Theorem mgcval 33541

Description: Monotone Galois connection between two functions 𝐹 and 𝐺. If this relation is satisfied, 𝐹 is called the lower adjoint of 𝐺, and 𝐺 is called the upper adjoint of 𝐹.

Technically, this is implemented as an operation taking a pair of structures 𝑉 and 𝑊, expected to be posets, which gives a relation between pairs of functions 𝐹 and 𝐺.

If such a relation exists, it can be proven to be unique.

Galois connections generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields. (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)
mgcval.1	⊢ 𝐻 = (𝑉MGalConn𝑊)
mgcval.2	⊢ (𝜑 → 𝑉 ∈ Proset )
mgcval.3	⊢ (𝜑 → 𝑊 ∈ Proset )

Assertion

Ref	Expression
mgcval	⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐻(𝑥, 𝑦)   ≤ (𝑥, 𝑦)   ≲ (𝑥, 𝑦)

Proof of Theorem mgcval

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgcval.1	. . . 4 ⊢ 𝐻 = (𝑉MGalConn𝑊)
2		mgcval.2	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ Proset )
3		mgcval.3	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Proset )
4		mgcoval.1	. . . . . 6 ⊢ 𝐴 = (Base‘𝑉)
5		mgcoval.2	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
6		mgcoval.3	. . . . . 6 ⊢ ≤ = (le‘𝑉)
7		mgcoval.4	. . . . . 6 ⊢ ≲ = (le‘𝑊)
8	4, 5, 6, 7	mgcoval 33540	. . . . 5 ⊢ ((𝑉 ∈ Proset ∧ 𝑊 ∈ Proset ) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))})
9	2, 3, 8	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))})
10	1, 9	eqtrid 2808	. . 3 ⊢ (𝜑 → 𝐻 = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))})
11	10	breqd 5114	. 2 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ 𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))}𝐺))
12		fveq1 6882	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
13	12	adantr 486	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑥) = (𝐹‘𝑥))
14	13	breq1d 5113	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥) ≲ 𝑦 ↔ (𝐹‘𝑥) ≲ 𝑦))
15		fveq1 6882	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑦) = (𝐺‘𝑦))
16	15	adantl 487	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔‘𝑦) = (𝐺‘𝑦))
17	16	breq2d 5115	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥 ≤ (𝑔‘𝑦) ↔ 𝑥 ≤ (𝐺‘𝑦)))
18	14, 17	bibi12d 348	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)) ↔ ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
19	18	2ralbidv 3227	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
20		eqid 2761	. . . 4 ⊢ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))}
21	19, 20	brab2a 5744	. . 3 ⊢ (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))}𝐺 ↔ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
22	5	fvexi 6897	. . . . . 6 ⊢ 𝐵 ∈ V
23	4	fvexi 6897	. . . . . 6 ⊢ 𝐴 ∈ V
24	22, 23	elmap 8892	. . . . 5 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) ↔ 𝐹:𝐴⟶𝐵)
25	23, 22	elmap 8892	. . . . 5 ⊢ (𝐺 ∈ (𝐴 ↑m 𝐵) ↔ 𝐺:𝐵⟶𝐴)
26	24, 25	anbi12i 640	. . . 4 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐴 ↑m 𝐵)) ↔ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴))
27	26	anbi1i 636	. . 3 ⊢ (((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))) ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
28	21, 27	bitr2i 279	. 2 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))) ↔ 𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))}𝐺)
29	11, 28	bitr4di 292	1 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   class class class wbr 5103  {copab 5167  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418   ↑_m cmap 8840  Basecbs 17380  lecple 17428   Proset cproset 18459  MGalConncmgc 33533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-map 8842  df-mgc 33535

This theorem is used by:  mgcf1  33542  mgcf2  33543  mgccole1  33544  mgccole2  33545  mgcmnt1  33546  mgcmnt2  33547  dfmgc2lem  33549  dfmgc2  33550  mgccnv  33553  pwrssmgc  33554  nsgmgc  33956