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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihmeetlem5 | Structured version Visualization version GIF version |
Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.) |
Ref | Expression |
---|---|
dihmeetlem5.b | β’ π΅ = (BaseβπΎ) |
dihmeetlem5.l | β’ β€ = (leβπΎ) |
dihmeetlem5.j | β’ β¨ = (joinβπΎ) |
dihmeetlem5.m | β’ β§ = (meetβπΎ) |
dihmeetlem5.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
dihmeetlem5 | β’ (((πΎ β HL β§ π β π΅ β§ π β π΅) β§ (π β π΄ β§ π β€ π)) β (π β§ (π β¨ π)) = ((π β§ π) β¨ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1192 | . . 3 β’ (((πΎ β HL β§ π β π΅ β§ π β π΅) β§ (π β π΄ β§ π β€ π)) β πΎ β HL) | |
2 | simprl 770 | . . 3 β’ (((πΎ β HL β§ π β π΅ β§ π β π΅) β§ (π β π΄ β§ π β€ π)) β π β π΄) | |
3 | simpl2 1193 | . . 3 β’ (((πΎ β HL β§ π β π΅ β§ π β π΅) β§ (π β π΄ β§ π β€ π)) β π β π΅) | |
4 | simpl3 1194 | . . 3 β’ (((πΎ β HL β§ π β π΅ β§ π β π΅) β§ (π β π΄ β§ π β€ π)) β π β π΅) | |
5 | simprr 772 | . . 3 β’ (((πΎ β HL β§ π β π΅ β§ π β π΅) β§ (π β π΄ β§ π β€ π)) β π β€ π) | |
6 | dihmeetlem5.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
7 | dihmeetlem5.l | . . . 4 β’ β€ = (leβπΎ) | |
8 | dihmeetlem5.j | . . . 4 β’ β¨ = (joinβπΎ) | |
9 | dihmeetlem5.m | . . . 4 β’ β§ = (meetβπΎ) | |
10 | dihmeetlem5.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
11 | 6, 7, 8, 9, 10 | atmod2i1 38374 | . . 3 β’ ((πΎ β HL β§ (π β π΄ β§ π β π΅ β§ π β π΅) β§ π β€ π) β ((π β§ π) β¨ π) = (π β§ (π β¨ π))) |
12 | 1, 2, 3, 4, 5, 11 | syl131anc 1384 | . 2 β’ (((πΎ β HL β§ π β π΅ β§ π β π΅) β§ (π β π΄ β§ π β€ π)) β ((π β§ π) β¨ π) = (π β§ (π β¨ π))) |
13 | 12 | eqcomd 2739 | 1 β’ (((πΎ β HL β§ π β π΅ β§ π β π΅) β§ (π β π΄ β§ π β€ π)) β (π β§ (π β¨ π)) = ((π β§ π) β¨ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5109 βcfv 6500 (class class class)co 7361 Basecbs 17091 lecple 17148 joincjn 18208 meetcmee 18209 Atomscatm 37775 HLchlt 37862 |
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-1st 7925 df-2nd 7926 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-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-psubsp 38016 df-pmap 38017 df-padd 38309 |
This theorem is referenced by: dihmeetlem6 39822 dihjatc1 39824 dihmeetlem10N 39829 |
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