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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mgcval | Structured version Visualization version GIF version | ||
| 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.) | 
| 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 ) | 
| Ref | Expression | 
|---|---|
| mgcval | ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))) | 
| 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 32976 | . . . . 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 584 | . . . 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 2789 | . . 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 5154 | . 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 6905 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
| 13 | 12 | adantr 480 | . . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑥) = (𝐹‘𝑥)) | 
| 14 | 13 | breq1d 5153 | . . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥) ≲ 𝑦 ↔ (𝐹‘𝑥) ≲ 𝑦)) | 
| 15 | fveq1 6905 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑦) = (𝐺‘𝑦)) | |
| 16 | 15 | adantl 481 | . . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑔‘𝑦) = (𝐺‘𝑦)) | 
| 17 | 16 | breq2d 5155 | . . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥 ≤ (𝑔‘𝑦) ↔ 𝑥 ≤ (𝐺‘𝑦))) | 
| 18 | 14, 17 | bibi12d 345 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)) ↔ ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))) | 
| 19 | 18 | 2ralbidv 3221 | . . . 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 2737 | . . . 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 5779 | . . 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 6920 | . . . . . 6 ⊢ 𝐵 ∈ V | 
| 23 | 4 | fvexi 6920 | . . . . . 6 ⊢ 𝐴 ∈ V | 
| 24 | 22, 23 | elmap 8911 | . . . . 5 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) ↔ 𝐹:𝐴⟶𝐵) | 
| 25 | 23, 22 | elmap 8911 | . . . . 5 ⊢ (𝐺 ∈ (𝐴 ↑m 𝐵) ↔ 𝐺:𝐵⟶𝐴) | 
| 26 | 24, 25 | anbi12i 628 | . . . 4 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐴 ↑m 𝐵)) ↔ (𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴)) | 
| 27 | 26 | anbi1i 624 | . . 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 276 | . 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 289 | 1 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 {copab 5205 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Basecbs 17247 lecple 17304 Proset cproset 18338 MGalConncmgc 32969 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-mgc 32971 | 
| This theorem is referenced by: mgcf1 32978 mgcf2 32979 mgccole1 32980 mgccole2 32981 mgcmnt1 32982 mgcmnt2 32983 dfmgc2lem 32985 dfmgc2 32986 mgccnv 32989 pwrssmgc 32990 nsgmgc 33440 | 
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