Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| 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 1191 | . . 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 769 | . . 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 1192 | . . 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 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΅) | |
| 5 | simprr 771 | . . 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 38727 | . . 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 1383 | . 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 2738 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5148 βcfv 6543 (class class class)co 7408 Basecbs 17143 lecple 17203 joincjn 18263 meetcmee 18264 Atomscatm 38128 HLchlt 38215 |
| 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-rep 5285 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-rmo 3376 df-reu 3377 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-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-proset 18247 df-poset 18265 df-plt 18282 df-lub 18298 df-glb 18299 df-join 18300 df-meet 18301 df-p0 18377 df-lat 18384 df-clat 18451 df-oposet 38041 df-ol 38043 df-oml 38044 df-covers 38131 df-ats 38132 df-atl 38163 df-cvlat 38187 df-hlat 38216 df-psubsp 38369 df-pmap 38370 df-padd 38662 |
| This theorem is referenced by: dihmeetlem6 40175 dihjatc1 40177 dihmeetlem10N 40182 |
| Copyright terms: Public domain | W3C validator |