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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihglblem3aN | Structured version Visualization version GIF version |
Description: Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dihglblem.b | β’ π΅ = (BaseβπΎ) |
dihglblem.l | β’ β€ = (leβπΎ) |
dihglblem.m | β’ β§ = (meetβπΎ) |
dihglblem.g | β’ πΊ = (glbβπΎ) |
dihglblem.h | β’ π» = (LHypβπΎ) |
dihglblem.t | β’ π = {π’ β π΅ β£ βπ£ β π π’ = (π£ β§ π)} |
dihglblem.i | β’ π½ = ((DIsoBβπΎ)βπ) |
dihglblem.ih | β’ πΌ = ((DIsoHβπΎ)βπ) |
Ref | Expression |
---|---|
dihglblem3aN | β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β ) β§ (πΊβπ) β€ π) β (πΌβ(πΊβπ)) = β© π₯ β π (πΌβπ₯)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihglblem.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
2 | dihglblem.l | . . . . 5 β’ β€ = (leβπΎ) | |
3 | dihglblem.m | . . . . 5 β’ β§ = (meetβπΎ) | |
4 | dihglblem.g | . . . . 5 β’ πΊ = (glbβπΎ) | |
5 | dihglblem.h | . . . . 5 β’ π» = (LHypβπΎ) | |
6 | dihglblem.t | . . . . 5 β’ π = {π’ β π΅ β£ βπ£ β π π’ = (π£ β§ π)} | |
7 | 1, 2, 3, 4, 5, 6 | dihglblem2N 39807 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ π β π΅ β§ (πΊβπ) β€ π) β (πΊβπ) = (πΊβπ)) |
8 | 7 | 3adant2r 1180 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β ) β§ (πΊβπ) β€ π) β (πΊβπ) = (πΊβπ)) |
9 | 8 | fveq2d 6850 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β ) β§ (πΊβπ) β€ π) β (πΌβ(πΊβπ)) = (πΌβ(πΊβπ))) |
10 | dihglblem.i | . . 3 β’ π½ = ((DIsoBβπΎ)βπ) | |
11 | dihglblem.ih | . . 3 β’ πΌ = ((DIsoHβπΎ)βπ) | |
12 | 1, 2, 3, 4, 5, 6, 10, 11 | dihglblem3N 39808 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β ) β§ (πΊβπ) β€ π) β (πΌβ(πΊβπ)) = β© π₯ β π (πΌβπ₯)) |
13 | 9, 12 | eqtrd 2773 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β β ) β§ (πΊβπ) β€ π) β (πΌβ(πΊβπ)) = β© π₯ β π (πΌβπ₯)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 βwrex 3070 {crab 3406 β wss 3914 β c0 4286 β© ciin 4959 class class class wbr 5109 βcfv 6500 (class class class)co 7361 Basecbs 17091 lecple 17148 glbcglb 18207 meetcmee 18209 HLchlt 37862 LHypclh 38497 DIsoBcdib 39651 DIsoHcdih 39741 |
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-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-map 8773 df-proset 18192 df-poset 18210 df-plt 18227 df-lub 18243 df-glb 18244 df-join 18245 df-meet 18246 df-p0 18322 df-p1 18323 df-lat 18329 df-clat 18396 df-oposet 37688 df-ol 37690 df-oml 37691 df-covers 37778 df-ats 37779 df-atl 37810 df-cvlat 37834 df-hlat 37863 df-lhyp 38501 df-laut 38502 df-ldil 38617 df-ltrn 38618 df-trl 38672 df-disoa 39542 df-dib 39652 df-dih 39742 |
This theorem is referenced by: (None) |
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