) → ( 𝑗 + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) |
406 |
395
|
eqcomd |
⊢ ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → 𝑗 = ( ( 𝑗 + 1 ) − 1 ) ) |
407 |
406
|
adantl |
⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) → 𝑗 = ( ( 𝑗 + 1 ) − 1 ) ) |
408 |
|
oveq1 |
⊢ ( 𝑛 = ( 𝑗 + 1 ) → ( 𝑛 − 1 ) = ( ( 𝑗 + 1 ) − 1 ) ) |
409 |
408
|
rspceeqv |
⊢ ( ( ( 𝑗 + 1 ) ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ∧ 𝑗 = ( ( 𝑗 + 1 ) − 1 ) ) → ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) ) |
410 |
405 407 409
|
syl2anc |
⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) → ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) ) |
411 |
410
|
ex |
⊢ ( 𝑦 ∈ ℤ → ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) → ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) ) ) |
412 |
|
simpr |
⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) |
413 |
|
elfzelz |
⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → 𝑛 ∈ ℤ ) |
414 |
413 115
|
jctir |
⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) ) |
415 |
|
fzsubel |
⊢ ( ( ( ( 1 + 1 ) ∈ ℤ ∧ ( 𝑦 + 1 ) ∈ ℤ ) ∧ ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) ) |
416 |
400 414 415
|
syl2an |
⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) ) |
417 |
412 416
|
mpbid |
⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) |
418 |
|
eleq1 |
⊢ ( 𝑗 = ( 𝑛 − 1 ) → ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) ) |
419 |
417 418
|
syl5ibrcom |
⊢ ( ( 𝑦 ∈ ℤ ∧ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) → ( 𝑗 = ( 𝑛 − 1 ) → 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) ) |
420 |
419
|
rexlimdva |
⊢ ( 𝑦 ∈ ℤ → ( ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) → 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ) ) |
421 |
411 420
|
impbid |
⊢ ( 𝑦 ∈ ℤ → ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ↔ ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) ) ) |
422 |
|
eqid |
⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) = ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) |
423 |
422
|
elrnmpt |
⊢ ( 𝑗 ∈ V → ( 𝑗 ∈ ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) ↔ ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) ) ) |
424 |
423
|
elv |
⊢ ( 𝑗 ∈ ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) ↔ ∃ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) 𝑗 = ( 𝑛 − 1 ) ) |
425 |
421 424
|
bitr4di |
⊢ ( 𝑦 ∈ ℤ → ( 𝑗 ∈ ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) ↔ 𝑗 ∈ ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) ) ) |
426 |
425
|
eqrdv |
⊢ ( 𝑦 ∈ ℤ → ( ( ( 1 + 1 ) − 1 ) ... ( ( 𝑦 + 1 ) − 1 ) ) = ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) ) |
427 |
391 426
|
eqtr3d |
⊢ ( 𝑦 ∈ ℤ → ( 1 ... 𝑦 ) = ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) ) |
428 |
386 427
|
syl |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 1 ... 𝑦 ) = ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) ) |
429 |
428
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑦 ) = ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) ) |
430 |
|
df-ima |
⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ran ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) |
431 |
|
uzid |
⊢ ( 1 ∈ ℤ → 1 ∈ ( ℤ≥ ‘ 1 ) ) |
432 |
|
peano2uz |
⊢ ( 1 ∈ ( ℤ≥ ‘ 1 ) → ( 1 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
433 |
115 431 432
|
mp2b |
⊢ ( 1 + 1 ) ∈ ( ℤ≥ ‘ 1 ) |
434 |
|
fzss1 |
⊢ ( ( 1 + 1 ) ∈ ( ℤ≥ ‘ 1 ) → ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... ( 𝑦 + 1 ) ) ) |
435 |
433 434
|
ax-mp |
⊢ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... ( 𝑦 + 1 ) ) |
436 |
435 272
|
sstrid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ⊆ ( 1 ... 𝑁 ) ) |
437 |
436
|
resmptd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) |
438 |
|
elfzle1 |
⊢ ( 1 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → ( 1 + 1 ) ≤ 1 ) |
439 |
171 438
|
mto |
⊢ ¬ 1 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) |
440 |
|
eleq1 |
⊢ ( 𝑛 = 1 → ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↔ 1 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) |
441 |
439 440
|
mtbiri |
⊢ ( 𝑛 = 1 → ¬ 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) |
442 |
441
|
necon2ai |
⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → 𝑛 ≠ 1 ) |
443 |
442 182
|
syl |
⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) → if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) = ( 𝑛 − 1 ) ) |
444 |
443
|
mpteq2ia |
⊢ ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) = ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) |
445 |
437 444
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) ) |
446 |
445
|
rneqd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ran ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) ) |
447 |
430 446
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) = ran ( 𝑛 ∈ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ↦ ( 𝑛 − 1 ) ) ) |
448 |
429 447
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 1 ... 𝑦 ) = ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) |
449 |
448
|
imaeq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) ) |
450 |
266
|
sneqd |
⊢ ( 𝜑 → { ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) } = { 𝑁 } ) |
451 |
|
fnsnfv |
⊢ ( ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) Fn ( 1 ... 𝑁 ) ∧ 1 ∈ ( 1 ... 𝑁 ) ) → { ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) } = ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) |
452 |
269 265 451
|
syl2anc |
⊢ ( 𝜑 → { ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ‘ 1 ) } = ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) |
453 |
450 452
|
eqtr3d |
⊢ ( 𝜑 → { 𝑁 } = ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) |
454 |
453
|
imaeq2d |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ { 𝑁 } ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) ) |
455 |
352 454
|
eqtrd |
⊢ ( 𝜑 → { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) ) |
456 |
455
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) ) |
457 |
449 456
|
uneq12d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( 1 + 1 ) ... ( 𝑦 + 1 ) ) ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ { 1 } ) ) ) ) |
458 |
379 385 457
|
3eqtr4a |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) |
459 |
458
|
xpeq1d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) × { 1 } ) ) |
460 |
|
xpundir |
⊢ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ∪ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) × { 1 } ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) |
461 |
459 460
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ) |
462 |
|
imaco |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) |
463 |
|
df-ima |
⊢ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ran ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) |
464 |
|
fzss1 |
⊢ ( ( ( 𝑦 + 1 ) + 1 ) ∈ ( ℤ≥ ‘ 1 ) → ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ) |
465 |
230 464
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ) |
466 |
465
|
resmptd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) |
467 |
|
1red |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 1 ∈ ℝ ) |
468 |
65
|
nnzd |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℤ ) |
469 |
468
|
peano2zd |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ℤ ) |
470 |
469
|
zred |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( 𝑦 + 1 ) + 1 ) ∈ ℝ ) |
471 |
65
|
nnge1d |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 1 ≤ ( 𝑦 + 1 ) ) |
472 |
467 217 470 471 218
|
lelttrd |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 1 < ( ( 𝑦 + 1 ) + 1 ) ) |
473 |
467 470
|
ltnled |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 1 < ( ( 𝑦 + 1 ) + 1 ) ↔ ¬ ( ( 𝑦 + 1 ) + 1 ) ≤ 1 ) ) |
474 |
472 473
|
mpbid |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ¬ ( ( 𝑦 + 1 ) + 1 ) ≤ 1 ) |
475 |
|
elfzle1 |
⊢ ( 1 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) → ( ( 𝑦 + 1 ) + 1 ) ≤ 1 ) |
476 |
474 475
|
nsyl |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ¬ 1 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) |
477 |
|
eleq1 |
⊢ ( 𝑛 = 1 → ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↔ 1 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) |
478 |
477
|
notbid |
⊢ ( 𝑛 = 1 → ( ¬ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↔ ¬ 1 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) |
479 |
476 478
|
syl5ibrcom |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑛 = 1 → ¬ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) |
480 |
479
|
necon2ad |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) → 𝑛 ≠ 1 ) ) |
481 |
480
|
imp |
⊢ ( ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ∧ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) → 𝑛 ≠ 1 ) |
482 |
481 182
|
syl |
⊢ ( ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ∧ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) → if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) = ( 𝑛 − 1 ) ) |
483 |
482
|
mpteq2dva |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) = ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ) |
484 |
483
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) = ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ) |
485 |
466 484
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ) |
486 |
485
|
rneqd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ran ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ↾ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ran ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ) |
487 |
463 486
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ran ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ) |
488 |
|
eqid |
⊢ ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) = ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) |
489 |
488
|
elrnmpt |
⊢ ( 𝑗 ∈ V → ( 𝑗 ∈ ran ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ↔ ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) ) ) |
490 |
489
|
elv |
⊢ ( 𝑗 ∈ ran ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ↔ ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) ) |
491 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) → 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) |
492 |
114 469
|
anim12ci |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 𝑦 + 1 ) + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) |
493 |
|
elfzelz |
⊢ ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) → 𝑛 ∈ ℤ ) |
494 |
493 115
|
jctir |
⊢ ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) → ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) ) |
495 |
|
fzsubel |
⊢ ( ( ( ( ( 𝑦 + 1 ) + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) ) |
496 |
492 494 495
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) → ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) ) |
497 |
491 496
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) → ( 𝑛 − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) |
498 |
|
eleq1 |
⊢ ( 𝑗 = ( 𝑛 − 1 ) → ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ↔ ( 𝑛 − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) ) |
499 |
497 498
|
syl5ibrcom |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) → ( 𝑗 = ( 𝑛 − 1 ) → 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) ) |
500 |
499
|
rexlimdva |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) → 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) ) |
501 |
|
elfzelz |
⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → 𝑗 ∈ ℤ ) |
502 |
501
|
zcnd |
⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → 𝑗 ∈ ℂ ) |
503 |
502 394
|
syl |
⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → ( ( 𝑗 + 1 ) − 1 ) = 𝑗 ) |
504 |
503
|
eleq1d |
⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → ( ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ↔ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) ) |
505 |
504
|
ibir |
⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) |
506 |
505
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) → ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) |
507 |
501
|
peano2zd |
⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → ( 𝑗 + 1 ) ∈ ℤ ) |
508 |
507 115
|
jctir |
⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → ( ( 𝑗 + 1 ) ∈ ℤ ∧ 1 ∈ ℤ ) ) |
509 |
|
fzsubel |
⊢ ( ( ( ( ( 𝑦 + 1 ) + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( ( 𝑗 + 1 ) ∈ ℤ ∧ 1 ∈ ℤ ) ) → ( ( 𝑗 + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↔ ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) ) |
510 |
492 508 509
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) → ( ( 𝑗 + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↔ ( ( 𝑗 + 1 ) − 1 ) ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) ) |
511 |
506 510
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) → ( 𝑗 + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) |
512 |
503
|
eqcomd |
⊢ ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → 𝑗 = ( ( 𝑗 + 1 ) − 1 ) ) |
513 |
512
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) → 𝑗 = ( ( 𝑗 + 1 ) − 1 ) ) |
514 |
408
|
rspceeqv |
⊢ ( ( ( 𝑗 + 1 ) ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ∧ 𝑗 = ( ( 𝑗 + 1 ) − 1 ) ) → ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) ) |
515 |
511 513 514
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) → ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) ) |
516 |
515
|
ex |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) → ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) ) ) |
517 |
500 516
|
impbid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ∃ 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) 𝑗 = ( 𝑛 − 1 ) ↔ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) ) |
518 |
490 517
|
syl5bb |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑗 ∈ ran ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) ↔ 𝑗 ∈ ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) ) |
519 |
518
|
eqrdv |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ran ( 𝑛 ∈ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ↦ ( 𝑛 − 1 ) ) = ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) ) |
520 |
65
|
nncnd |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑦 + 1 ) ∈ ℂ ) |
521 |
|
pncan1 |
⊢ ( ( 𝑦 + 1 ) ∈ ℂ → ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) = ( 𝑦 + 1 ) ) |
522 |
520 521
|
syl |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) = ( 𝑦 + 1 ) ) |
523 |
522
|
oveq1d |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) = ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) |
524 |
523
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 𝑦 + 1 ) + 1 ) − 1 ) ... ( 𝑁 − 1 ) ) = ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) |
525 |
487 519 524
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) |
526 |
525
|
imaeq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) |
527 |
462 526
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ) |
528 |
527
|
xpeq1d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) |
529 |
461 528
|
uneq12d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ) |
530 |
|
un23 |
⊢ ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) |
531 |
529 530
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ) |
532 |
531
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) ) |
533 |
532
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 1 } ) ) ‘ 𝑛 ) ) |
534 |
|
imaundi |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ { 𝑁 } ) ) |
535 |
|
fzsplit2 |
⊢ ( ( ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ ( 𝑦 + 1 ) ) ∧ 𝑁 ∈ ( ℤ≥ ‘ ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) |
536 |
232 200 535
|
syl2anr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) ) |
537 |
206
|
uneq2d |
⊢ ( 𝜑 → ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ) |
538 |
537
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ ( ( ( 𝑁 − 1 ) + 1 ) ... 𝑁 ) ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ) |
539 |
536 538
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑦 + 1 ) ... 𝑁 ) = ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ) |
540 |
539
|
imaeq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ) ) |
541 |
352
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ { 𝑁 } ) ) |
542 |
541
|
uneq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ∪ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ { 𝑁 } ) ) ) |
543 |
534 540 542
|
3eqtr4a |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ∪ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) ) |
544 |
543
|
xpeq1d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ∪ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) × { 0 } ) ) |
545 |
|
xpundir |
⊢ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) ∪ { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } ) × { 0 } ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) |
546 |
544 545
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ) |
547 |
546
|
uneq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ) ) |
548 |
|
unass |
⊢ ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ) |
549 |
547 548
|
eqtr4di |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ) |
550 |
549
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) ) |
551 |
550
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... ( 𝑁 − 1 ) ) ) × { 0 } ) ) ∪ ( { ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) } × { 0 } ) ) ‘ 𝑛 ) ) |
552 |
374 533 551
|
3eqtr4d |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) |
553 |
316 552
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) ∧ ¬ 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) ) → ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − 0 ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) |
554 |
250 252 314 553
|
ifbothda |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) = ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) |
555 |
554
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) = ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) |
556 |
248 555
|
eqtr2d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) = ( ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) |
557 |
556
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) + ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ) |
558 |
93 557
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ) |
559 |
53
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑦 ∈ ℝ ) |
560 |
160
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑁 − 1 ) ∈ ℝ ) |
561 |
158
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑁 ∈ ℝ ) |
562 |
|
elfzle2 |
⊢ ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑦 ≤ ( 𝑁 − 1 ) ) |
563 |
562
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑦 ≤ ( 𝑁 − 1 ) ) |
564 |
159
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑁 − 1 ) < 𝑁 ) |
565 |
559 560 561 563 564
|
lelttrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑦 < 𝑁 ) |
566 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 2nd ‘ 𝑇 ) = 𝑁 ) |
567 |
565 566
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → 𝑦 < ( 2nd ‘ 𝑇 ) ) |
568 |
567
|
iftrued |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) = 𝑦 ) |
569 |
568
|
csbeq1d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ⦋ if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ 𝑦 / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
570 |
|
vex |
⊢ 𝑦 ∈ V |
571 |
|
oveq2 |
⊢ ( 𝑗 = 𝑦 → ( 1 ... 𝑗 ) = ( 1 ... 𝑦 ) ) |
572 |
571
|
imaeq2d |
⊢ ( 𝑗 = 𝑦 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) ) |
573 |
572
|
xpeq1d |
⊢ ( 𝑗 = 𝑦 → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ) |
574 |
|
oveq1 |
⊢ ( 𝑗 = 𝑦 → ( 𝑗 + 1 ) = ( 𝑦 + 1 ) ) |
575 |
574
|
oveq1d |
⊢ ( 𝑗 = 𝑦 → ( ( 𝑗 + 1 ) ... 𝑁 ) = ( ( 𝑦 + 1 ) ... 𝑁 ) ) |
576 |
575
|
imaeq2d |
⊢ ( 𝑗 = 𝑦 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) ) |
577 |
576
|
xpeq1d |
⊢ ( 𝑗 = 𝑦 → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) |
578 |
573 577
|
uneq12d |
⊢ ( 𝑗 = 𝑦 → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) |
579 |
578
|
oveq2d |
⊢ ( 𝑗 = 𝑦 → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
580 |
570 579
|
csbie |
⊢ ⦋ 𝑦 / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) |
581 |
569 580
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ⦋ if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑦 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑦 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
582 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ∈ V ) |
583 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) ∧ 𝑛 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ∈ V ) |
584 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) ) |
585 |
245
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ) |
586 |
|
nfcv |
⊢ Ⅎ 𝑛 ( 2nd ‘ ( 1st ‘ 𝑇 ) ) |
587 |
|
nfmpt1 |
⊢ Ⅎ 𝑛 ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) |
588 |
586 587
|
nfco |
⊢ Ⅎ 𝑛 ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) |
589 |
|
nfcv |
⊢ Ⅎ 𝑛 ( 1 ... ( 𝑦 + 1 ) ) |
590 |
588 589
|
nfima |
⊢ Ⅎ 𝑛 ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) |
591 |
|
nfcv |
⊢ Ⅎ 𝑛 { 1 } |
592 |
590 591
|
nfxp |
⊢ Ⅎ 𝑛 ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) |
593 |
|
nfcv |
⊢ Ⅎ 𝑛 ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) |
594 |
588 593
|
nfima |
⊢ Ⅎ 𝑛 ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) |
595 |
|
nfcv |
⊢ Ⅎ 𝑛 { 0 } |
596 |
594 595
|
nfxp |
⊢ Ⅎ 𝑛 ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) |
597 |
592 596
|
nfun |
⊢ Ⅎ 𝑛 ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) |
598 |
597
|
dffn5f |
⊢ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ↔ ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) |
599 |
585 598
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) |
600 |
89 582 583 584 599
|
offval2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) ∘f + ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) + ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑛 ) ) ) ) |
601 |
558 581 600
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) → ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) ∘f + ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
602 |
601
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) ∘f + ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) |
603 |
23 602
|
eqtr4d |
⊢ ( 𝜑 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ( ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑛 ) − if ( 𝑛 = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑁 ) , 1 , 0 ) ) ) ∘f + ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( 1 ... ( 𝑦 + 1 ) ) ) × { 1 } ) ∪ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∘ ( 𝑛 ∈ ( 1 ... 𝑁 ) ↦ if ( 𝑛 = 1 , 𝑁 , ( 𝑛 − 1 ) ) ) ) “ ( ( ( 𝑦 + 1 ) + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) |