| Step | Hyp | Ref
| Expression |
| 1 | | mgcval.3 |
. . 3
⊢ (𝜑 → 𝑊 ∈ Proset ) |
| 2 | | mgccole.1 |
. . . . . 6
⊢ (𝜑 → 𝐹𝐻𝐺) |
| 3 | | mgcoval.1 |
. . . . . . 7
⊢ 𝐴 = (Base‘𝑉) |
| 4 | | mgcoval.2 |
. . . . . . 7
⊢ 𝐵 = (Base‘𝑊) |
| 5 | | mgcoval.3 |
. . . . . . 7
⊢ ≤ =
(le‘𝑉) |
| 6 | | mgcoval.4 |
. . . . . . 7
⊢ ≲ =
(le‘𝑊) |
| 7 | | mgcval.1 |
. . . . . . 7
⊢ 𝐻 = (𝑉MGalConn𝑊) |
| 8 | | mgcval.2 |
. . . . . . 7
⊢ (𝜑 → 𝑉 ∈ Proset ) |
| 9 | 3, 4, 5, 6, 7, 8, 1 | mgcval 32977 |
. . . . . 6
⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))) |
| 10 | 2, 9 | mpbid 232 |
. . . . 5
⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))) |
| 11 | 10 | simplld 768 |
. . . 4
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
| 12 | 10 | simplrd 770 |
. . . . 5
⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
| 13 | | mgcmnt2.1 |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| 14 | 12, 13 | ffvelcdmd 7105 |
. . . 4
⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴) |
| 15 | 11, 14 | ffvelcdmd 7105 |
. . 3
⊢ (𝜑 → (𝐹‘(𝐺‘𝑋)) ∈ 𝐵) |
| 16 | | mgcmnt2.2 |
. . 3
⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| 17 | 3, 4, 5, 6, 7, 8, 1, 2, 13 | mgccole2 32981 |
. . 3
⊢ (𝜑 → (𝐹‘(𝐺‘𝑋)) ≲ 𝑋) |
| 18 | | mgcmnt2.3 |
. . 3
⊢ (𝜑 → 𝑋 ≲ 𝑌) |
| 19 | 4, 6 | prstr 18345 |
. . 3
⊢ ((𝑊 ∈ Proset ∧ ((𝐹‘(𝐺‘𝑋)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐹‘(𝐺‘𝑋)) ≲ 𝑋 ∧ 𝑋 ≲ 𝑌)) → (𝐹‘(𝐺‘𝑋)) ≲ 𝑌) |
| 20 | 1, 15, 13, 16, 17, 18, 19 | syl132anc 1390 |
. 2
⊢ (𝜑 → (𝐹‘(𝐺‘𝑋)) ≲ 𝑌) |
| 21 | | breq2 5147 |
. . . 4
⊢ (𝑦 = 𝑌 → ((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑋)) ≲ 𝑌)) |
| 22 | | fveq2 6906 |
. . . . 5
⊢ (𝑦 = 𝑌 → (𝐺‘𝑦) = (𝐺‘𝑌)) |
| 23 | 22 | breq2d 5155 |
. . . 4
⊢ (𝑦 = 𝑌 → ((𝐺‘𝑋) ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑌))) |
| 24 | 21, 23 | bibi12d 345 |
. . 3
⊢ (𝑦 = 𝑌 → (((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑋)) ≲ 𝑌 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑌)))) |
| 25 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑥 = (𝐺‘𝑋) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑋))) |
| 26 | 25 | breq1d 5153 |
. . . . . 6
⊢ (𝑥 = (𝐺‘𝑋) → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑋)) ≲ 𝑦)) |
| 27 | | breq1 5146 |
. . . . . 6
⊢ (𝑥 = (𝐺‘𝑋) → (𝑥 ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦))) |
| 28 | 26, 27 | bibi12d 345 |
. . . . 5
⊢ (𝑥 = (𝐺‘𝑋) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦)))) |
| 29 | 28 | ralbidv 3178 |
. . . 4
⊢ (𝑥 = (𝐺‘𝑋) → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦)))) |
| 30 | 10 | simprd 495 |
. . . 4
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))) |
| 31 | 29, 30, 14 | rspcdva 3623 |
. . 3
⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦))) |
| 32 | 24, 31, 16 | rspcdva 3623 |
. 2
⊢ (𝜑 → ((𝐹‘(𝐺‘𝑋)) ≲ 𝑌 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑌))) |
| 33 | 20, 32 | mpbid 232 |
1
⊢ (𝜑 → (𝐺‘𝑋) ≤ (𝐺‘𝑌)) |