| Step |
Hyp |
Ref |
Expression |
| 1 |
|
eqidd |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) = ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) ) |
| 2 |
|
simprr |
⊢ ( ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑟 = 𝑅 ∧ 𝑛 = ( 𝑁 + 1 ) ) ) → 𝑛 = ( 𝑁 + 1 ) ) |
| 3 |
|
dmeq |
⊢ ( 𝑟 = 𝑅 → dom 𝑟 = dom 𝑅 ) |
| 4 |
|
rneq |
⊢ ( 𝑟 = 𝑅 → ran 𝑟 = ran 𝑅 ) |
| 5 |
3 4
|
uneq12d |
⊢ ( 𝑟 = 𝑅 → ( dom 𝑟 ∪ ran 𝑟 ) = ( dom 𝑅 ∪ ran 𝑅 ) ) |
| 6 |
5
|
reseq2d |
⊢ ( 𝑟 = 𝑅 → ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) = ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
| 7 |
|
eqidd |
⊢ ( 𝑟 = 𝑅 → 1 = 1 ) |
| 8 |
|
coeq2 |
⊢ ( 𝑟 = 𝑅 → ( 𝑥 ∘ 𝑟 ) = ( 𝑥 ∘ 𝑅 ) ) |
| 9 |
8
|
mpoeq3dv |
⊢ ( 𝑟 = 𝑅 → ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) = ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) ) |
| 10 |
|
id |
⊢ ( 𝑟 = 𝑅 → 𝑟 = 𝑅 ) |
| 11 |
10
|
mpteq2dv |
⊢ ( 𝑟 = 𝑅 → ( 𝑧 ∈ V ↦ 𝑟 ) = ( 𝑧 ∈ V ↦ 𝑅 ) ) |
| 12 |
7 9 11
|
seqeq123d |
⊢ ( 𝑟 = 𝑅 → seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) = seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ) |
| 13 |
12
|
fveq1d |
⊢ ( 𝑟 = 𝑅 → ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ ( 𝑁 + 1 ) ) = ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) ) |
| 14 |
6 13
|
ifeq12d |
⊢ ( 𝑟 = 𝑅 → if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ ( 𝑁 + 1 ) ) ) = if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) ) ) |
| 15 |
14
|
ad2antrl |
⊢ ( ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑟 = 𝑅 ∧ ( 𝑁 + 1 ) = ( 𝑁 + 1 ) ) ) → if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ ( 𝑁 + 1 ) ) ) = if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) ) ) |
| 16 |
15
|
a1i |
⊢ ( 𝑛 = ( 𝑁 + 1 ) → ( ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑟 = 𝑅 ∧ ( 𝑁 + 1 ) = ( 𝑁 + 1 ) ) ) → if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ ( 𝑁 + 1 ) ) ) = if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) ) ) ) |
| 17 |
|
eqeq1 |
⊢ ( 𝑛 = ( 𝑁 + 1 ) → ( 𝑛 = ( 𝑁 + 1 ) ↔ ( 𝑁 + 1 ) = ( 𝑁 + 1 ) ) ) |
| 18 |
17
|
anbi2d |
⊢ ( 𝑛 = ( 𝑁 + 1 ) → ( ( 𝑟 = 𝑅 ∧ 𝑛 = ( 𝑁 + 1 ) ) ↔ ( 𝑟 = 𝑅 ∧ ( 𝑁 + 1 ) = ( 𝑁 + 1 ) ) ) ) |
| 19 |
18
|
anbi2d |
⊢ ( 𝑛 = ( 𝑁 + 1 ) → ( ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑟 = 𝑅 ∧ 𝑛 = ( 𝑁 + 1 ) ) ) ↔ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑟 = 𝑅 ∧ ( 𝑁 + 1 ) = ( 𝑁 + 1 ) ) ) ) ) |
| 20 |
|
eqeq1 |
⊢ ( 𝑛 = ( 𝑁 + 1 ) → ( 𝑛 = 0 ↔ ( 𝑁 + 1 ) = 0 ) ) |
| 21 |
|
fveq2 |
⊢ ( 𝑛 = ( 𝑁 + 1 ) → ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) = ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ ( 𝑁 + 1 ) ) ) |
| 22 |
20 21
|
ifbieq2d |
⊢ ( 𝑛 = ( 𝑁 + 1 ) → if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) = if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ ( 𝑁 + 1 ) ) ) ) |
| 23 |
22
|
eqeq1d |
⊢ ( 𝑛 = ( 𝑁 + 1 ) → ( if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) = if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) ) ↔ if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ ( 𝑁 + 1 ) ) ) = if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) ) ) ) |
| 24 |
16 19 23
|
3imtr4d |
⊢ ( 𝑛 = ( 𝑁 + 1 ) → ( ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑟 = 𝑅 ∧ 𝑛 = ( 𝑁 + 1 ) ) ) → if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) = if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) ) ) ) |
| 25 |
2 24
|
mpcom |
⊢ ( ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑟 = 𝑅 ∧ 𝑛 = ( 𝑁 + 1 ) ) ) → if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) = if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) ) ) |
| 26 |
|
elex |
⊢ ( 𝑅 ∈ 𝑉 → 𝑅 ∈ V ) |
| 27 |
26
|
adantr |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → 𝑅 ∈ V ) |
| 28 |
|
simpr |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → 𝑁 ∈ ℕ ) |
| 29 |
28
|
peano2nnd |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑁 + 1 ) ∈ ℕ ) |
| 30 |
29
|
nnnn0d |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑁 + 1 ) ∈ ℕ0 ) |
| 31 |
|
dmexg |
⊢ ( 𝑅 ∈ 𝑉 → dom 𝑅 ∈ V ) |
| 32 |
|
rnexg |
⊢ ( 𝑅 ∈ 𝑉 → ran 𝑅 ∈ V ) |
| 33 |
|
unexg |
⊢ ( ( dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V ) → ( dom 𝑅 ∪ ran 𝑅 ) ∈ V ) |
| 34 |
31 32 33
|
syl2anc |
⊢ ( 𝑅 ∈ 𝑉 → ( dom 𝑅 ∪ ran 𝑅 ) ∈ V ) |
| 35 |
|
resiexg |
⊢ ( ( dom 𝑅 ∪ ran 𝑅 ) ∈ V → ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ∈ V ) |
| 36 |
34 35
|
syl |
⊢ ( 𝑅 ∈ 𝑉 → ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ∈ V ) |
| 37 |
36
|
adantr |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ∈ V ) |
| 38 |
|
fvexd |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) ∈ V ) |
| 39 |
37 38
|
ifcld |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) ) ∈ V ) |
| 40 |
1 25 27 30 39
|
ovmpod |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) ( 𝑁 + 1 ) ) = if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) ) ) |
| 41 |
|
nnne0 |
⊢ ( ( 𝑁 + 1 ) ∈ ℕ → ( 𝑁 + 1 ) ≠ 0 ) |
| 42 |
41
|
neneqd |
⊢ ( ( 𝑁 + 1 ) ∈ ℕ → ¬ ( 𝑁 + 1 ) = 0 ) |
| 43 |
29 42
|
syl |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ¬ ( 𝑁 + 1 ) = 0 ) |
| 44 |
43
|
iffalsed |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → if ( ( 𝑁 + 1 ) = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) ) = ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) ) |
| 45 |
|
elnnuz |
⊢ ( 𝑁 ∈ ℕ ↔ 𝑁 ∈ ( ℤ≥ ‘ 1 ) ) |
| 46 |
45
|
biimpi |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ( ℤ≥ ‘ 1 ) ) |
| 47 |
46
|
adantl |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → 𝑁 ∈ ( ℤ≥ ‘ 1 ) ) |
| 48 |
|
seqp1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 1 ) → ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) = ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) ( ( 𝑧 ∈ V ↦ 𝑅 ) ‘ ( 𝑁 + 1 ) ) ) ) |
| 49 |
47 48
|
syl |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) = ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) ( ( 𝑧 ∈ V ↦ 𝑅 ) ‘ ( 𝑁 + 1 ) ) ) ) |
| 50 |
|
ovex |
⊢ ( 𝑁 + 1 ) ∈ V |
| 51 |
|
simpl |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → 𝑅 ∈ 𝑉 ) |
| 52 |
|
eqidd |
⊢ ( 𝑧 = ( 𝑁 + 1 ) → 𝑅 = 𝑅 ) |
| 53 |
|
eqid |
⊢ ( 𝑧 ∈ V ↦ 𝑅 ) = ( 𝑧 ∈ V ↦ 𝑅 ) |
| 54 |
52 53
|
fvmptg |
⊢ ( ( ( 𝑁 + 1 ) ∈ V ∧ 𝑅 ∈ 𝑉 ) → ( ( 𝑧 ∈ V ↦ 𝑅 ) ‘ ( 𝑁 + 1 ) ) = 𝑅 ) |
| 55 |
50 51 54
|
sylancr |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ( 𝑧 ∈ V ↦ 𝑅 ) ‘ ( 𝑁 + 1 ) ) = 𝑅 ) |
| 56 |
55
|
oveq2d |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) ( ( 𝑧 ∈ V ↦ 𝑅 ) ‘ ( 𝑁 + 1 ) ) ) = ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) 𝑅 ) ) |
| 57 |
|
nfcv |
⊢ Ⅎ 𝑎 ( 𝑥 ∘ 𝑅 ) |
| 58 |
|
nfcv |
⊢ Ⅎ 𝑏 ( 𝑥 ∘ 𝑅 ) |
| 59 |
|
nfcv |
⊢ Ⅎ 𝑥 ( 𝑎 ∘ 𝑅 ) |
| 60 |
|
nfcv |
⊢ Ⅎ 𝑦 ( 𝑎 ∘ 𝑅 ) |
| 61 |
|
simpl |
⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → 𝑥 = 𝑎 ) |
| 62 |
61
|
coeq1d |
⊢ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) → ( 𝑥 ∘ 𝑅 ) = ( 𝑎 ∘ 𝑅 ) ) |
| 63 |
57 58 59 60 62
|
cbvmpo |
⊢ ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑎 ∘ 𝑅 ) ) |
| 64 |
|
oveq |
⊢ ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑎 ∘ 𝑅 ) ) → ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) 𝑅 ) = ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑎 ∘ 𝑅 ) ) 𝑅 ) ) |
| 65 |
63 64
|
mp1i |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) 𝑅 ) = ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑎 ∘ 𝑅 ) ) 𝑅 ) ) |
| 66 |
|
eqidd |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑎 ∘ 𝑅 ) ) = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑎 ∘ 𝑅 ) ) ) |
| 67 |
|
simprl |
⊢ ( ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑎 = ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ∧ 𝑏 = 𝑅 ) ) → 𝑎 = ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ) |
| 68 |
67
|
coeq1d |
⊢ ( ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑎 = ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ∧ 𝑏 = 𝑅 ) ) → ( 𝑎 ∘ 𝑅 ) = ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ∘ 𝑅 ) ) |
| 69 |
|
fvexd |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ∈ V ) |
| 70 |
|
fvex |
⊢ ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ∈ V |
| 71 |
|
coexg |
⊢ ( ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ∈ V ∧ 𝑅 ∈ 𝑉 ) → ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ∘ 𝑅 ) ∈ V ) |
| 72 |
70 51 71
|
sylancr |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ∘ 𝑅 ) ∈ V ) |
| 73 |
66 68 69 27 72
|
ovmpod |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑎 ∘ 𝑅 ) ) 𝑅 ) = ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ∘ 𝑅 ) ) |
| 74 |
|
simpr |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑛 = 𝑁 ) → 𝑛 = 𝑁 ) |
| 75 |
74
|
eqeq1d |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑛 = 𝑁 ) → ( 𝑛 = 0 ↔ 𝑁 = 0 ) ) |
| 76 |
6
|
adantr |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑛 = 𝑁 ) → ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) = ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
| 77 |
12
|
adantr |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑛 = 𝑁 ) → seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) = seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ) |
| 78 |
77 74
|
fveq12d |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑛 = 𝑁 ) → ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) = ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ) |
| 79 |
75 76 78
|
ifbieq12d |
⊢ ( ( 𝑟 = 𝑅 ∧ 𝑛 = 𝑁 ) → if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) = if ( 𝑁 = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ) ) |
| 80 |
79
|
adantl |
⊢ ( ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑟 = 𝑅 ∧ 𝑛 = 𝑁 ) ) → if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) = if ( 𝑁 = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ) ) |
| 81 |
28
|
nnnn0d |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → 𝑁 ∈ ℕ0 ) |
| 82 |
37 69
|
ifcld |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → if ( 𝑁 = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ) ∈ V ) |
| 83 |
1 80 27 81 82
|
ovmpod |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) 𝑁 ) = if ( 𝑁 = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ) ) |
| 84 |
|
nnne0 |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ≠ 0 ) |
| 85 |
84
|
adantl |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → 𝑁 ≠ 0 ) |
| 86 |
85
|
neneqd |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ¬ 𝑁 = 0 ) |
| 87 |
86
|
iffalsed |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → if ( 𝑁 = 0 , ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ) = ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ) |
| 88 |
83 87
|
eqtr2d |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) = ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) 𝑁 ) ) |
| 89 |
88
|
coeq1d |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ∘ 𝑅 ) = ( ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) 𝑁 ) ∘ 𝑅 ) ) |
| 90 |
65 73 89
|
3eqtrd |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ 𝑁 ) ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) 𝑅 ) = ( ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) 𝑁 ) ∘ 𝑅 ) ) |
| 91 |
49 56 90
|
3eqtrd |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑅 ) ) , ( 𝑧 ∈ V ↦ 𝑅 ) ) ‘ ( 𝑁 + 1 ) ) = ( ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) 𝑁 ) ∘ 𝑅 ) ) |
| 92 |
40 44 91
|
3eqtrd |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) ( 𝑁 + 1 ) ) = ( ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) 𝑁 ) ∘ 𝑅 ) ) |
| 93 |
|
df-relexp |
⊢ ↑𝑟 = ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) |
| 94 |
|
oveq |
⊢ ( ↑𝑟 = ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) → ( 𝑅 ↑𝑟 ( 𝑁 + 1 ) ) = ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) ( 𝑁 + 1 ) ) ) |
| 95 |
|
oveq |
⊢ ( ↑𝑟 = ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) → ( 𝑅 ↑𝑟 𝑁 ) = ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) 𝑁 ) ) |
| 96 |
95
|
coeq1d |
⊢ ( ↑𝑟 = ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) → ( ( 𝑅 ↑𝑟 𝑁 ) ∘ 𝑅 ) = ( ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) 𝑁 ) ∘ 𝑅 ) ) |
| 97 |
94 96
|
eqeq12d |
⊢ ( ↑𝑟 = ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) → ( ( 𝑅 ↑𝑟 ( 𝑁 + 1 ) ) = ( ( 𝑅 ↑𝑟 𝑁 ) ∘ 𝑅 ) ↔ ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) ( 𝑁 + 1 ) ) = ( ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) 𝑁 ) ∘ 𝑅 ) ) ) |
| 98 |
97
|
imbi2d |
⊢ ( ↑𝑟 = ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) → ( ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑅 ↑𝑟 ( 𝑁 + 1 ) ) = ( ( 𝑅 ↑𝑟 𝑁 ) ∘ 𝑅 ) ) ↔ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) ( 𝑁 + 1 ) ) = ( ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) 𝑁 ) ∘ 𝑅 ) ) ) ) |
| 99 |
93 98
|
ax-mp |
⊢ ( ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑅 ↑𝑟 ( 𝑁 + 1 ) ) = ( ( 𝑅 ↑𝑟 𝑁 ) ∘ 𝑅 ) ) ↔ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) ( 𝑁 + 1 ) ) = ( ( 𝑅 ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) 𝑁 ) ∘ 𝑅 ) ) ) |
| 100 |
92 99
|
mpbir |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑅 ↑𝑟 ( 𝑁 + 1 ) ) = ( ( 𝑅 ↑𝑟 𝑁 ) ∘ 𝑅 ) ) |