Metamath Proof Explorer


Theorem mgcmnt2

Description: The upper adjoint G 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 ( 𝜑𝑉 ∈ Proset )
mgcval.3 ( 𝜑𝑊 ∈ Proset )
mgccole.1 ( 𝜑𝐹 𝐻 𝐺 )
mgcmnt2.1 ( 𝜑𝑋𝐵 )
mgcmnt2.2 ( 𝜑𝑌𝐵 )
mgcmnt2.3 ( 𝜑𝑋 𝑌 )
Assertion mgcmnt2 ( 𝜑 → ( 𝐺𝑋 ) ( 𝐺𝑌 ) )

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 ( 𝜑𝑉 ∈ Proset )
7 mgcval.3 ( 𝜑𝑊 ∈ Proset )
8 mgccole.1 ( 𝜑𝐹 𝐻 𝐺 )
9 mgcmnt2.1 ( 𝜑𝑋𝐵 )
10 mgcmnt2.2 ( 𝜑𝑌𝐵 )
11 mgcmnt2.3 ( 𝜑𝑋 𝑌 )
12 1 2 3 4 5 6 7 mgcval ( 𝜑 → ( 𝐹 𝐻 𝐺 ↔ ( ( 𝐹 : 𝐴𝐵𝐺 : 𝐵𝐴 ) ∧ ∀ 𝑥𝐴𝑦𝐵 ( ( 𝐹𝑥 ) 𝑦𝑥 ( 𝐺𝑦 ) ) ) ) )
13 8 12 mpbid ( 𝜑 → ( ( 𝐹 : 𝐴𝐵𝐺 : 𝐵𝐴 ) ∧ ∀ 𝑥𝐴𝑦𝐵 ( ( 𝐹𝑥 ) 𝑦𝑥 ( 𝐺𝑦 ) ) ) )
14 13 simplld ( 𝜑𝐹 : 𝐴𝐵 )
15 13 simplrd ( 𝜑𝐺 : 𝐵𝐴 )
16 15 9 ffvelrnd ( 𝜑 → ( 𝐺𝑋 ) ∈ 𝐴 )
17 14 16 ffvelrnd ( 𝜑 → ( 𝐹 ‘ ( 𝐺𝑋 ) ) ∈ 𝐵 )
18 1 2 3 4 5 6 7 8 9 mgccole2 ( 𝜑 → ( 𝐹 ‘ ( 𝐺𝑋 ) ) 𝑋 )
19 2 4 prstr ( ( 𝑊 ∈ Proset ∧ ( ( 𝐹 ‘ ( 𝐺𝑋 ) ) ∈ 𝐵𝑋𝐵𝑌𝐵 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑋 ) ) 𝑋𝑋 𝑌 ) ) → ( 𝐹 ‘ ( 𝐺𝑋 ) ) 𝑌 )
20 7 17 9 10 18 11 19 syl132anc ( 𝜑 → ( 𝐹 ‘ ( 𝐺𝑋 ) ) 𝑌 )
21 breq2 ( 𝑦 = 𝑌 → ( ( 𝐹 ‘ ( 𝐺𝑋 ) ) 𝑦 ↔ ( 𝐹 ‘ ( 𝐺𝑋 ) ) 𝑌 ) )
22 fveq2 ( 𝑦 = 𝑌 → ( 𝐺𝑦 ) = ( 𝐺𝑌 ) )
23 22 breq2d ( 𝑦 = 𝑌 → ( ( 𝐺𝑋 ) ( 𝐺𝑦 ) ↔ ( 𝐺𝑋 ) ( 𝐺𝑌 ) ) )
24 21 23 bibi12d ( 𝑦 = 𝑌 → ( ( ( 𝐹 ‘ ( 𝐺𝑋 ) ) 𝑦 ↔ ( 𝐺𝑋 ) ( 𝐺𝑦 ) ) ↔ ( ( 𝐹 ‘ ( 𝐺𝑋 ) ) 𝑌 ↔ ( 𝐺𝑋 ) ( 𝐺𝑌 ) ) ) )
25 fveq2 ( 𝑥 = ( 𝐺𝑋 ) → ( 𝐹𝑥 ) = ( 𝐹 ‘ ( 𝐺𝑋 ) ) )
26 25 breq1d ( 𝑥 = ( 𝐺𝑋 ) → ( ( 𝐹𝑥 ) 𝑦 ↔ ( 𝐹 ‘ ( 𝐺𝑋 ) ) 𝑦 ) )
27 breq1 ( 𝑥 = ( 𝐺𝑋 ) → ( 𝑥 ( 𝐺𝑦 ) ↔ ( 𝐺𝑋 ) ( 𝐺𝑦 ) ) )
28 26 27 bibi12d ( 𝑥 = ( 𝐺𝑋 ) → ( ( ( 𝐹𝑥 ) 𝑦𝑥 ( 𝐺𝑦 ) ) ↔ ( ( 𝐹 ‘ ( 𝐺𝑋 ) ) 𝑦 ↔ ( 𝐺𝑋 ) ( 𝐺𝑦 ) ) ) )
29 28 ralbidv ( 𝑥 = ( 𝐺𝑋 ) → ( ∀ 𝑦𝐵 ( ( 𝐹𝑥 ) 𝑦𝑥 ( 𝐺𝑦 ) ) ↔ ∀ 𝑦𝐵 ( ( 𝐹 ‘ ( 𝐺𝑋 ) ) 𝑦 ↔ ( 𝐺𝑋 ) ( 𝐺𝑦 ) ) ) )
30 13 simprd ( 𝜑 → ∀ 𝑥𝐴𝑦𝐵 ( ( 𝐹𝑥 ) 𝑦𝑥 ( 𝐺𝑦 ) ) )
31 29 30 16 rspcdva ( 𝜑 → ∀ 𝑦𝐵 ( ( 𝐹 ‘ ( 𝐺𝑋 ) ) 𝑦 ↔ ( 𝐺𝑋 ) ( 𝐺𝑦 ) ) )
32 24 31 10 rspcdva ( 𝜑 → ( ( 𝐹 ‘ ( 𝐺𝑋 ) ) 𝑌 ↔ ( 𝐺𝑋 ) ( 𝐺𝑌 ) ) )
33 20 32 mpbid ( 𝜑 → ( 𝐺𝑋 ) ( 𝐺𝑌 ) )