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| 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 38238 | . . 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 7420 | . . . 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 7427 | . 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 2792 | 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 1088 = wceq 1542 β wcel 2107 βcfv 6544 (class class class)co 7409 Basecbs 17144 lecple 17204 joincjn 18264 meetcmee 18265 Atomscatm 38133 HLchlt 38220 LHypclh 38855 |
| 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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
| 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 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-lub 18299 df-join 18301 df-lat 18385 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 |
| This theorem is referenced by: (None) |
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