Step |
Hyp |
Ref |
Expression |
1 |
|
lautlt.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
lautlt.s |
⊢ < = ( lt ‘ 𝐾 ) |
3 |
|
lautlt.i |
⊢ 𝐼 = ( LAut ‘ 𝐾 ) |
4 |
|
simpl |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐾 ∈ 𝐴 ) |
5 |
|
simpr1 |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐹 ∈ 𝐼 ) |
6 |
|
simpr2 |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 ) |
7 |
|
simpr3 |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 ) |
8 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
9 |
1 8 3
|
lautle |
⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ) ) |
10 |
4 5 6 7 9
|
syl22anc |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ) ) |
11 |
1 3
|
laut11 |
⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) ) |
12 |
4 5 6 7 11
|
syl22anc |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) ) |
13 |
12
|
bicomd |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 = 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ) ) |
14 |
13
|
necon3bid |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ≠ 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝑌 ) ) ) |
15 |
10 14
|
anbi12d |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑋 ≠ 𝑌 ) ↔ ( ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝑌 ) ) ) ) |
16 |
8 2
|
pltval |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 ↔ ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) ) |
17 |
16
|
3adant3r1 |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 < 𝑌 ↔ ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) ) |
18 |
1 3
|
lautcl |
⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝐹 ∈ 𝐼 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ) |
19 |
4 5 6 18
|
syl21anc |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ) |
20 |
1 3
|
lautcl |
⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝐹 ∈ 𝐼 ) ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 ) |
21 |
4 5 7 20
|
syl21anc |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 ) |
22 |
8 2
|
pltval |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑋 ) < ( 𝐹 ‘ 𝑌 ) ↔ ( ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝑌 ) ) ) ) |
23 |
4 19 21 22
|
syl3anc |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) < ( 𝐹 ‘ 𝑌 ) ↔ ( ( 𝐹 ‘ 𝑋 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑌 ) ∧ ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝑌 ) ) ) ) |
24 |
15 17 23
|
3bitr4d |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝐹 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 < 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) < ( 𝐹 ‘ 𝑌 ) ) ) |