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Theorem mgcval 31896
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 π‘Š, 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 = (Baseβ€˜π‘‰)
mgcoval.2 𝐡 = (Baseβ€˜π‘Š)
mgcoval.3 ≀ = (leβ€˜π‘‰)
mgcoval.4 ≲ = (leβ€˜π‘Š)
mgcval.1 𝐻 = (𝑉MGalConnπ‘Š)
mgcval.2 (πœ‘ β†’ 𝑉 ∈ Proset )
mgcval.3 (πœ‘ β†’ π‘Š ∈ Proset )
Assertion
Ref Expression
mgcval (πœ‘ β†’ (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)))))
Distinct variable groups:   π‘₯,𝐴,𝑦   π‘₯,𝐡,𝑦   π‘₯,𝑉,𝑦   π‘₯,π‘Š,𝑦   π‘₯,𝐹,𝑦   π‘₯,𝐺,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦)   𝐻(π‘₯,𝑦)   ≀ (π‘₯,𝑦)   ≲ (π‘₯,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 = (𝑉MGalConnπ‘Š)
2 mgcval.2 . . . . 5 (πœ‘ β†’ 𝑉 ∈ Proset )
3 mgcval.3 . . . . 5 (πœ‘ β†’ π‘Š ∈ Proset )
4 mgcoval.1 . . . . . 6 𝐴 = (Baseβ€˜π‘‰)
5 mgcoval.2 . . . . . 6 𝐡 = (Baseβ€˜π‘Š)
6 mgcoval.3 . . . . . 6 ≀ = (leβ€˜π‘‰)
7 mgcoval.4 . . . . . 6 ≲ = (leβ€˜π‘Š)
84, 5, 6, 7mgcoval 31895 . . . . 5 ((𝑉 ∈ Proset ∧ π‘Š ∈ Proset ) β†’ (𝑉MGalConnπ‘Š) = {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))})
92, 3, 8syl2anc 585 . . . 4 (πœ‘ β†’ (𝑉MGalConnπ‘Š) = {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))})
101, 9eqtrid 2785 . . 3 (πœ‘ β†’ 𝐻 = {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))})
1110breqd 5117 . 2 (πœ‘ β†’ (𝐹𝐻𝐺 ↔ 𝐹{βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))}𝐺))
12 fveq1 6842 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
1312adantr 482 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
1413breq1d 5116 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ (πΉβ€˜π‘₯) ≲ 𝑦))
15 fveq1 6842 . . . . . . . 8 (𝑔 = 𝐺 β†’ (π‘”β€˜π‘¦) = (πΊβ€˜π‘¦))
1615adantl 483 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (π‘”β€˜π‘¦) = (πΊβ€˜π‘¦))
1716breq2d 5118 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (π‘₯ ≀ (π‘”β€˜π‘¦) ↔ π‘₯ ≀ (πΊβ€˜π‘¦)))
1814, 17bibi12d 346 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)) ↔ ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦))))
19182ralbidv 3209 . . . 4 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦))))
20 eqid 2733 . . . 4 {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))} = {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))}
2119, 20brab2a 5726 . . 3 (𝐹{βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))}𝐺 ↔ ((𝐹 ∈ (𝐡 ↑m 𝐴) ∧ 𝐺 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦))))
225fvexi 6857 . . . . . 6 𝐡 ∈ V
234fvexi 6857 . . . . . 6 𝐴 ∈ V
2422, 23elmap 8812 . . . . 5 (𝐹 ∈ (𝐡 ↑m 𝐴) ↔ 𝐹:𝐴⟢𝐡)
2523, 22elmap 8812 . . . . 5 (𝐺 ∈ (𝐴 ↑m 𝐡) ↔ 𝐺:𝐡⟢𝐴)
2624, 25anbi12i 628 . . . 4 ((𝐹 ∈ (𝐡 ↑m 𝐴) ∧ 𝐺 ∈ (𝐴 ↑m 𝐡)) ↔ (𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴))
2726anbi1i 625 . . 3 (((𝐹 ∈ (𝐡 ↑m 𝐴) ∧ 𝐺 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦))) ↔ ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦))))
2821, 27bitr2i 276 . 2 (((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦))) ↔ 𝐹{βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ (𝐡 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐡)) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)))}𝐺)
2911, 28bitr4di 289 1 (πœ‘ β†’ (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   class class class wbr 5106  {copab 5168  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ↑m cmap 8768  Basecbs 17088  lecple 17145   Proset cproset 18187  MGalConncmgc 31888
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-mgc 31890
This theorem is referenced by:  mgcf1  31897  mgcf2  31898  mgccole1  31899  mgccole2  31900  mgcmnt1  31901  mgcmnt2  31902  dfmgc2lem  31904  dfmgc2  31905  mgccnv  31908  pwrssmgc  31909  nsgmgc  32238
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