Theorem mgcoval 31264

Description: Operation value of the monotone Galois connection. (Contributed by Thierry Arnoux, 23-Apr-2024.)

Hypotheses

Ref	Expression
mgcoval.1	⊢ 𝐴 = (Base‘𝑉)
mgcoval.2	⊢ 𝐵 = (Base‘𝑊)
mgcoval.3	⊢ ≤ = (le‘𝑉)
mgcoval.4	⊢ ≲ = (le‘𝑊)

Assertion

Ref	Expression
mgcoval	⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))})

Distinct variable groups:   𝐴,𝑓,𝑔,𝑥,𝑦   𝐵,𝑓,𝑔,𝑥,𝑦   𝑓,𝑉,𝑔,𝑥,𝑦   𝑓,𝑊,𝑔,𝑥,𝑦   𝑓,𝑋,𝑔,𝑥,𝑦   𝑓,𝑌,𝑔,𝑥,𝑦

Allowed substitution hints:   ≤ (𝑥,𝑦,𝑓,𝑔)   ≲ (𝑥,𝑦,𝑓,𝑔)

Proof of Theorem mgcoval

Dummy variables 𝑎 𝑏 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mgc 31259	. . 3 ⊢ MGalConn = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌⦋(Base‘𝑤) / 𝑏⦌{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏 ↑m 𝑎) ∧ 𝑔 ∈ (𝑎 ↑m 𝑏)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)))})
2	1	a1i 11	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → MGalConn = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌⦋(Base‘𝑤) / 𝑏⦌{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏 ↑m 𝑎) ∧ 𝑔 ∈ (𝑎 ↑m 𝑏)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)))}))
3		fvexd 6789	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) → (Base‘𝑣) ∈ V)
4		simprl 768	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) → 𝑣 = 𝑉)
5	4	fveq2d 6778	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) → (Base‘𝑣) = (Base‘𝑉))
6		mgcoval.1	. . . 4 ⊢ 𝐴 = (Base‘𝑉)
7	5, 6	eqtr4di 2796	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) → (Base‘𝑣) = 𝐴)
8		fvexd 6789	. . . 4 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) ∈ V)
9		simplrr 775	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → 𝑤 = 𝑊)
10	9	fveq2d 6778	. . . . 5 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
11		mgcoval.2	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
12	10, 11	eqtr4di 2796	. . . 4 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
13		simpr 485	. . . . . . . . 9 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
14		simplr 766	. . . . . . . . 9 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑎 = 𝐴)
15	13, 14	oveq12d 7293	. . . . . . . 8 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑏 ↑m 𝑎) = (𝐵 ↑m 𝐴))
16	15	eleq2d 2824	. . . . . . 7 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑓 ∈ (𝑏 ↑m 𝑎) ↔ 𝑓 ∈ (𝐵 ↑m 𝐴)))
17	14, 13	oveq12d 7293	. . . . . . . 8 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑎 ↑m 𝑏) = (𝐴 ↑m 𝐵))
18	17	eleq2d 2824	. . . . . . 7 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑔 ∈ (𝑎 ↑m 𝑏) ↔ 𝑔 ∈ (𝐴 ↑m 𝐵)))
19	16, 18	anbi12d 631	. . . . . 6 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ((𝑓 ∈ (𝑏 ↑m 𝑎) ∧ 𝑔 ∈ (𝑎 ↑m 𝑏)) ↔ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵))))
20	9	adantr 481	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑤 = 𝑊)
21	20	fveq2d 6778	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑤) = (le‘𝑊))
22		mgcoval.4	. . . . . . . . . . 11 ⊢ ≲ = (le‘𝑊)
23	21, 22	eqtr4di 2796	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑤) = ≲ )
24	23	breqd 5085	. . . . . . . . 9 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ (𝑓‘𝑥) ≲ 𝑦))
25	4	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑣 = 𝑉)
26	25	fveq2d 6778	. . . . . . . . . . 11 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑣) = (le‘𝑉))
27		mgcoval.3	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝑉)
28	26, 27	eqtr4di 2796	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑣) = ≤ )
29	28	breqd 5085	. . . . . . . . 9 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑥(le‘𝑣)(𝑔‘𝑦) ↔ 𝑥 ≤ (𝑔‘𝑦)))
30	24, 29	bibi12d 346	. . . . . . . 8 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)) ↔ ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦))))
31	13, 30	raleqbidv 3336	. . . . . . 7 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦))))
32	14, 31	raleqbidv 3336	. . . . . 6 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦))))
33	19, 32	anbi12d 631	. . . . 5 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (((𝑓 ∈ (𝑏 ↑m 𝑎) ∧ 𝑔 ∈ (𝑎 ↑m 𝑏)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦))) ↔ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))))
34	33	opabbidv 5140	. . . 4 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏 ↑m 𝑎) ∧ 𝑔 ∈ (𝑎 ↑m 𝑏)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))})
35	8, 12, 34	csbied2 3872	. . 3 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → ⦋(Base‘𝑤) / 𝑏⦌{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏 ↑m 𝑎) ∧ 𝑔 ∈ (𝑎 ↑m 𝑏)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))})
36	3, 7, 35	csbied2 3872	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) → ⦋(Base‘𝑣) / 𝑎⦌⦋(Base‘𝑤) / 𝑏⦌{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏 ↑m 𝑎) ∧ 𝑔 ∈ (𝑎 ↑m 𝑏)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))})
37		simpl 483	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑉 ∈ 𝑋)
38	37	elexd 3452	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑉 ∈ V)
39		simpr 485	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑊 ∈ 𝑌)
40	39	elexd 3452	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑊 ∈ V)
41		ovexd 7310	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐵 ↑m 𝐴) ∈ V)
42		ovexd 7310	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐴 ↑m 𝐵) ∈ V)
43		simprll 776	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))) → 𝑓 ∈ (𝐵 ↑m 𝐴))
44		simprlr 777	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))) → 𝑔 ∈ (𝐴 ↑m 𝐵))
45	41, 42, 43, 44	opabex2 7897	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))} ∈ V)
46	2, 36, 38, 40, 45	ovmpod 7425	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432  ⦋csb 3832   class class class wbr 5074  {copab 5136  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   ↑_m cmap 8615  Basecbs 16912  lecple 16969  MGalConncmgc 31257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-mgc 31259

This theorem is referenced by:  mgcval  31265