Step |
Hyp |
Ref |
Expression |
1 |
|
opsqrlem2.1 |
⊢ 𝑇 ∈ HrmOp |
2 |
|
opsqrlem2.2 |
⊢ 𝑆 = ( 𝑥 ∈ HrmOp , 𝑦 ∈ HrmOp ↦ ( 𝑥 +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( 𝑥 ∘ 𝑥 ) ) ) ) ) |
3 |
|
opsqrlem2.3 |
⊢ 𝐹 = seq 1 ( 𝑆 , ( ℕ × { 0hop } ) ) |
4 |
|
opsqrlem6.4 |
⊢ 𝑇 ≤op Iop |
5 |
|
fveq2 |
⊢ ( 𝑗 = 1 → ( 𝐹 ‘ 𝑗 ) = ( 𝐹 ‘ 1 ) ) |
6 |
5
|
breq1d |
⊢ ( 𝑗 = 1 → ( ( 𝐹 ‘ 𝑗 ) ≤op Iop ↔ ( 𝐹 ‘ 1 ) ≤op Iop ) ) |
7 |
|
fveq2 |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
8 |
7
|
breq1d |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( ( 𝐹 ‘ 𝑗 ) ≤op Iop ↔ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ≤op Iop ) ) |
9 |
|
fveq2 |
⊢ ( 𝑗 = 𝑁 → ( 𝐹 ‘ 𝑗 ) = ( 𝐹 ‘ 𝑁 ) ) |
10 |
9
|
breq1d |
⊢ ( 𝑗 = 𝑁 → ( ( 𝐹 ‘ 𝑗 ) ≤op Iop ↔ ( 𝐹 ‘ 𝑁 ) ≤op Iop ) ) |
11 |
1 2 3
|
opsqrlem2 |
⊢ ( 𝐹 ‘ 1 ) = 0hop |
12 |
|
idleop |
⊢ 0hop ≤op Iop |
13 |
11 12
|
eqbrtri |
⊢ ( 𝐹 ‘ 1 ) ≤op Iop |
14 |
|
idhmop |
⊢ Iop ∈ HrmOp |
15 |
1 2 3
|
opsqrlem4 |
⊢ 𝐹 : ℕ ⟶ HrmOp |
16 |
15
|
ffvelrni |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ 𝑘 ) ∈ HrmOp ) |
17 |
|
hmopd |
⊢ ( ( Iop ∈ HrmOp ∧ ( 𝐹 ‘ 𝑘 ) ∈ HrmOp ) → ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∈ HrmOp ) |
18 |
14 16 17
|
sylancr |
⊢ ( 𝑘 ∈ ℕ → ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∈ HrmOp ) |
19 |
|
eqid |
⊢ ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) |
20 |
|
hmopco |
⊢ ( ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∈ HrmOp ∧ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∈ HrmOp ∧ ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ∈ HrmOp ) |
21 |
19 20
|
mp3an3 |
⊢ ( ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∈ HrmOp ∧ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∈ HrmOp ) → ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ∈ HrmOp ) |
22 |
18 18 21
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ∈ HrmOp ) |
23 |
|
leopsq |
⊢ ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∈ HrmOp → 0hop ≤op ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ) |
24 |
18 23
|
syl |
⊢ ( 𝑘 ∈ ℕ → 0hop ≤op ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ) |
25 |
|
leop3 |
⊢ ( ( 𝑇 ∈ HrmOp ∧ Iop ∈ HrmOp ) → ( 𝑇 ≤op Iop ↔ 0hop ≤op ( Iop −op 𝑇 ) ) ) |
26 |
1 14 25
|
mp2an |
⊢ ( 𝑇 ≤op Iop ↔ 0hop ≤op ( Iop −op 𝑇 ) ) |
27 |
4 26
|
mpbi |
⊢ 0hop ≤op ( Iop −op 𝑇 ) |
28 |
|
hmopd |
⊢ ( ( Iop ∈ HrmOp ∧ 𝑇 ∈ HrmOp ) → ( Iop −op 𝑇 ) ∈ HrmOp ) |
29 |
14 1 28
|
mp2an |
⊢ ( Iop −op 𝑇 ) ∈ HrmOp |
30 |
|
leopadd |
⊢ ( ( ( ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ∈ HrmOp ∧ ( Iop −op 𝑇 ) ∈ HrmOp ) ∧ ( 0hop ≤op ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ∧ 0hop ≤op ( Iop −op 𝑇 ) ) ) → 0hop ≤op ( ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) +op ( Iop −op 𝑇 ) ) ) |
31 |
29 30
|
mpanl2 |
⊢ ( ( ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ∈ HrmOp ∧ ( 0hop ≤op ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ∧ 0hop ≤op ( Iop −op 𝑇 ) ) ) → 0hop ≤op ( ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) +op ( Iop −op 𝑇 ) ) ) |
32 |
27 31
|
mpanr2 |
⊢ ( ( ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ∈ HrmOp ∧ 0hop ≤op ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ) → 0hop ≤op ( ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) +op ( Iop −op 𝑇 ) ) ) |
33 |
22 24 32
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → 0hop ≤op ( ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) +op ( Iop −op 𝑇 ) ) ) |
34 |
|
2cn |
⊢ 2 ∈ ℂ |
35 |
|
hmopf |
⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ HrmOp → ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ) |
36 |
16 35
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ) |
37 |
|
homulcl |
⊢ ( ( 2 ∈ ℂ ∧ ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ) → ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) |
38 |
34 36 37
|
sylancr |
⊢ ( 𝑘 ∈ ℕ → ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) |
39 |
|
hmopf |
⊢ ( 𝑇 ∈ HrmOp → 𝑇 : ℋ ⟶ ℋ ) |
40 |
1 39
|
ax-mp |
⊢ 𝑇 : ℋ ⟶ ℋ |
41 |
|
fco |
⊢ ( ( ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ∧ ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ) → ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) |
42 |
36 36 41
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) |
43 |
|
hosubcl |
⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) → ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ) |
44 |
40 42 43
|
sylancr |
⊢ ( 𝑘 ∈ ℕ → ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ) |
45 |
|
hmopf |
⊢ ( Iop ∈ HrmOp → Iop : ℋ ⟶ ℋ ) |
46 |
14 45
|
ax-mp |
⊢ Iop : ℋ ⟶ ℋ |
47 |
|
homulcl |
⊢ ( ( 2 ∈ ℂ ∧ Iop : ℋ ⟶ ℋ ) → ( 2 ·op Iop ) : ℋ ⟶ ℋ ) |
48 |
34 46 47
|
mp2an |
⊢ ( 2 ·op Iop ) : ℋ ⟶ ℋ |
49 |
|
hosubsub4 |
⊢ ( ( ( 2 ·op Iop ) : ℋ ⟶ ℋ ∧ ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ∧ ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ) → ( ( ( 2 ·op Iop ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) −op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( 2 ·op Iop ) −op ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
50 |
48 49
|
mp3an1 |
⊢ ( ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ∧ ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ) → ( ( ( 2 ·op Iop ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) −op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( 2 ·op Iop ) −op ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
51 |
38 44 50
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → ( ( ( 2 ·op Iop ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) −op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( 2 ·op Iop ) −op ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
52 |
|
hosubcl |
⊢ ( ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ∧ ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) → ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ) |
53 |
42 38 52
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ) |
54 |
|
hoadd32 |
⊢ ( ( Iop : ℋ ⟶ ℋ ∧ ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ∧ Iop : ℋ ⟶ ℋ ) → ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op Iop ) = ( ( Iop +op Iop ) +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
55 |
46 46 54
|
mp3an13 |
⊢ ( ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ → ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op Iop ) = ( ( Iop +op Iop ) +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
56 |
53 55
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op Iop ) = ( ( Iop +op Iop ) +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
57 |
|
ho2times |
⊢ ( Iop : ℋ ⟶ ℋ → ( 2 ·op Iop ) = ( Iop +op Iop ) ) |
58 |
46 57
|
ax-mp |
⊢ ( 2 ·op Iop ) = ( Iop +op Iop ) |
59 |
58
|
oveq1i |
⊢ ( ( 2 ·op Iop ) +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( Iop +op Iop ) +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) |
60 |
56 59
|
eqtr4di |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op Iop ) = ( ( 2 ·op Iop ) +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
61 |
|
hoaddsubass |
⊢ ( ( ( 2 ·op Iop ) : ℋ ⟶ ℋ ∧ ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ∧ ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) → ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( 2 ·op Iop ) +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
62 |
48 61
|
mp3an1 |
⊢ ( ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ∧ ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) → ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( 2 ·op Iop ) +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
63 |
42 38 62
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( 2 ·op Iop ) +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
64 |
60 63
|
eqtr4d |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op Iop ) = ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) |
65 |
64
|
oveq1d |
⊢ ( 𝑘 ∈ ℕ → ( ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op Iop ) −op 𝑇 ) = ( ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) −op 𝑇 ) ) |
66 |
|
hoaddcl |
⊢ ( ( Iop : ℋ ⟶ ℋ ∧ ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ) → ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) : ℋ ⟶ ℋ ) |
67 |
46 53 66
|
sylancr |
⊢ ( 𝑘 ∈ ℕ → ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) : ℋ ⟶ ℋ ) |
68 |
|
hoaddsubass |
⊢ ( ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) : ℋ ⟶ ℋ ∧ Iop : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op Iop ) −op 𝑇 ) = ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op ( Iop −op 𝑇 ) ) ) |
69 |
46 40 68
|
mp3an23 |
⊢ ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) : ℋ ⟶ ℋ → ( ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op Iop ) −op 𝑇 ) = ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op ( Iop −op 𝑇 ) ) ) |
70 |
67 69
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op Iop ) −op 𝑇 ) = ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op ( Iop −op 𝑇 ) ) ) |
71 |
|
hoaddcl |
⊢ ( ( ( 2 ·op Iop ) : ℋ ⟶ ℋ ∧ ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) → ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ) |
72 |
48 42 71
|
sylancr |
⊢ ( 𝑘 ∈ ℕ → ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ) |
73 |
|
hosubsub4 |
⊢ ( ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ∧ ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) −op 𝑇 ) = ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op 𝑇 ) ) ) |
74 |
40 73
|
mp3an3 |
⊢ ( ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ∧ ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) → ( ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) −op 𝑇 ) = ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op 𝑇 ) ) ) |
75 |
72 38 74
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → ( ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) −op 𝑇 ) = ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op 𝑇 ) ) ) |
76 |
65 70 75
|
3eqtr3d |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op ( Iop −op 𝑇 ) ) = ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op 𝑇 ) ) ) |
77 |
|
hosubadd4 |
⊢ ( ( ( ( 2 ·op Iop ) : ℋ ⟶ ℋ ∧ ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) ∧ ( 𝑇 : ℋ ⟶ ℋ ∧ ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) ) → ( ( ( 2 ·op Iop ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) −op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op 𝑇 ) ) ) |
78 |
40 77
|
mpanr1 |
⊢ ( ( ( ( 2 ·op Iop ) : ℋ ⟶ ℋ ∧ ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) ∧ ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) → ( ( ( 2 ·op Iop ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) −op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op 𝑇 ) ) ) |
79 |
48 78
|
mpanl1 |
⊢ ( ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ∧ ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) → ( ( ( 2 ·op Iop ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) −op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op 𝑇 ) ) ) |
80 |
38 42 79
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → ( ( ( 2 ·op Iop ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) −op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( ( 2 ·op Iop ) +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op 𝑇 ) ) ) |
81 |
76 80
|
eqtr4d |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op ( Iop −op 𝑇 ) ) = ( ( ( 2 ·op Iop ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) −op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
82 |
|
halfcn |
⊢ ( 1 / 2 ) ∈ ℂ |
83 |
|
homulcl |
⊢ ( ( ( 1 / 2 ) ∈ ℂ ∧ ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ) → ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) : ℋ ⟶ ℋ ) |
84 |
82 44 83
|
sylancr |
⊢ ( 𝑘 ∈ ℕ → ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) : ℋ ⟶ ℋ ) |
85 |
|
hoadddi |
⊢ ( ( 2 ∈ ℂ ∧ ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ∧ ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) : ℋ ⟶ ℋ ) → ( 2 ·op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) = ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op ( 2 ·op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) |
86 |
34 85
|
mp3an1 |
⊢ ( ( ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ∧ ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) : ℋ ⟶ ℋ ) → ( 2 ·op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) = ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op ( 2 ·op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) |
87 |
36 84 86
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → ( 2 ·op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) = ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op ( 2 ·op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) |
88 |
|
2ne0 |
⊢ 2 ≠ 0 |
89 |
34 88
|
recidi |
⊢ ( 2 · ( 1 / 2 ) ) = 1 |
90 |
89
|
oveq1i |
⊢ ( ( 2 · ( 1 / 2 ) ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( 1 ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
91 |
|
homulass |
⊢ ( ( 2 ∈ ℂ ∧ ( 1 / 2 ) ∈ ℂ ∧ ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ ) → ( ( 2 · ( 1 / 2 ) ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( 2 ·op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
92 |
34 82 91
|
mp3an12 |
⊢ ( ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ → ( ( 2 · ( 1 / 2 ) ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( 2 ·op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
93 |
44 92
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( ( 2 · ( 1 / 2 ) ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( 2 ·op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
94 |
|
homulid2 |
⊢ ( ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) : ℋ ⟶ ℋ → ( 1 ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
95 |
44 94
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( 1 ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
96 |
90 93 95
|
3eqtr3a |
⊢ ( 𝑘 ∈ ℕ → ( 2 ·op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) = ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
97 |
96
|
oveq2d |
⊢ ( 𝑘 ∈ ℕ → ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op ( 2 ·op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) = ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
98 |
87 97
|
eqtrd |
⊢ ( 𝑘 ∈ ℕ → ( 2 ·op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) = ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
99 |
98
|
oveq2d |
⊢ ( 𝑘 ∈ ℕ → ( ( 2 ·op Iop ) −op ( 2 ·op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) = ( ( 2 ·op Iop ) −op ( ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) +op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
100 |
51 81 99
|
3eqtr4d |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op ( Iop −op 𝑇 ) ) = ( ( 2 ·op Iop ) −op ( 2 ·op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) ) |
101 |
|
hoaddcl |
⊢ ( ( ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ∧ ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) : ℋ ⟶ ℋ ) → ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) : ℋ ⟶ ℋ ) |
102 |
36 84 101
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) : ℋ ⟶ ℋ ) |
103 |
|
hosubdi |
⊢ ( ( 2 ∈ ℂ ∧ Iop : ℋ ⟶ ℋ ∧ ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) : ℋ ⟶ ℋ ) → ( 2 ·op ( Iop −op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) = ( ( 2 ·op Iop ) −op ( 2 ·op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) ) |
104 |
34 46 103
|
mp3an12 |
⊢ ( ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) : ℋ ⟶ ℋ → ( 2 ·op ( Iop −op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) = ( ( 2 ·op Iop ) −op ( 2 ·op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) ) |
105 |
102 104
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( 2 ·op ( Iop −op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) = ( ( 2 ·op Iop ) −op ( 2 ·op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) ) |
106 |
100 105
|
eqtr4d |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op ( Iop −op 𝑇 ) ) = ( 2 ·op ( Iop −op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) ) |
107 |
|
hosubcl |
⊢ ( ( Iop : ℋ ⟶ ℋ ∧ ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ) → ( Iop −op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) |
108 |
46 36 107
|
sylancr |
⊢ ( 𝑘 ∈ ℕ → ( Iop −op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) |
109 |
|
hocsubdir |
⊢ ( ( Iop : ℋ ⟶ ℋ ∧ ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ∧ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) → ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( Iop ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
110 |
46 109
|
mp3an1 |
⊢ ( ( ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ∧ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) → ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( Iop ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
111 |
36 108 110
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( Iop ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
112 |
|
hmoplin |
⊢ ( Iop ∈ HrmOp → Iop ∈ LinOp ) |
113 |
14 112
|
ax-mp |
⊢ Iop ∈ LinOp |
114 |
|
hoddi |
⊢ ( ( Iop ∈ LinOp ∧ Iop : ℋ ⟶ ℋ ∧ ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ) → ( Iop ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( Iop ∘ Iop ) −op ( Iop ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
115 |
113 46 114
|
mp3an12 |
⊢ ( ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ → ( Iop ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( Iop ∘ Iop ) −op ( Iop ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
116 |
36 115
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( Iop ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( Iop ∘ Iop ) −op ( Iop ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
117 |
46
|
hoid1i |
⊢ ( Iop ∘ Iop ) = Iop |
118 |
117
|
a1i |
⊢ ( 𝑘 ∈ ℕ → ( Iop ∘ Iop ) = Iop ) |
119 |
|
hoico2 |
⊢ ( ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ → ( Iop ∘ ( 𝐹 ‘ 𝑘 ) ) = ( 𝐹 ‘ 𝑘 ) ) |
120 |
36 119
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( Iop ∘ ( 𝐹 ‘ 𝑘 ) ) = ( 𝐹 ‘ 𝑘 ) ) |
121 |
118 120
|
oveq12d |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop ∘ Iop ) −op ( Iop ∘ ( 𝐹 ‘ 𝑘 ) ) ) = ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) |
122 |
116 121
|
eqtrd |
⊢ ( 𝑘 ∈ ℕ → ( Iop ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) |
123 |
|
hmoplin |
⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ HrmOp → ( 𝐹 ‘ 𝑘 ) ∈ LinOp ) |
124 |
16 123
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ 𝑘 ) ∈ LinOp ) |
125 |
|
hoddi |
⊢ ( ( ( 𝐹 ‘ 𝑘 ) ∈ LinOp ∧ Iop : ℋ ⟶ ℋ ∧ ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ) → ( ( 𝐹 ‘ 𝑘 ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( ( 𝐹 ‘ 𝑘 ) ∘ Iop ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
126 |
46 125
|
mp3an2 |
⊢ ( ( ( 𝐹 ‘ 𝑘 ) ∈ LinOp ∧ ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ) → ( ( 𝐹 ‘ 𝑘 ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( ( 𝐹 ‘ 𝑘 ) ∘ Iop ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
127 |
124 36 126
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝐹 ‘ 𝑘 ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( ( 𝐹 ‘ 𝑘 ) ∘ Iop ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
128 |
|
hoico1 |
⊢ ( ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ → ( ( 𝐹 ‘ 𝑘 ) ∘ Iop ) = ( 𝐹 ‘ 𝑘 ) ) |
129 |
36 128
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝐹 ‘ 𝑘 ) ∘ Iop ) = ( 𝐹 ‘ 𝑘 ) ) |
130 |
129
|
oveq1d |
⊢ ( 𝑘 ∈ ℕ → ( ( ( 𝐹 ‘ 𝑘 ) ∘ Iop ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) = ( ( 𝐹 ‘ 𝑘 ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
131 |
127 130
|
eqtrd |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝐹 ‘ 𝑘 ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( 𝐹 ‘ 𝑘 ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
132 |
122 131
|
oveq12d |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) −op ( ( 𝐹 ‘ 𝑘 ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
133 |
36 46
|
jctil |
⊢ ( 𝑘 ∈ ℕ → ( Iop : ℋ ⟶ ℋ ∧ ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ) ) |
134 |
|
hosubadd4 |
⊢ ( ( ( Iop : ℋ ⟶ ℋ ∧ ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ) ∧ ( ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ ∧ ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) ) → ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) −op ( ( 𝐹 ‘ 𝑘 ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( Iop +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 𝐹 ‘ 𝑘 ) +op ( 𝐹 ‘ 𝑘 ) ) ) ) |
135 |
133 36 42 134
|
syl12anc |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) −op ( ( 𝐹 ‘ 𝑘 ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( Iop +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 𝐹 ‘ 𝑘 ) +op ( 𝐹 ‘ 𝑘 ) ) ) ) |
136 |
132 135
|
eqtrd |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ) = ( ( Iop +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 𝐹 ‘ 𝑘 ) +op ( 𝐹 ‘ 𝑘 ) ) ) ) |
137 |
|
ho2times |
⊢ ( ( 𝐹 ‘ 𝑘 ) : ℋ ⟶ ℋ → ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) = ( ( 𝐹 ‘ 𝑘 ) +op ( 𝐹 ‘ 𝑘 ) ) ) |
138 |
36 137
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) = ( ( 𝐹 ‘ 𝑘 ) +op ( 𝐹 ‘ 𝑘 ) ) ) |
139 |
138
|
oveq2d |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) = ( ( Iop +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 𝐹 ‘ 𝑘 ) +op ( 𝐹 ‘ 𝑘 ) ) ) ) |
140 |
|
hoaddsubass |
⊢ ( ( Iop : ℋ ⟶ ℋ ∧ ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ∧ ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) → ( ( Iop +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) = ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
141 |
46 140
|
mp3an1 |
⊢ ( ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ∧ ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) : ℋ ⟶ ℋ ) → ( ( Iop +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) = ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
142 |
42 38 141
|
syl2anc |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop +op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) = ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
143 |
136 139 142
|
3eqtr2d |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) ) = ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
144 |
111 143
|
eqtrd |
⊢ ( 𝑘 ∈ ℕ → ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) = ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
145 |
144
|
oveq1d |
⊢ ( 𝑘 ∈ ℕ → ( ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) +op ( Iop −op 𝑇 ) ) = ( ( Iop +op ( ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) −op ( 2 ·op ( 𝐹 ‘ 𝑘 ) ) ) ) +op ( Iop −op 𝑇 ) ) ) |
146 |
1 2 3
|
opsqrlem5 |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
147 |
146
|
oveq2d |
⊢ ( 𝑘 ∈ ℕ → ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) = ( Iop −op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) |
148 |
147
|
oveq2d |
⊢ ( 𝑘 ∈ ℕ → ( 2 ·op ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) = ( 2 ·op ( Iop −op ( ( 𝐹 ‘ 𝑘 ) +op ( ( 1 / 2 ) ·op ( 𝑇 −op ( ( 𝐹 ‘ 𝑘 ) ∘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) ) ) |
149 |
106 145 148
|
3eqtr4d |
⊢ ( 𝑘 ∈ ℕ → ( ( ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ∘ ( Iop −op ( 𝐹 ‘ 𝑘 ) ) ) +op ( Iop −op 𝑇 ) ) = ( 2 ·op ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) |
150 |
33 149
|
breqtrd |
⊢ ( 𝑘 ∈ ℕ → 0hop ≤op ( 2 ·op ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) |
151 |
|
peano2nn |
⊢ ( 𝑘 ∈ ℕ → ( 𝑘 + 1 ) ∈ ℕ ) |
152 |
15
|
ffvelrni |
⊢ ( ( 𝑘 + 1 ) ∈ ℕ → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ HrmOp ) |
153 |
151 152
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ HrmOp ) |
154 |
|
hmopd |
⊢ ( ( Iop ∈ HrmOp ∧ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ HrmOp ) → ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ∈ HrmOp ) |
155 |
14 153 154
|
sylancr |
⊢ ( 𝑘 ∈ ℕ → ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ∈ HrmOp ) |
156 |
|
2re |
⊢ 2 ∈ ℝ |
157 |
|
2pos |
⊢ 0 < 2 |
158 |
|
leopmul |
⊢ ( ( 2 ∈ ℝ ∧ ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ∈ HrmOp ∧ 0 < 2 ) → ( 0hop ≤op ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ↔ 0hop ≤op ( 2 ·op ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) ) |
159 |
156 157 158
|
mp3an13 |
⊢ ( ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ∈ HrmOp → ( 0hop ≤op ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ↔ 0hop ≤op ( 2 ·op ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) ) |
160 |
155 159
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( 0hop ≤op ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ↔ 0hop ≤op ( 2 ·op ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) ) |
161 |
150 160
|
mpbird |
⊢ ( 𝑘 ∈ ℕ → 0hop ≤op ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
162 |
|
leop3 |
⊢ ( ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ HrmOp ∧ Iop ∈ HrmOp ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ≤op Iop ↔ 0hop ≤op ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) |
163 |
153 14 162
|
sylancl |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ≤op Iop ↔ 0hop ≤op ( Iop −op ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) |
164 |
161 163
|
mpbird |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ≤op Iop ) |
165 |
6 8 10 13 164
|
nn1suc |
⊢ ( 𝑁 ∈ ℕ → ( 𝐹 ‘ 𝑁 ) ≤op Iop ) |