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Theorem mgcval 32962
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 (𝜑𝑉 ∈ Proset )
mgcval.3 (𝜑𝑊 ∈ Proset )
Assertion
Ref Expression
mgcval (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
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.
StepHypRef Expression
1 mgcval.1 . . . 4 𝐻 = (𝑉MGalConn𝑊)
2 mgcval.2 . . . . 5 (𝜑𝑉 ∈ Proset )
3 mgcval.3 . . . . 5 (𝜑𝑊 ∈ 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)
84, 5, 6, 7mgcoval 32961 . . . . 5 ((𝑉 ∈ Proset ∧ 𝑊 ∈ Proset ) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
92, 3, 8syl2anc 584 . . . 4 (𝜑 → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
101, 9eqtrid 2787 . . 3 (𝜑𝐻 = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
1110breqd 5159 . 2 (𝜑 → (𝐹𝐻𝐺𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))}𝐺))
12 fveq1 6906 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
1312adantr 480 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑥) = (𝐹𝑥))
1413breq1d 5158 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥) 𝑦 ↔ (𝐹𝑥) 𝑦))
15 fveq1 6906 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔𝑦) = (𝐺𝑦))
1615adantl 481 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔𝑦) = (𝐺𝑦))
1716breq2d 5160 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥 (𝑔𝑦) ↔ 𝑥 (𝐺𝑦)))
1814, 17bibi12d 345 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)) ↔ ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
19182ralbidv 3219 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)) ↔ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
20 eqid 2735 . . . 4 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))}
2119, 20brab2a 5782 . . 3 (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))}𝐺 ↔ ((𝐹 ∈ (𝐵m 𝐴) ∧ 𝐺 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
225fvexi 6921 . . . . . 6 𝐵 ∈ V
234fvexi 6921 . . . . . 6 𝐴 ∈ V
2422, 23elmap 8910 . . . . 5 (𝐹 ∈ (𝐵m 𝐴) ↔ 𝐹:𝐴𝐵)
2523, 22elmap 8910 . . . . 5 (𝐺 ∈ (𝐴m 𝐵) ↔ 𝐺:𝐵𝐴)
2624, 25anbi12i 628 . . . 4 ((𝐹 ∈ (𝐵m 𝐴) ∧ 𝐺 ∈ (𝐴m 𝐵)) ↔ (𝐹:𝐴𝐵𝐺:𝐵𝐴))
2726anbi1i 624 . . 3 (((𝐹 ∈ (𝐵m 𝐴) ∧ 𝐺 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))) ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))))
2821, 27bitr2i 276 . 2 (((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦))) ↔ 𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))}𝐺)
2911, 28bitr4di 289 1 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥) 𝑦𝑥 (𝐺𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059   class class class wbr 5148  {copab 5210  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  Basecbs 17245  lecple 17305   Proset cproset 18350  MGalConncmgc 32954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-mgc 32956
This theorem is referenced by:  mgcf1  32963  mgcf2  32964  mgccole1  32965  mgccole2  32966  mgcmnt1  32967  mgcmnt2  32968  dfmgc2lem  32970  dfmgc2  32971  mgccnv  32974  pwrssmgc  32975  nsgmgc  33420
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