Metamath Proof Explorer

Theorem mgcmnt1

Description: The lower adjoint F of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024)

Ref Expression
Hypotheses mgcoval.1 ⊢ 𝐴 = ( Base ‘ 𝑉 )
mgcoval.2 ⊢ 𝐵 = ( Base ‘ 𝑊 )
mgcoval.3 ⊢ ≤ = ( le ‘ 𝑉 )
mgcoval.4 ⊢ ≲ = ( le ‘ 𝑊 )
mgcval.1 ⊢ 𝐻 = ( 𝑉 MGalConn 𝑊 )
mgcval.2 ⊢ ( 𝜑 → 𝑉 ∈ Proset )
mgcval.3 ⊢ ( 𝜑 → 𝑊 ∈ Proset )
mgccole.1 ⊢ ( 𝜑 → 𝐹 𝐻 𝐺 )
mgcmnt1.1 ⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
mgcmnt1.2 ⊢ ( 𝜑 → 𝑌 ∈ 𝐴 )
mgcmnt1.3 ⊢ ( 𝜑 → 𝑋 ≤ 𝑌 )
Assertion mgcmnt1 ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑌 ) )

Proof

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 mgcmnt1.1 ⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
10 mgcmnt1.2 ⊢ ( 𝜑 → 𝑌 ∈ 𝐴 )
11 mgcmnt1.3 ⊢ ( 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 simplrd ⊢ ( 𝜑 → 𝐺 : 𝐵 ⟶ 𝐴 )
15 13 simplld ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
16 15 10 ffvelcdmd ⊢ ( 𝜑 → ( 𝐹 ‘ 𝑌 ) ∈ 𝐵 )
17 14 16 ffvelcdmd ⊢ ( 𝜑 → ( 𝐺 ‘ ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐴 )
18 1 2 3 4 5 6 7 8 10 mgccole1 ⊢ ( 𝜑 → 𝑌 ≤ ( 𝐺 ‘ ( 𝐹 ‘ 𝑌 ) ) )
19 1 3 prstr ⊢ ( ( 𝑉 ∈ Proset ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ ( 𝐺 ‘ ( 𝐹 ‘ 𝑌 ) ) ∈ 𝐴 ) ∧ ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ ( 𝐺 ‘ ( 𝐹 ‘ 𝑌 ) ) ) ) → 𝑋 ≤ ( 𝐺 ‘ ( 𝐹 ‘ 𝑌 ) ) )
20 6 9 10 17 11 18 19 syl132anc ⊢ ( 𝜑 → 𝑋 ≤ ( 𝐺 ‘ ( 𝐹 ‘ 𝑌 ) ) )
21 13 simprd ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝐺 ‘ 𝑦 ) ) )
22 fveq2 ⊢ ( 𝑥 = 𝑋 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑋 ) )
23 22 breq1d ⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ‘ 𝑥 ) ≲ 𝑦 ↔ ( 𝐹 ‘ 𝑋 ) ≲ 𝑦 ) )
24 breq1 ⊢ ( 𝑥 = 𝑋 → ( 𝑥 ≤ ( 𝐺 ‘ 𝑦 ) ↔ 𝑋 ≤ ( 𝐺 ‘ 𝑦 ) ) )
25 23 24 bibi12d ⊢ ( 𝑥 = 𝑋 → ( ( ( 𝐹 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝐺 ‘ 𝑦 ) ) ↔ ( ( 𝐹 ‘ 𝑋 ) ≲ 𝑦 ↔ 𝑋 ≤ ( 𝐺 ‘ 𝑦 ) ) ) )
26 25 adantl ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( ( ( 𝐹 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝐺 ‘ 𝑦 ) ) ↔ ( ( 𝐹 ‘ 𝑋 ) ≲ 𝑦 ↔ 𝑋 ≤ ( 𝐺 ‘ 𝑦 ) ) ) )
27 26 ralbidv ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝐺 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑋 ) ≲ 𝑦 ↔ 𝑋 ≤ ( 𝐺 ‘ 𝑦 ) ) ) )
28 9 27 rspcdv ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝐺 ‘ 𝑦 ) ) → ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑋 ) ≲ 𝑦 ↔ 𝑋 ≤ ( 𝐺 ‘ 𝑦 ) ) ) )
29 21 28 mpd ⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑋 ) ≲ 𝑦 ↔ 𝑋 ≤ ( 𝐺 ‘ 𝑦 ) ) )
30 simpr ⊢ ( ( 𝜑 ∧ 𝑦 = ( 𝐹 ‘ 𝑌 ) ) → 𝑦 = ( 𝐹 ‘ 𝑌 ) )
31 30 breq2d ⊢ ( ( 𝜑 ∧ 𝑦 = ( 𝐹 ‘ 𝑌 ) ) → ( ( 𝐹 ‘ 𝑋 ) ≲ 𝑦 ↔ ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑌 ) ) )
32 30 fveq2d ⊢ ( ( 𝜑 ∧ 𝑦 = ( 𝐹 ‘ 𝑌 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐹 ‘ 𝑌 ) ) )
33 32 breq2d ⊢ ( ( 𝜑 ∧ 𝑦 = ( 𝐹 ‘ 𝑌 ) ) → ( 𝑋 ≤ ( 𝐺 ‘ 𝑦 ) ↔ 𝑋 ≤ ( 𝐺 ‘ ( 𝐹 ‘ 𝑌 ) ) ) )
34 31 33 bibi12d ⊢ ( ( 𝜑 ∧ 𝑦 = ( 𝐹 ‘ 𝑌 ) ) → ( ( ( 𝐹 ‘ 𝑋 ) ≲ 𝑦 ↔ 𝑋 ≤ ( 𝐺 ‘ 𝑦 ) ) ↔ ( ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 ≤ ( 𝐺 ‘ ( 𝐹 ‘ 𝑌 ) ) ) ) )
35 16 34 rspcdv ⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐵 ( ( 𝐹 ‘ 𝑋 ) ≲ 𝑦 ↔ 𝑋 ≤ ( 𝐺 ‘ 𝑦 ) ) → ( ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 ≤ ( 𝐺 ‘ ( 𝐹 ‘ 𝑌 ) ) ) ) )
36 29 35 mpd ⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 ≤ ( 𝐺 ‘ ( 𝐹 ‘ 𝑌 ) ) ) )
37 20 36 mpbird ⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑌 ) )