Step |
Hyp |
Ref |
Expression |
1 |
|
mgcoval.1 |
⊢ 𝐴 = ( Base ‘ 𝑉 ) |
2 |
|
mgcoval.2 |
⊢ 𝐵 = ( Base ‘ 𝑊 ) |
3 |
|
mgcoval.3 |
⊢ ≤ = ( le ‘ 𝑉 ) |
4 |
|
mgcoval.4 |
⊢ ≲ = ( le ‘ 𝑊 ) |
5 |
|
mgcval.1 |
⊢ 𝐻 = ( 𝑉 MGalConn 𝑊 ) |
6 |
|
mgcval.2 |
⊢ ( 𝜑 → 𝑉 ∈ Proset ) |
7 |
|
mgcval.3 |
⊢ ( 𝜑 → 𝑊 ∈ Proset ) |
8 |
|
mgccole.1 |
⊢ ( 𝜑 → 𝐹 𝐻 𝐺 ) |
9 |
|
mgcmnt2.1 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
10 |
|
mgcmnt2.2 |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
11 |
|
mgcmnt2.3 |
⊢ ( 𝜑 → 𝑋 ≲ 𝑌 ) |
12 |
1 2 3 4 5 6 7
|
mgcval |
⊢ ( 𝜑 → ( 𝐹 𝐻 𝐺 ↔ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 : 𝐵 ⟶ 𝐴 ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝐺 ‘ 𝑦 ) ) ) ) ) |
13 |
8 12
|
mpbid |
⊢ ( 𝜑 → ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐺 : 𝐵 ⟶ 𝐴 ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝐺 ‘ 𝑦 ) ) ) ) |
14 |
13
|
simplld |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
15 |
13
|
simplrd |
⊢ ( 𝜑 → 𝐺 : 𝐵 ⟶ 𝐴 ) |
16 |
15 9
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) ∈ 𝐴 ) |
17 |
14 16
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ∈ 𝐵 ) |
18 |
1 2 3 4 5 6 7 8 9
|
mgccole2 |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ≲ 𝑋 ) |
19 |
2 4
|
prstr |
⊢ ( ( 𝑊 ∈ Proset ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ≲ 𝑋 ∧ 𝑋 ≲ 𝑌 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ≲ 𝑌 ) |
20 |
7 17 9 10 18 11 19
|
syl132anc |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ≲ 𝑌 ) |
21 |
|
breq2 |
⊢ ( 𝑦 = 𝑌 → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ≲ 𝑦 ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ≲ 𝑌 ) ) |
22 |
|
fveq2 |
⊢ ( 𝑦 = 𝑌 → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑌 ) ) |
23 |
22
|
breq2d |
⊢ ( 𝑦 = 𝑌 → ( ( 𝐺 ‘ 𝑋 ) ≤ ( 𝐺 ‘ 𝑦 ) ↔ ( 𝐺 ‘ 𝑋 ) ≤ ( 𝐺 ‘ 𝑌 ) ) ) |
24 |
21 23
|
bibi12d |
⊢ ( 𝑦 = 𝑌 → ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ≲ 𝑦 ↔ ( 𝐺 ‘ 𝑋 ) ≤ ( 𝐺 ‘ 𝑦 ) ) ↔ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ≲ 𝑌 ↔ ( 𝐺 ‘ 𝑋 ) ≤ ( 𝐺 ‘ 𝑌 ) ) ) ) |
25 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝐺 ‘ 𝑋 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ) |
26 |
25
|
breq1d |
⊢ ( 𝑥 = ( 𝐺 ‘ 𝑋 ) → ( ( 𝐹 ‘ 𝑥 ) ≲ 𝑦 ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ≲ 𝑦 ) ) |
27 |
|
breq1 |
⊢ ( 𝑥 = ( 𝐺 ‘ 𝑋 ) → ( 𝑥 ≤ ( 𝐺 ‘ 𝑦 ) ↔ ( 𝐺 ‘ 𝑋 ) ≤ ( 𝐺 ‘ 𝑦 ) ) ) |
28 |
26 27
|
bibi12d |
⊢ ( 𝑥 = ( 𝐺 ‘ 𝑋 ) → ( ( ( 𝐹 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝐺 ‘ 𝑦 ) ) ↔ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ≲ 𝑦 ↔ ( 𝐺 ‘ 𝑋 ) ≤ ( 𝐺 ‘ 𝑦 ) ) ) ) |
29 |
28
|
ralbidv |
⊢ ( 𝑥 = ( 𝐺 ‘ 𝑋 ) → ( ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝐺 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ≲ 𝑦 ↔ ( 𝐺 ‘ 𝑋 ) ≤ ( 𝐺 ‘ 𝑦 ) ) ) ) |
30 |
13
|
simprd |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝐺 ‘ 𝑦 ) ) ) |
31 |
29 30 16
|
rspcdva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ≲ 𝑦 ↔ ( 𝐺 ‘ 𝑋 ) ≤ ( 𝐺 ‘ 𝑦 ) ) ) |
32 |
24 31 10
|
rspcdva |
⊢ ( 𝜑 → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) ≲ 𝑌 ↔ ( 𝐺 ‘ 𝑋 ) ≤ ( 𝐺 ‘ 𝑌 ) ) ) |
33 |
20 32
|
mpbid |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) ≤ ( 𝐺 ‘ 𝑌 ) ) |