Step |
Hyp |
Ref |
Expression |
1 |
|
elpri |
⊢ ( 𝐽 ∈ { 0 , 1 } → ( 𝐽 = 0 ∨ 𝐽 = 1 ) ) |
2 |
|
elpri |
⊢ ( 𝐾 ∈ { 0 , 1 } → ( 𝐾 = 0 ∨ 𝐾 = 1 ) ) |
3 |
|
dmresi |
⊢ dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) = ( dom 𝑅 ∪ ran 𝑅 ) |
4 |
|
rnresi |
⊢ ran ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) = ( dom 𝑅 ∪ ran 𝑅 ) |
5 |
3 4
|
uneq12i |
⊢ ( dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ∪ ran ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) = ( ( dom 𝑅 ∪ ran 𝑅 ) ∪ ( dom 𝑅 ∪ ran 𝑅 ) ) |
6 |
|
unidm |
⊢ ( ( dom 𝑅 ∪ ran 𝑅 ) ∪ ( dom 𝑅 ∪ ran 𝑅 ) ) = ( dom 𝑅 ∪ ran 𝑅 ) |
7 |
5 6
|
eqtri |
⊢ ( dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ∪ ran ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) = ( dom 𝑅 ∪ ran 𝑅 ) |
8 |
7
|
reseq2i |
⊢ ( I ↾ ( dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ∪ ran ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) ) = ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) |
9 |
|
simp1 |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐽 = 0 ) |
10 |
9
|
oveq2d |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 𝐽 ) = ( 𝑅 ↑𝑟 0 ) ) |
11 |
|
simp3l |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝑅 ∈ 𝑉 ) |
12 |
|
relexp0g |
⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ↑𝑟 0 ) = ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
13 |
11 12
|
syl |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 0 ) = ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
14 |
10 13
|
eqtrd |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 𝐽 ) = ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
15 |
|
simp2 |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐾 = 0 ) |
16 |
14 15
|
oveq12d |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ↑𝑟 0 ) ) |
17 |
|
dmexg |
⊢ ( 𝑅 ∈ 𝑉 → dom 𝑅 ∈ V ) |
18 |
|
rnexg |
⊢ ( 𝑅 ∈ 𝑉 → ran 𝑅 ∈ V ) |
19 |
|
unexg |
⊢ ( ( dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V ) → ( dom 𝑅 ∪ ran 𝑅 ) ∈ V ) |
20 |
17 18 19
|
syl2anc |
⊢ ( 𝑅 ∈ 𝑉 → ( dom 𝑅 ∪ ran 𝑅 ) ∈ V ) |
21 |
20
|
resiexd |
⊢ ( 𝑅 ∈ 𝑉 → ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ∈ V ) |
22 |
|
relexp0g |
⊢ ( ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ∈ V → ( ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ↑𝑟 0 ) = ( I ↾ ( dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ∪ ran ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) ) ) |
23 |
11 21 22
|
3syl |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ↑𝑟 0 ) = ( I ↾ ( dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ∪ ran ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) ) ) |
24 |
16 23
|
eqtrd |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( I ↾ ( dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ∪ ran ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) ) ) |
25 |
|
simp3r |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) |
26 |
|
0re |
⊢ 0 ∈ ℝ |
27 |
26
|
ltnri |
⊢ ¬ 0 < 0 |
28 |
9 15
|
breq12d |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝐽 < 𝐾 ↔ 0 < 0 ) ) |
29 |
27 28
|
mtbiri |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ¬ 𝐽 < 𝐾 ) |
30 |
29
|
iffalsed |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) = 𝐾 ) |
31 |
25 30 15
|
3eqtrd |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐼 = 0 ) |
32 |
31
|
oveq2d |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 𝐼 ) = ( 𝑅 ↑𝑟 0 ) ) |
33 |
32 13
|
eqtrd |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 𝐼 ) = ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
34 |
8 24 33
|
3eqtr4a |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) |
35 |
34
|
3exp |
⊢ ( 𝐽 = 0 → ( 𝐾 = 0 → ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) ) ) |
36 |
|
simp1 |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐽 = 1 ) |
37 |
36
|
oveq2d |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 𝐽 ) = ( 𝑅 ↑𝑟 1 ) ) |
38 |
|
simp3l |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝑅 ∈ 𝑉 ) |
39 |
|
relexp1g |
⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ↑𝑟 1 ) = 𝑅 ) |
40 |
38 39
|
syl |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 1 ) = 𝑅 ) |
41 |
37 40
|
eqtrd |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 𝐽 ) = 𝑅 ) |
42 |
|
simp2 |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐾 = 0 ) |
43 |
41 42
|
oveq12d |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 0 ) ) |
44 |
|
simp3r |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) |
45 |
|
0lt1 |
⊢ 0 < 1 |
46 |
|
1re |
⊢ 1 ∈ ℝ |
47 |
26 46
|
ltnsymi |
⊢ ( 0 < 1 → ¬ 1 < 0 ) |
48 |
45 47
|
mp1i |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ¬ 1 < 0 ) |
49 |
36 42
|
breq12d |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝐽 < 𝐾 ↔ 1 < 0 ) ) |
50 |
48 49
|
mtbird |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ¬ 𝐽 < 𝐾 ) |
51 |
50
|
iffalsed |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) = 𝐾 ) |
52 |
44 51 42
|
3eqtrd |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐼 = 0 ) |
53 |
52
|
oveq2d |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 𝐼 ) = ( 𝑅 ↑𝑟 0 ) ) |
54 |
43 53
|
eqtr4d |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 0 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) |
55 |
54
|
3exp |
⊢ ( 𝐽 = 1 → ( 𝐾 = 0 → ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) ) ) |
56 |
35 55
|
jaoi |
⊢ ( ( 𝐽 = 0 ∨ 𝐽 = 1 ) → ( 𝐾 = 0 → ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) ) ) |
57 |
|
ovex |
⊢ ( 𝑅 ↑𝑟 0 ) ∈ V |
58 |
|
relexp1g |
⊢ ( ( 𝑅 ↑𝑟 0 ) ∈ V → ( ( 𝑅 ↑𝑟 0 ) ↑𝑟 1 ) = ( 𝑅 ↑𝑟 0 ) ) |
59 |
57 58
|
mp1i |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( ( 𝑅 ↑𝑟 0 ) ↑𝑟 1 ) = ( 𝑅 ↑𝑟 0 ) ) |
60 |
|
simp1 |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐽 = 0 ) |
61 |
60
|
oveq2d |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 𝐽 ) = ( 𝑅 ↑𝑟 0 ) ) |
62 |
|
simp2 |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐾 = 1 ) |
63 |
61 62
|
oveq12d |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( ( 𝑅 ↑𝑟 0 ) ↑𝑟 1 ) ) |
64 |
|
simp3r |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) |
65 |
60 62
|
breq12d |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝐽 < 𝐾 ↔ 0 < 1 ) ) |
66 |
45 65
|
mpbiri |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐽 < 𝐾 ) |
67 |
66
|
iftrued |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) = 𝐽 ) |
68 |
64 67 60
|
3eqtrd |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐼 = 0 ) |
69 |
68
|
oveq2d |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 𝐼 ) = ( 𝑅 ↑𝑟 0 ) ) |
70 |
59 63 69
|
3eqtr4d |
⊢ ( ( 𝐽 = 0 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) |
71 |
70
|
3exp |
⊢ ( 𝐽 = 0 → ( 𝐾 = 1 → ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) ) ) |
72 |
|
simp1 |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐽 = 1 ) |
73 |
72
|
oveq2d |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 𝐽 ) = ( 𝑅 ↑𝑟 1 ) ) |
74 |
|
simp3l |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝑅 ∈ 𝑉 ) |
75 |
74 39
|
syl |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 1 ) = 𝑅 ) |
76 |
73 75
|
eqtrd |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 𝐽 ) = 𝑅 ) |
77 |
|
simp2 |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐾 = 1 ) |
78 |
76 77
|
oveq12d |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 1 ) ) |
79 |
|
simp3r |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) |
80 |
46
|
ltnri |
⊢ ¬ 1 < 1 |
81 |
72 77
|
breq12d |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝐽 < 𝐾 ↔ 1 < 1 ) ) |
82 |
80 81
|
mtbiri |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ¬ 𝐽 < 𝐾 ) |
83 |
82
|
iffalsed |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) = 𝐾 ) |
84 |
79 83 77
|
3eqtrd |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → 𝐼 = 1 ) |
85 |
84
|
oveq2d |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( 𝑅 ↑𝑟 𝐼 ) = ( 𝑅 ↑𝑟 1 ) ) |
86 |
78 85
|
eqtr4d |
⊢ ( ( 𝐽 = 1 ∧ 𝐾 = 1 ∧ ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) |
87 |
86
|
3exp |
⊢ ( 𝐽 = 1 → ( 𝐾 = 1 → ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) ) ) |
88 |
71 87
|
jaoi |
⊢ ( ( 𝐽 = 0 ∨ 𝐽 = 1 ) → ( 𝐾 = 1 → ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) ) ) |
89 |
56 88
|
jaod |
⊢ ( ( 𝐽 = 0 ∨ 𝐽 = 1 ) → ( ( 𝐾 = 0 ∨ 𝐾 = 1 ) → ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) ) ) |
90 |
89
|
imp |
⊢ ( ( ( 𝐽 = 0 ∨ 𝐽 = 1 ) ∧ ( 𝐾 = 0 ∨ 𝐾 = 1 ) ) → ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) ) |
91 |
1 2 90
|
syl2an |
⊢ ( ( 𝐽 ∈ { 0 , 1 } ∧ 𝐾 ∈ { 0 , 1 } ) → ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) ) |
92 |
91
|
impcom |
⊢ ( ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 = if ( 𝐽 < 𝐾 , 𝐽 , 𝐾 ) ) ∧ ( 𝐽 ∈ { 0 , 1 } ∧ 𝐾 ∈ { 0 , 1 } ) ) → ( ( 𝑅 ↑𝑟 𝐽 ) ↑𝑟 𝐾 ) = ( 𝑅 ↑𝑟 𝐼 ) ) |