Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme43aN | Structured version Visualization version GIF version | ||
| Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT p. 115 penultimate line: g(f(r)) = (p v q) ^ (g(s) v v1). (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| cdleme43.b | β’ π΅ = (BaseβπΎ) |
| cdleme43.l | β’ β€ = (leβπΎ) |
| cdleme43.j | β’ β¨ = (joinβπΎ) |
| cdleme43.m | β’ β§ = (meetβπΎ) |
| cdleme43.a | β’ π΄ = (AtomsβπΎ) |
| cdleme43.h | β’ π» = (LHypβπΎ) |
| cdleme43.u | β’ π = ((π β¨ π) β§ π) |
| cdleme43.x | β’ π = ((π β¨ π) β§ π) |
| cdleme43.c | β’ πΆ = ((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) |
| cdleme43.f | β’ π = ((π β¨ π) β§ (πΆ β¨ ((π β¨ π) β§ π))) |
| cdleme43.d | β’ π· = ((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) |
| cdleme43.g | β’ πΊ = ((π β¨ π) β§ (π· β¨ ((π β¨ π) β§ π))) |
| cdleme43.e | β’ πΈ = ((π· β¨ π) β§ (π β¨ ((π β¨ π·) β§ π))) |
| cdleme43.v | β’ π = ((π β¨ π) β§ π) |
| cdleme43.y | β’ π = ((π β¨ π·) β§ π) |
| Ref | Expression |
|---|---|
| cdleme43aN | β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β πΊ = ((π β¨ π) β§ (π· β¨ π))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdleme43.g | . 2 β’ πΊ = ((π β¨ π) β§ (π· β¨ ((π β¨ π) β§ π))) | |
| 2 | cdleme43.j | . . . 4 β’ β¨ = (joinβπΎ) | |
| 3 | cdleme43.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
| 4 | 2, 3 | hlatjcom 38541 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) = (π β¨ π)) |
| 5 | cdleme43.v | . . . . 5 β’ π = ((π β¨ π) β§ π) | |
| 6 | 5 | oveq2i 7422 | . . . 4 β’ (π· β¨ π) = (π· β¨ ((π β¨ π) β§ π)) |
| 7 | 6 | a1i 11 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π· β¨ π) = (π· β¨ ((π β¨ π) β§ π))) |
| 8 | 4, 7 | oveq12d 7429 | . 2 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β ((π β¨ π) β§ (π· β¨ π)) = ((π β¨ π) β§ (π· β¨ ((π β¨ π) β§ π)))) |
| 9 | 1, 8 | eqtr4id 2789 | 1 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β πΊ = ((π β¨ π) β§ (π· β¨ π))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1085 = wceq 1539 β wcel 2104 βcfv 6542 (class class class)co 7411 Basecbs 17148 lecple 17208 joincjn 18268 meetcmee 18269 Atomscatm 38436 HLchlt 38523 LHypclh 39158 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-lub 18303 df-join 18305 df-lat 18389 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |