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Theorem mgcval 32152
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 32151 . . . . 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 584 . . . 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 2784 . . 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 5159 . 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 6890 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
1312adantr 481 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
1413breq1d 5158 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ ((π‘“β€˜π‘₯) ≲ 𝑦 ↔ (πΉβ€˜π‘₯) ≲ 𝑦))
15 fveq1 6890 . . . . . . . 8 (𝑔 = 𝐺 β†’ (π‘”β€˜π‘¦) = (πΊβ€˜π‘¦))
1615adantl 482 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (π‘”β€˜π‘¦) = (πΊβ€˜π‘¦))
1716breq2d 5160 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (π‘₯ ≀ (π‘”β€˜π‘¦) ↔ π‘₯ ≀ (πΊβ€˜π‘¦)))
1814, 17bibi12d 345 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (((π‘“β€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (π‘”β€˜π‘¦)) ↔ ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦))))
19182ralbidv 3218 . . . 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 2732 . . . 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 5769 . . 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 6905 . . . . . 6 𝐡 ∈ V
234fvexi 6905 . . . . . 6 𝐴 ∈ V
2422, 23elmap 8864 . . . . 5 (𝐹 ∈ (𝐡 ↑m 𝐴) ↔ 𝐹:𝐴⟢𝐡)
2523, 22elmap 8864 . . . . 5 (𝐺 ∈ (𝐴 ↑m 𝐡) ↔ 𝐺:𝐡⟢𝐴)
2624, 25anbi12i 627 . . . 4 ((𝐹 ∈ (𝐡 ↑m 𝐴) ∧ 𝐺 ∈ (𝐴 ↑m 𝐡)) ↔ (𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴))
2726anbi1i 624 . . 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 275 . 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 288 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   class class class wbr 5148  {copab 5210  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ↑m cmap 8819  Basecbs 17143  lecple 17203   Proset cproset 18245  MGalConncmgc 32144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-mgc 32146
This theorem is referenced by:  mgcf1  32153  mgcf2  32154  mgccole1  32155  mgccole2  32156  mgcmnt1  32157  mgcmnt2  32158  dfmgc2lem  32160  dfmgc2  32161  mgccnv  32164  pwrssmgc  32165  nsgmgc  32518
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