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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme35sn3a | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013.) |
Ref | Expression |
---|---|
cdleme32s.b | β’ π΅ = (BaseβπΎ) |
cdleme32s.l | β’ β€ = (leβπΎ) |
cdleme32s.j | β’ β¨ = (joinβπΎ) |
cdleme32s.m | β’ β§ = (meetβπΎ) |
cdleme32s.a | β’ π΄ = (AtomsβπΎ) |
cdleme32s.h | β’ π» = (LHypβπΎ) |
cdleme32s.u | β’ π = ((π β¨ π) β§ π) |
cdleme32s.d | β’ π· = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) |
cdleme32s.n | β’ π = if(π β€ (π β¨ π), πΌ, π·) |
Ref | Expression |
---|---|
cdleme35sn3a | β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π)) β§ Β¬ π β€ (π β¨ π)) β Β¬ β¦π / π β¦π β€ (π β¨ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme32s.l | . . 3 β’ β€ = (leβπΎ) | |
2 | cdleme32s.j | . . 3 β’ β¨ = (joinβπΎ) | |
3 | cdleme32s.m | . . 3 β’ β§ = (meetβπΎ) | |
4 | cdleme32s.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
5 | cdleme32s.h | . . 3 β’ π» = (LHypβπΎ) | |
6 | cdleme32s.u | . . 3 β’ π = ((π β¨ π) β§ π) | |
7 | eqid 2731 | . . 3 β’ ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | cdleme35fnpq 39624 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π)) β§ Β¬ π β€ (π β¨ π)) β Β¬ ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) β€ (π β¨ π)) |
9 | simp2rl 1241 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π)) β§ Β¬ π β€ (π β¨ π)) β π β π΄) | |
10 | simp3 1137 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π)) β§ Β¬ π β€ (π β¨ π)) β Β¬ π β€ (π β¨ π)) | |
11 | cdleme32s.d | . . . . 5 β’ π· = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) | |
12 | cdleme32s.n | . . . . 5 β’ π = if(π β€ (π β¨ π), πΌ, π·) | |
13 | 11, 12, 7 | cdleme31sn2 39564 | . . . 4 β’ ((π β π΄ β§ Β¬ π β€ (π β¨ π)) β β¦π / π β¦π = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π)))) |
14 | 9, 10, 13 | syl2anc 583 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π)) β§ Β¬ π β€ (π β¨ π)) β β¦π / π β¦π = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π)))) |
15 | 14 | breq1d 5158 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π)) β§ Β¬ π β€ (π β¨ π)) β (β¦π / π β¦π β€ (π β¨ π) β ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) β€ (π β¨ π))) |
16 | 8, 15 | mtbird 325 | 1 β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π)) β§ Β¬ π β€ (π β¨ π)) β Β¬ β¦π / π β¦π β€ (π β¨ π)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2939 β¦csb 3893 ifcif 4528 class class class wbr 5148 βcfv 6543 (class class class)co 7412 Basecbs 17149 lecple 17209 joincjn 18269 meetcmee 18270 Atomscatm 38437 HLchlt 38524 LHypclh 39159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-proset 18253 df-poset 18271 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18390 df-clat 18457 df-oposet 38350 df-ol 38352 df-oml 38353 df-covers 38440 df-ats 38441 df-atl 38472 df-cvlat 38496 df-hlat 38525 df-lines 38676 df-psubsp 38678 df-pmap 38679 df-padd 38971 df-lhyp 39163 |
This theorem is referenced by: cdleme41sn3aw 39649 |
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