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 |
|
poimirlem12.2 |
⊢ ( 𝜑 → 𝑇 ∈ 𝑆 ) |
5 |
|
poimirlem12.3 |
⊢ ( 𝜑 → ( 2nd ‘ 𝑇 ) = 𝑁 ) |
6 |
|
poimirlem12.4 |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
7 |
|
poimirlem12.5 |
⊢ ( 𝜑 → ( 2nd ‘ 𝑈 ) = 𝑁 ) |
8 |
|
poimirlem12.6 |
⊢ ( 𝜑 → 𝑀 ∈ ( 0 ... ( 𝑁 − 1 ) ) ) |
9 |
|
eldif |
⊢ ( 𝑦 ∈ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∖ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ↔ ( 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∧ ¬ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) |
10 |
|
imassrn |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ⊆ ran ( 2nd ‘ ( 1st ‘ 𝑇 ) ) |
11 |
|
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 ) ) ) |
12 |
11 2
|
eleq2s |
⊢ ( 𝑇 ∈ 𝑆 → 𝑇 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ) |
13 |
|
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 ) } ) ) |
14 |
4 12 13
|
3syl |
⊢ ( 𝜑 → ( 1st ‘ 𝑇 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) |
15 |
|
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 ) } ) |
16 |
14 15
|
syl |
⊢ ( 𝜑 → ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) |
17 |
|
fvex |
⊢ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∈ V |
18 |
|
f1oeq1 |
⊢ ( 𝑓 = ( 2nd ‘ ( 1st ‘ 𝑇 ) ) → ( 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) ) |
19 |
17 18
|
elab |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ↔ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) |
20 |
16 19
|
sylib |
⊢ ( 𝜑 → ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) |
21 |
|
f1of |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) ) |
22 |
|
frn |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) → ran ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ⊆ ( 1 ... 𝑁 ) ) |
23 |
20 21 22
|
3syl |
⊢ ( 𝜑 → ran ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ⊆ ( 1 ... 𝑁 ) ) |
24 |
10 23
|
sstrid |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ⊆ ( 1 ... 𝑁 ) ) |
25 |
|
elrabi |
⊢ ( 𝑈 ∈ { 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 } ) ) ) ) } → 𝑈 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ) |
26 |
25 2
|
eleq2s |
⊢ ( 𝑈 ∈ 𝑆 → 𝑈 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ) |
27 |
|
xp1st |
⊢ ( 𝑈 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) → ( 1st ‘ 𝑈 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) |
28 |
6 26 27
|
3syl |
⊢ ( 𝜑 → ( 1st ‘ 𝑈 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) ) |
29 |
|
xp2nd |
⊢ ( ( 1st ‘ 𝑈 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) → ( 2nd ‘ ( 1st ‘ 𝑈 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) |
30 |
28 29
|
syl |
⊢ ( 𝜑 → ( 2nd ‘ ( 1st ‘ 𝑈 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) |
31 |
|
fvex |
⊢ ( 2nd ‘ ( 1st ‘ 𝑈 ) ) ∈ V |
32 |
|
f1oeq1 |
⊢ ( 𝑓 = ( 2nd ‘ ( 1st ‘ 𝑈 ) ) → ( 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( 2nd ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) ) |
33 |
31 32
|
elab |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) ∈ { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ↔ ( 2nd ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) |
34 |
30 33
|
sylib |
⊢ ( 𝜑 → ( 2nd ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) |
35 |
|
f1ofo |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 2nd ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) ) |
36 |
|
foima |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) ) |
37 |
34 35 36
|
3syl |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) ) |
38 |
24 37
|
sseqtrrd |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ⊆ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑁 ) ) ) |
39 |
38
|
ssdifd |
⊢ ( 𝜑 → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∖ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ⊆ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑁 ) ) ∖ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) |
40 |
|
dff1o3 |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) ∧ Fun ◡ ( 2nd ‘ ( 1st ‘ 𝑈 ) ) ) ) |
41 |
40
|
simprbi |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → Fun ◡ ( 2nd ‘ ( 1st ‘ 𝑈 ) ) ) |
42 |
|
imadif |
⊢ ( Fun ◡ ( 2nd ‘ ( 1st ‘ 𝑈 ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 1 ... 𝑁 ) ∖ ( 1 ... 𝑀 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑁 ) ) ∖ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) |
43 |
34 41 42
|
3syl |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 1 ... 𝑁 ) ∖ ( 1 ... 𝑀 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑁 ) ) ∖ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) |
44 |
|
difun2 |
⊢ ( ( ( ( 𝑀 + 1 ) ... 𝑁 ) ∪ ( 1 ... 𝑀 ) ) ∖ ( 1 ... 𝑀 ) ) = ( ( ( 𝑀 + 1 ) ... 𝑁 ) ∖ ( 1 ... 𝑀 ) ) |
45 |
|
elfznn0 |
⊢ ( 𝑀 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑀 ∈ ℕ0 ) |
46 |
|
nn0p1nn |
⊢ ( 𝑀 ∈ ℕ0 → ( 𝑀 + 1 ) ∈ ℕ ) |
47 |
8 45 46
|
3syl |
⊢ ( 𝜑 → ( 𝑀 + 1 ) ∈ ℕ ) |
48 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
49 |
47 48
|
eleqtrdi |
⊢ ( 𝜑 → ( 𝑀 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
50 |
1
|
nncnd |
⊢ ( 𝜑 → 𝑁 ∈ ℂ ) |
51 |
|
npcan1 |
⊢ ( 𝑁 ∈ ℂ → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 ) |
52 |
50 51
|
syl |
⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) = 𝑁 ) |
53 |
|
elfzuz3 |
⊢ ( 𝑀 ∈ ( 0 ... ( 𝑁 − 1 ) ) → ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
54 |
|
peano2uz |
⊢ ( ( 𝑁 − 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
55 |
8 53 54
|
3syl |
⊢ ( 𝜑 → ( ( 𝑁 − 1 ) + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
56 |
52 55
|
eqeltrrd |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
57 |
|
fzsplit2 |
⊢ ( ( ( 𝑀 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( 1 ... 𝑁 ) = ( ( 1 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
58 |
49 56 57
|
syl2anc |
⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( ( 1 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
59 |
|
uncom |
⊢ ( ( 1 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ( ( ( 𝑀 + 1 ) ... 𝑁 ) ∪ ( 1 ... 𝑀 ) ) |
60 |
58 59
|
eqtrdi |
⊢ ( 𝜑 → ( 1 ... 𝑁 ) = ( ( ( 𝑀 + 1 ) ... 𝑁 ) ∪ ( 1 ... 𝑀 ) ) ) |
61 |
60
|
difeq1d |
⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) ∖ ( 1 ... 𝑀 ) ) = ( ( ( ( 𝑀 + 1 ) ... 𝑁 ) ∪ ( 1 ... 𝑀 ) ) ∖ ( 1 ... 𝑀 ) ) ) |
62 |
|
incom |
⊢ ( ( ( 𝑀 + 1 ) ... 𝑁 ) ∩ ( 1 ... 𝑀 ) ) = ( ( 1 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
63 |
8 45
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
64 |
63
|
nn0red |
⊢ ( 𝜑 → 𝑀 ∈ ℝ ) |
65 |
64
|
ltp1d |
⊢ ( 𝜑 → 𝑀 < ( 𝑀 + 1 ) ) |
66 |
|
fzdisj |
⊢ ( 𝑀 < ( 𝑀 + 1 ) → ( ( 1 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ∅ ) |
67 |
65 66
|
syl |
⊢ ( 𝜑 → ( ( 1 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ∅ ) |
68 |
62 67
|
syl5eq |
⊢ ( 𝜑 → ( ( ( 𝑀 + 1 ) ... 𝑁 ) ∩ ( 1 ... 𝑀 ) ) = ∅ ) |
69 |
|
disj3 |
⊢ ( ( ( ( 𝑀 + 1 ) ... 𝑁 ) ∩ ( 1 ... 𝑀 ) ) = ∅ ↔ ( ( 𝑀 + 1 ) ... 𝑁 ) = ( ( ( 𝑀 + 1 ) ... 𝑁 ) ∖ ( 1 ... 𝑀 ) ) ) |
70 |
68 69
|
sylib |
⊢ ( 𝜑 → ( ( 𝑀 + 1 ) ... 𝑁 ) = ( ( ( 𝑀 + 1 ) ... 𝑁 ) ∖ ( 1 ... 𝑀 ) ) ) |
71 |
44 61 70
|
3eqtr4a |
⊢ ( 𝜑 → ( ( 1 ... 𝑁 ) ∖ ( 1 ... 𝑀 ) ) = ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
72 |
71
|
imaeq2d |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 1 ... 𝑁 ) ∖ ( 1 ... 𝑀 ) ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
73 |
43 72
|
eqtr3d |
⊢ ( 𝜑 → ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑁 ) ) ∖ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
74 |
39 73
|
sseqtrd |
⊢ ( 𝜑 → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∖ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ⊆ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
75 |
74
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∖ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) → 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
76 |
9 75
|
sylan2br |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∧ ¬ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) → 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
77 |
|
fveq2 |
⊢ ( 𝑡 = 𝑈 → ( 2nd ‘ 𝑡 ) = ( 2nd ‘ 𝑈 ) ) |
78 |
77
|
breq2d |
⊢ ( 𝑡 = 𝑈 → ( 𝑦 < ( 2nd ‘ 𝑡 ) ↔ 𝑦 < ( 2nd ‘ 𝑈 ) ) ) |
79 |
78
|
ifbid |
⊢ ( 𝑡 = 𝑈 → if ( 𝑦 < ( 2nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) = if ( 𝑦 < ( 2nd ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) ) |
80 |
79
|
csbeq1d |
⊢ ( 𝑡 = 𝑈 → ⦋ 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 ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑡 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
81 |
|
2fveq3 |
⊢ ( 𝑡 = 𝑈 → ( 1st ‘ ( 1st ‘ 𝑡 ) ) = ( 1st ‘ ( 1st ‘ 𝑈 ) ) ) |
82 |
|
2fveq3 |
⊢ ( 𝑡 = 𝑈 → ( 2nd ‘ ( 1st ‘ 𝑡 ) ) = ( 2nd ‘ ( 1st ‘ 𝑈 ) ) ) |
83 |
82
|
imaeq1d |
⊢ ( 𝑡 = 𝑈 → ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) ) |
84 |
83
|
xpeq1d |
⊢ ( 𝑡 = 𝑈 → ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ) |
85 |
82
|
imaeq1d |
⊢ ( 𝑡 = 𝑈 → ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) |
86 |
85
|
xpeq1d |
⊢ ( 𝑡 = 𝑈 → ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) |
87 |
84 86
|
uneq12d |
⊢ ( 𝑡 = 𝑈 → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) |
88 |
81 87
|
oveq12d |
⊢ ( 𝑡 = 𝑈 → ( ( 1st ‘ ( 1st ‘ 𝑡 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
89 |
88
|
csbeq2dv |
⊢ ( 𝑡 = 𝑈 → ⦋ if ( 𝑦 < ( 2nd ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑡 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ if ( 𝑦 < ( 2nd ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
90 |
80 89
|
eqtrd |
⊢ ( 𝑡 = 𝑈 → ⦋ 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 ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
91 |
90
|
mpteq2dv |
⊢ ( 𝑡 = 𝑈 → ( 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 ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) |
92 |
91
|
eqeq2d |
⊢ ( 𝑡 = 𝑈 → ( 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 ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) ) |
93 |
92 2
|
elrab2 |
⊢ ( 𝑈 ∈ 𝑆 ↔ ( 𝑈 ∈ ( ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) × ( 0 ... 𝑁 ) ) ∧ 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2nd ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) ) |
94 |
93
|
simprbi |
⊢ ( 𝑈 ∈ 𝑆 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2nd ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) |
95 |
6 94
|
syl |
⊢ ( 𝜑 → 𝐹 = ( 𝑦 ∈ ( 0 ... ( 𝑁 − 1 ) ) ↦ ⦋ if ( 𝑦 < ( 2nd ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) ) |
96 |
|
breq1 |
⊢ ( 𝑦 = 𝑀 → ( 𝑦 < ( 2nd ‘ 𝑈 ) ↔ 𝑀 < ( 2nd ‘ 𝑈 ) ) ) |
97 |
|
id |
⊢ ( 𝑦 = 𝑀 → 𝑦 = 𝑀 ) |
98 |
96 97
|
ifbieq1d |
⊢ ( 𝑦 = 𝑀 → if ( 𝑦 < ( 2nd ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) = if ( 𝑀 < ( 2nd ‘ 𝑈 ) , 𝑀 , ( 𝑦 + 1 ) ) ) |
99 |
1
|
nnred |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
100 |
|
peano2rem |
⊢ ( 𝑁 ∈ ℝ → ( 𝑁 − 1 ) ∈ ℝ ) |
101 |
99 100
|
syl |
⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ℝ ) |
102 |
|
elfzle2 |
⊢ ( 𝑀 ∈ ( 0 ... ( 𝑁 − 1 ) ) → 𝑀 ≤ ( 𝑁 − 1 ) ) |
103 |
8 102
|
syl |
⊢ ( 𝜑 → 𝑀 ≤ ( 𝑁 − 1 ) ) |
104 |
99
|
ltm1d |
⊢ ( 𝜑 → ( 𝑁 − 1 ) < 𝑁 ) |
105 |
64 101 99 103 104
|
lelttrd |
⊢ ( 𝜑 → 𝑀 < 𝑁 ) |
106 |
105 7
|
breqtrrd |
⊢ ( 𝜑 → 𝑀 < ( 2nd ‘ 𝑈 ) ) |
107 |
106
|
iftrued |
⊢ ( 𝜑 → if ( 𝑀 < ( 2nd ‘ 𝑈 ) , 𝑀 , ( 𝑦 + 1 ) ) = 𝑀 ) |
108 |
98 107
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝑀 ) → if ( 𝑦 < ( 2nd ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) = 𝑀 ) |
109 |
108
|
csbeq1d |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝑀 ) → ⦋ if ( 𝑦 < ( 2nd ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ⦋ 𝑀 / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
110 |
|
oveq2 |
⊢ ( 𝑗 = 𝑀 → ( 1 ... 𝑗 ) = ( 1 ... 𝑀 ) ) |
111 |
110
|
imaeq2d |
⊢ ( 𝑗 = 𝑀 → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) |
112 |
111
|
xpeq1d |
⊢ ( 𝑗 = 𝑀 → ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ) |
113 |
|
oveq1 |
⊢ ( 𝑗 = 𝑀 → ( 𝑗 + 1 ) = ( 𝑀 + 1 ) ) |
114 |
113
|
oveq1d |
⊢ ( 𝑗 = 𝑀 → ( ( 𝑗 + 1 ) ... 𝑁 ) = ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
115 |
114
|
imaeq2d |
⊢ ( 𝑗 = 𝑀 → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
116 |
115
|
xpeq1d |
⊢ ( 𝑗 = 𝑀 → ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) |
117 |
112 116
|
uneq12d |
⊢ ( 𝑗 = 𝑀 → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) |
118 |
117
|
oveq2d |
⊢ ( 𝑗 = 𝑀 → ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
119 |
118
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 = 𝑀 ) → ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
120 |
8 119
|
csbied |
⊢ ( 𝜑 → ⦋ 𝑀 / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
121 |
120
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝑀 ) → ⦋ 𝑀 / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
122 |
109 121
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝑀 ) → ⦋ if ( 𝑦 < ( 2nd ‘ 𝑈 ) , 𝑦 , ( 𝑦 + 1 ) ) / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) = ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
123 |
|
ovexd |
⊢ ( 𝜑 → ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∈ V ) |
124 |
95 122 8 123
|
fvmptd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) = ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
125 |
124
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝑀 ) ‘ 𝑦 ) = ( ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑦 ) ) |
126 |
125
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( 𝐹 ‘ 𝑀 ) ‘ 𝑦 ) = ( ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑦 ) ) |
127 |
|
imassrn |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ⊆ ran ( 2nd ‘ ( 1st ‘ 𝑈 ) ) |
128 |
|
f1of |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 2nd ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) ) |
129 |
|
frn |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) ⟶ ( 1 ... 𝑁 ) → ran ( 2nd ‘ ( 1st ‘ 𝑈 ) ) ⊆ ( 1 ... 𝑁 ) ) |
130 |
34 128 129
|
3syl |
⊢ ( 𝜑 → ran ( 2nd ‘ ( 1st ‘ 𝑈 ) ) ⊆ ( 1 ... 𝑁 ) ) |
131 |
127 130
|
sstrid |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ⊆ ( 1 ... 𝑁 ) ) |
132 |
131
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → 𝑦 ∈ ( 1 ... 𝑁 ) ) |
133 |
|
xp1st |
⊢ ( ( 1st ‘ 𝑈 ) ∈ ( ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) × { 𝑓 ∣ 𝑓 : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) } ) → ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) ) |
134 |
|
elmapfn |
⊢ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) → ( 1st ‘ ( 1st ‘ 𝑈 ) ) Fn ( 1 ... 𝑁 ) ) |
135 |
28 133 134
|
3syl |
⊢ ( 𝜑 → ( 1st ‘ ( 1st ‘ 𝑈 ) ) Fn ( 1 ... 𝑁 ) ) |
136 |
135
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( 1st ‘ ( 1st ‘ 𝑈 ) ) Fn ( 1 ... 𝑁 ) ) |
137 |
|
1ex |
⊢ 1 ∈ V |
138 |
|
fnconstg |
⊢ ( 1 ∈ V → ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) |
139 |
137 138
|
ax-mp |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) |
140 |
|
c0ex |
⊢ 0 ∈ V |
141 |
|
fnconstg |
⊢ ( 0 ∈ V → ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
142 |
140 141
|
ax-mp |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
143 |
139 142
|
pm3.2i |
⊢ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
144 |
|
imain |
⊢ ( Fun ◡ ( 2nd ‘ ( 1st ‘ 𝑈 ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 1 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ) |
145 |
34 41 144
|
3syl |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 1 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ) |
146 |
67
|
imaeq2d |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 1 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ∅ ) ) |
147 |
|
ima0 |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ∅ ) = ∅ |
148 |
146 147
|
eqtrdi |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 1 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ∅ ) |
149 |
145 148
|
eqtr3d |
⊢ ( 𝜑 → ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ∅ ) |
150 |
|
fnun |
⊢ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ∅ ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ) |
151 |
143 149 150
|
sylancr |
⊢ ( 𝜑 → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ) |
152 |
|
imaundi |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 1 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
153 |
58
|
imaeq2d |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑁 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 1 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ) |
154 |
153 37
|
eqtr3d |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 1 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) ) |
155 |
152 154
|
syl5eqr |
⊢ ( 𝜑 → ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) ) |
156 |
155
|
fneq2d |
⊢ ( 𝜑 → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ↔ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ) ) |
157 |
151 156
|
mpbid |
⊢ ( 𝜑 → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ) |
158 |
157
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ) |
159 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( 1 ... 𝑁 ) ∈ V ) |
160 |
|
inidm |
⊢ ( ( 1 ... 𝑁 ) ∩ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) |
161 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) = ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) ) |
162 |
|
fvun2 |
⊢ ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ∧ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ∅ ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑦 ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ 𝑦 ) ) |
163 |
139 142 162
|
mp3an12 |
⊢ ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ∅ ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑦 ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ 𝑦 ) ) |
164 |
149 163
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑦 ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ 𝑦 ) ) |
165 |
140
|
fvconst2 |
⊢ ( 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ 𝑦 ) = 0 ) |
166 |
165
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ‘ 𝑦 ) = 0 ) |
167 |
164 166
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑦 ) = 0 ) |
168 |
167
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑦 ) = 0 ) |
169 |
136 158 159 159 160 161 168
|
ofval |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑦 ) = ( ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) + 0 ) ) |
170 |
132 169
|
mpdan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑦 ) = ( ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) + 0 ) ) |
171 |
|
elmapi |
⊢ ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) → ( 1st ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ..^ 𝐾 ) ) |
172 |
28 133 171
|
3syl |
⊢ ( 𝜑 → ( 1st ‘ ( 1st ‘ 𝑈 ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ..^ 𝐾 ) ) |
173 |
172
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) ∈ ( 0 ..^ 𝐾 ) ) |
174 |
|
elfzonn0 |
⊢ ( ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) ∈ ( 0 ..^ 𝐾 ) → ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) ∈ ℕ0 ) |
175 |
173 174
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) ∈ ℕ0 ) |
176 |
175
|
nn0cnd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) ∈ ℂ ) |
177 |
176
|
addid1d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) + 0 ) = ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) ) |
178 |
132 177
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) + 0 ) = ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) ) |
179 |
126 170 178
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) → ( ( 𝐹 ‘ 𝑀 ) ‘ 𝑦 ) = ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) ) |
180 |
76 179
|
syldan |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∧ ¬ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) → ( ( 𝐹 ‘ 𝑀 ) ‘ 𝑦 ) = ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) ) |
181 |
|
fveq2 |
⊢ ( 𝑡 = 𝑇 → ( 2nd ‘ 𝑡 ) = ( 2nd ‘ 𝑇 ) ) |
182 |
181
|
breq2d |
⊢ ( 𝑡 = 𝑇 → ( 𝑦 < ( 2nd ‘ 𝑡 ) ↔ 𝑦 < ( 2nd ‘ 𝑇 ) ) ) |
183 |
182
|
ifbid |
⊢ ( 𝑡 = 𝑇 → if ( 𝑦 < ( 2nd ‘ 𝑡 ) , 𝑦 , ( 𝑦 + 1 ) ) = if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) ) |
184 |
183
|
csbeq1d |
⊢ ( 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 } ) ) ) ) |
185 |
|
2fveq3 |
⊢ ( 𝑡 = 𝑇 → ( 1st ‘ ( 1st ‘ 𝑡 ) ) = ( 1st ‘ ( 1st ‘ 𝑇 ) ) ) |
186 |
|
2fveq3 |
⊢ ( 𝑡 = 𝑇 → ( 2nd ‘ ( 1st ‘ 𝑡 ) ) = ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ) |
187 |
186
|
imaeq1d |
⊢ ( 𝑡 = 𝑇 → ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) ) |
188 |
187
|
xpeq1d |
⊢ ( 𝑡 = 𝑇 → ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ) |
189 |
186
|
imaeq1d |
⊢ ( 𝑡 = 𝑇 → ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) ) |
190 |
189
|
xpeq1d |
⊢ ( 𝑡 = 𝑇 → ( ( ( 2nd ‘ ( 1st ‘ 𝑡 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) |
191 |
188 190
|
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 } ) ) ) |
192 |
185 191
|
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 } ) ) ) ) |
193 |
192
|
csbeq2dv |
⊢ ( 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 } ) ) ) ) |
194 |
184 193
|
eqtrd |
⊢ ( 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 } ) ) ) ) |
195 |
194
|
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 } ) ) ) ) ) |
196 |
195
|
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 } ) ) ) ) ) ) |
197 |
196 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 } ) ) ) ) ) ) |
198 |
197
|
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 } ) ) ) ) ) |
199 |
4 198
|
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 } ) ) ) ) ) |
200 |
|
breq1 |
⊢ ( 𝑦 = 𝑀 → ( 𝑦 < ( 2nd ‘ 𝑇 ) ↔ 𝑀 < ( 2nd ‘ 𝑇 ) ) ) |
201 |
200 97
|
ifbieq1d |
⊢ ( 𝑦 = 𝑀 → if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) = if ( 𝑀 < ( 2nd ‘ 𝑇 ) , 𝑀 , ( 𝑦 + 1 ) ) ) |
202 |
105 5
|
breqtrrd |
⊢ ( 𝜑 → 𝑀 < ( 2nd ‘ 𝑇 ) ) |
203 |
202
|
iftrued |
⊢ ( 𝜑 → if ( 𝑀 < ( 2nd ‘ 𝑇 ) , 𝑀 , ( 𝑦 + 1 ) ) = 𝑀 ) |
204 |
201 203
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝑀 ) → if ( 𝑦 < ( 2nd ‘ 𝑇 ) , 𝑦 , ( 𝑦 + 1 ) ) = 𝑀 ) |
205 |
204
|
csbeq1d |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝑀 ) → ⦋ 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 } ) ) ) = ⦋ 𝑀 / 𝑗 ⦌ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
206 |
110
|
imaeq2d |
⊢ ( 𝑗 = 𝑀 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) |
207 |
206
|
xpeq1d |
⊢ ( 𝑗 = 𝑀 → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ) |
208 |
114
|
imaeq2d |
⊢ ( 𝑗 = 𝑀 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
209 |
208
|
xpeq1d |
⊢ ( 𝑗 = 𝑀 → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) |
210 |
207 209
|
uneq12d |
⊢ ( 𝑗 = 𝑀 → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑗 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑗 + 1 ) ... 𝑁 ) ) × { 0 } ) ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) |
211 |
210
|
oveq2d |
⊢ ( 𝑗 = 𝑀 → ( ( 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 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
212 |
211
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 = 𝑀 ) → ( ( 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 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
213 |
8 212
|
csbied |
⊢ ( 𝜑 → ⦋ 𝑀 / 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 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
214 |
213
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝑀 ) → ⦋ 𝑀 / 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 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
215 |
205 214
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝑀 ) → ⦋ 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 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
216 |
|
ovexd |
⊢ ( 𝜑 → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ∈ V ) |
217 |
199 215 8 216
|
fvmptd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) = ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ) |
218 |
217
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝑀 ) ‘ 𝑦 ) = ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑦 ) ) |
219 |
218
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) → ( ( 𝐹 ‘ 𝑀 ) ‘ 𝑦 ) = ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑦 ) ) |
220 |
24
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) → 𝑦 ∈ ( 1 ... 𝑁 ) ) |
221 |
|
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 ) ) ) |
222 |
|
elmapfn |
⊢ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) → ( 1st ‘ ( 1st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ) |
223 |
14 221 222
|
3syl |
⊢ ( 𝜑 → ( 1st ‘ ( 1st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ) |
224 |
223
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) → ( 1st ‘ ( 1st ‘ 𝑇 ) ) Fn ( 1 ... 𝑁 ) ) |
225 |
|
fnconstg |
⊢ ( 1 ∈ V → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) |
226 |
137 225
|
ax-mp |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) |
227 |
|
fnconstg |
⊢ ( 0 ∈ V → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
228 |
140 227
|
ax-mp |
⊢ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
229 |
226 228
|
pm3.2i |
⊢ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
230 |
|
dff1o3 |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ↔ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) ∧ Fun ◡ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ) ) |
231 |
230
|
simprbi |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → Fun ◡ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) ) |
232 |
|
imain |
⊢ ( Fun ◡ ( 2nd ‘ ( 1st ‘ 𝑇 ) ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ) |
233 |
20 231 232
|
3syl |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ) |
234 |
67
|
imaeq2d |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ∅ ) ) |
235 |
|
ima0 |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ∅ ) = ∅ |
236 |
234 235
|
eqtrdi |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ∅ ) |
237 |
233 236
|
eqtr3d |
⊢ ( 𝜑 → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ∅ ) |
238 |
|
fnun |
⊢ ( ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ∅ ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ) |
239 |
229 237 238
|
sylancr |
⊢ ( 𝜑 → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ) |
240 |
|
imaundi |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) |
241 |
58
|
imaeq2d |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ) |
242 |
|
f1ofo |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) → ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) ) |
243 |
|
foima |
⊢ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) –onto→ ( 1 ... 𝑁 ) → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) ) |
244 |
20 242 243
|
3syl |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑁 ) ) = ( 1 ... 𝑁 ) ) |
245 |
241 244
|
eqtr3d |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 1 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) ) |
246 |
240 245
|
syl5eqr |
⊢ ( 𝜑 → ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ( 1 ... 𝑁 ) ) |
247 |
246
|
fneq2d |
⊢ ( 𝜑 → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∪ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) ↔ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ) ) |
248 |
239 247
|
mpbid |
⊢ ( 𝜑 → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ) |
249 |
248
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) Fn ( 1 ... 𝑁 ) ) |
250 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) → ( 1 ... 𝑁 ) ∈ V ) |
251 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) = ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) ) |
252 |
|
fvun1 |
⊢ ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∧ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) Fn ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ∧ ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ∅ ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑦 ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ‘ 𝑦 ) ) |
253 |
226 228 252
|
mp3an12 |
⊢ ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∩ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) ) = ∅ ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑦 ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ‘ 𝑦 ) ) |
254 |
237 253
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑦 ) = ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ‘ 𝑦 ) ) |
255 |
137
|
fvconst2 |
⊢ ( 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ‘ 𝑦 ) = 1 ) |
256 |
255
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) → ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ‘ 𝑦 ) = 1 ) |
257 |
254 256
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑦 ) = 1 ) |
258 |
257
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ‘ 𝑦 ) = 1 ) |
259 |
224 249 250 250 160 251 258
|
ofval |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑦 ) = ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) + 1 ) ) |
260 |
220 259
|
mpdan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∘f + ( ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) × { 1 } ) ∪ ( ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( ( 𝑀 + 1 ) ... 𝑁 ) ) × { 0 } ) ) ) ‘ 𝑦 ) = ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) + 1 ) ) |
261 |
219 260
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) → ( ( 𝐹 ‘ 𝑀 ) ‘ 𝑦 ) = ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) + 1 ) ) |
262 |
261
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∧ ¬ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) → ( ( 𝐹 ‘ 𝑀 ) ‘ 𝑦 ) = ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) + 1 ) ) |
263 |
1
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝑁 ) |
264 |
263 7
|
breqtrrd |
⊢ ( 𝜑 → 0 < ( 2nd ‘ 𝑈 ) ) |
265 |
1 2 6 264
|
poimirlem5 |
⊢ ( 𝜑 → ( 𝐹 ‘ 0 ) = ( 1st ‘ ( 1st ‘ 𝑈 ) ) ) |
266 |
263 5
|
breqtrrd |
⊢ ( 𝜑 → 0 < ( 2nd ‘ 𝑇 ) ) |
267 |
1 2 4 266
|
poimirlem5 |
⊢ ( 𝜑 → ( 𝐹 ‘ 0 ) = ( 1st ‘ ( 1st ‘ 𝑇 ) ) ) |
268 |
265 267
|
eqtr3d |
⊢ ( 𝜑 → ( 1st ‘ ( 1st ‘ 𝑈 ) ) = ( 1st ‘ ( 1st ‘ 𝑇 ) ) ) |
269 |
268
|
fveq1d |
⊢ ( 𝜑 → ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) = ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) ) |
270 |
269
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∧ ¬ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) → ( ( 1st ‘ ( 1st ‘ 𝑈 ) ) ‘ 𝑦 ) = ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) ) |
271 |
180 262 270
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∧ ¬ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) + 1 ) = ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) ) |
272 |
|
elmapi |
⊢ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ∈ ( ( 0 ..^ 𝐾 ) ↑m ( 1 ... 𝑁 ) ) → ( 1st ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ..^ 𝐾 ) ) |
273 |
14 221 272
|
3syl |
⊢ ( 𝜑 → ( 1st ‘ ( 1st ‘ 𝑇 ) ) : ( 1 ... 𝑁 ) ⟶ ( 0 ..^ 𝐾 ) ) |
274 |
273
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) ∈ ( 0 ..^ 𝐾 ) ) |
275 |
|
elfzonn0 |
⊢ ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) ∈ ( 0 ..^ 𝐾 ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) ∈ ℕ0 ) |
276 |
274 275
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) ∈ ℕ0 ) |
277 |
276
|
nn0red |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) ∈ ℝ ) |
278 |
277
|
ltp1d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) < ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) + 1 ) ) |
279 |
277 278
|
gtned |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) + 1 ) ≠ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) ) |
280 |
220 279
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) → ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) + 1 ) ≠ ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) ) |
281 |
280
|
neneqd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ) → ¬ ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) + 1 ) = ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) ) |
282 |
281
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∧ ¬ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) → ¬ ( ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) + 1 ) = ( ( 1st ‘ ( 1st ‘ 𝑇 ) ) ‘ 𝑦 ) ) |
283 |
271 282
|
pm2.65da |
⊢ ( 𝜑 → ¬ ( 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∧ ¬ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) |
284 |
|
iman |
⊢ ( ( 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) → 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ↔ ¬ ( 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ∧ ¬ 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) |
285 |
283 284
|
sylibr |
⊢ ( 𝜑 → ( 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) → 𝑦 ∈ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) ) |
286 |
285
|
ssrdv |
⊢ ( 𝜑 → ( ( 2nd ‘ ( 1st ‘ 𝑇 ) ) “ ( 1 ... 𝑀 ) ) ⊆ ( ( 2nd ‘ ( 1st ‘ 𝑈 ) ) “ ( 1 ... 𝑀 ) ) ) |