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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgcf2 | Structured version Visualization version GIF version |
Description: The upper adjoint πΊ of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.) |
Ref | Expression |
---|---|
mgcoval.1 | β’ π΄ = (Baseβπ) |
mgcoval.2 | β’ π΅ = (Baseβπ) |
mgcoval.3 | β’ β€ = (leβπ) |
mgcoval.4 | β’ β² = (leβπ) |
mgcval.1 | β’ π» = (πMGalConnπ) |
mgcval.2 | β’ (π β π β Proset ) |
mgcval.3 | β’ (π β π β Proset ) |
mgccole.1 | β’ (π β πΉπ»πΊ) |
Ref | Expression |
---|---|
mgcf2 | β’ (π β πΊ:π΅βΆπ΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgccole.1 | . . 3 β’ (π β πΉπ»πΊ) | |
2 | mgcoval.1 | . . . 4 β’ π΄ = (Baseβπ) | |
3 | mgcoval.2 | . . . 4 β’ π΅ = (Baseβπ) | |
4 | mgcoval.3 | . . . 4 β’ β€ = (leβπ) | |
5 | mgcoval.4 | . . . 4 β’ β² = (leβπ) | |
6 | mgcval.1 | . . . 4 β’ π» = (πMGalConnπ) | |
7 | mgcval.2 | . . . 4 β’ (π β π β Proset ) | |
8 | mgcval.3 | . . . 4 β’ (π β π β Proset ) | |
9 | 2, 3, 4, 5, 6, 7, 8 | mgcval 32152 | . . 3 β’ (π β (πΉπ»πΊ β ((πΉ:π΄βΆπ΅ β§ πΊ:π΅βΆπ΄) β§ βπ₯ β π΄ βπ¦ β π΅ ((πΉβπ₯) β² π¦ β π₯ β€ (πΊβπ¦))))) |
10 | 1, 9 | mpbid 231 | . 2 β’ (π β ((πΉ:π΄βΆπ΅ β§ πΊ:π΅βΆπ΄) β§ βπ₯ β π΄ βπ¦ β π΅ ((πΉβπ₯) β² π¦ β π₯ β€ (πΊβπ¦)))) |
11 | 10 | simplrd 768 | 1 β’ (π β πΊ:π΅βΆπ΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 class class class wbr 5148 βΆwf 6539 βcfv 6543 (class class class)co 7408 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: mgcmntco 32159 mgcmnt2d 32163 |
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