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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgcf1 | Structured version Visualization version GIF version |
Description: The lower adjoint πΉ 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 |
---|---|
mgcf1 | β’ (π β πΉ: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 31896 | . . 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 | simplld 767 | 1 β’ (π β πΉ:π΄βΆπ΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 class class class wbr 5106 βΆwf 6493 βcfv 6497 (class class class)co 7358 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: mgcmntco 31903 mgcmnt1d 31906 |
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