Theorem mgcoval 32971

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 32966	. . 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 6896	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) → (Base‘𝑣) ∈ V)
4		simprl 770	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) → 𝑣 = 𝑉)
5	4	fveq2d 6885	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) → (Base‘𝑣) = (Base‘𝑉))
6		mgcoval.1	. . . 4 ⊢ 𝐴 = (Base‘𝑉)
7	5, 6	eqtr4di 2789	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) → (Base‘𝑣) = 𝐴)
8		fvexd 6896	. . . 4 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) ∈ V)
9		simplrr 777	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → 𝑤 = 𝑊)
10	9	fveq2d 6885	. . . . 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 2789	. . . 4 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
13		simpr 484	. . . . . . . . 9 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
14		simplr 768	. . . . . . . . 9 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑎 = 𝐴)
15	13, 14	oveq12d 7428	. . . . . . . 8 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑏 ↑m 𝑎) = (𝐵 ↑m 𝐴))
16	15	eleq2d 2821	. . . . . . 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 7428	. . . . . . . 8 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑎 ↑m 𝑏) = (𝐴 ↑m 𝐵))
18	17	eleq2d 2821	. . . . . . 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 632	. . . . . 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 480	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑤 = 𝑊)
21	20	fveq2d 6885	. . . . . . . . . . 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 2789	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑤) = ≲ )
24	23	breqd 5135	. . . . . . . . 9 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ (𝑓‘𝑥) ≲ 𝑦))
25	4	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑣 = 𝑉)
26	25	fveq2d 6885	. . . . . . . . . . 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 2789	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑣) = ≤ )
29	28	breqd 5135	. . . . . . . . 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 345	. . . . . . . 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 3329	. . . . . . 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 3329	. . . . . 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 632	. . . . 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 5190	. . . 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 3916	. . 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 3916	. 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 482	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑉 ∈ 𝑋)
38	37	elexd 3488	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑉 ∈ V)
39		simpr 484	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑊 ∈ 𝑌)
40	39	elexd 3488	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑊 ∈ V)
41		ovexd 7445	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐵 ↑m 𝐴) ∈ V)
42		ovexd 7445	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐴 ↑m 𝐵) ∈ V)
43		simprll 778	. . 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 779	. . 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 8061	. 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 7564	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464  ⦋csb 3879   class class class wbr 5124  {copab 5186  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   ↑_m cmap 8845  Basecbs 17233  lecple 17283  MGalConncmgc 32964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-mgc 32966

This theorem is referenced by:  mgcval  32972