Step |
Hyp |
Ref |
Expression |
1 |
|
oicl.1 |
⊢ 𝐹 = OrdIso ( 𝑅 , 𝐴 ) |
2 |
|
simprr |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) |
3 |
1
|
ordtype |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ) |
4 |
3
|
adantr |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → 𝐹 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ) |
5 |
|
isocnv |
⊢ ( 𝐹 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) → ◡ 𝐹 Isom 𝑅 , E ( 𝐴 , dom 𝐹 ) ) |
6 |
4 5
|
syl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → ◡ 𝐹 Isom 𝑅 , E ( 𝐴 , dom 𝐹 ) ) |
7 |
|
isotr |
⊢ ( ( 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ∧ ◡ 𝐹 Isom 𝑅 , E ( 𝐴 , dom 𝐹 ) ) → ( ◡ 𝐹 ∘ 𝐺 ) Isom E , E ( 𝐵 , dom 𝐹 ) ) |
8 |
2 6 7
|
syl2anc |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → ( ◡ 𝐹 ∘ 𝐺 ) Isom E , E ( 𝐵 , dom 𝐹 ) ) |
9 |
|
simprl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → Ord 𝐵 ) |
10 |
1
|
oicl |
⊢ Ord dom 𝐹 |
11 |
10
|
a1i |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → Ord dom 𝐹 ) |
12 |
|
ordiso2 |
⊢ ( ( ( ◡ 𝐹 ∘ 𝐺 ) Isom E , E ( 𝐵 , dom 𝐹 ) ∧ Ord 𝐵 ∧ Ord dom 𝐹 ) → 𝐵 = dom 𝐹 ) |
13 |
8 9 11 12
|
syl3anc |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → 𝐵 = dom 𝐹 ) |
14 |
|
ordwe |
⊢ ( Ord 𝐵 → E We 𝐵 ) |
15 |
14
|
ad2antrl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → E We 𝐵 ) |
16 |
|
epse |
⊢ E Se 𝐵 |
17 |
16
|
a1i |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → E Se 𝐵 ) |
18 |
|
isoeq4 |
⊢ ( 𝐵 = dom 𝐹 → ( 𝐹 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ↔ 𝐹 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ) ) |
19 |
13 18
|
syl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → ( 𝐹 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ↔ 𝐹 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ) ) |
20 |
4 19
|
mpbird |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → 𝐹 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) |
21 |
|
weisoeq |
⊢ ( ( ( E We 𝐵 ∧ E Se 𝐵 ) ∧ ( 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ∧ 𝐹 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → 𝐺 = 𝐹 ) |
22 |
15 17 2 20 21
|
syl22anc |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → 𝐺 = 𝐹 ) |
23 |
13 22
|
jca |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) → ( 𝐵 = dom 𝐹 ∧ 𝐺 = 𝐹 ) ) |
24 |
23
|
ex |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ( ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) → ( 𝐵 = dom 𝐹 ∧ 𝐺 = 𝐹 ) ) ) |
25 |
3 10
|
jctil |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ( Ord dom 𝐹 ∧ 𝐹 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ) ) |
26 |
|
ordeq |
⊢ ( 𝐵 = dom 𝐹 → ( Ord 𝐵 ↔ Ord dom 𝐹 ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝐵 = dom 𝐹 ∧ 𝐺 = 𝐹 ) → ( Ord 𝐵 ↔ Ord dom 𝐹 ) ) |
28 |
|
isoeq4 |
⊢ ( 𝐵 = dom 𝐹 → ( 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ↔ 𝐺 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ) ) |
29 |
|
isoeq1 |
⊢ ( 𝐺 = 𝐹 → ( 𝐺 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ↔ 𝐹 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ) ) |
30 |
28 29
|
sylan9bb |
⊢ ( ( 𝐵 = dom 𝐹 ∧ 𝐺 = 𝐹 ) → ( 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ↔ 𝐹 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ) ) |
31 |
27 30
|
anbi12d |
⊢ ( ( 𝐵 = dom 𝐹 ∧ 𝐺 = 𝐹 ) → ( ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ↔ ( Ord dom 𝐹 ∧ 𝐹 Isom E , 𝑅 ( dom 𝐹 , 𝐴 ) ) ) ) |
32 |
25 31
|
syl5ibrcom |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ( ( 𝐵 = dom 𝐹 ∧ 𝐺 = 𝐹 ) → ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ) ) |
33 |
24 32
|
impbid |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ( ( Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 ( 𝐵 , 𝐴 ) ) ↔ ( 𝐵 = dom 𝐹 ∧ 𝐺 = 𝐹 ) ) ) |