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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme19f | Structured version Visualization version GIF version | ||
| Description: Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, line 3. π·, πΉ, π, π, πΊ, π represent s2, f(s), fs(r), t2, f(t), ft(r). We prove that if r β€ s β¨ t, then ft(r) = ft(r). (Contributed by NM, 14-Nov-2012.) |
| Ref | Expression |
|---|---|
| cdleme19.l | β’ β€ = (leβπΎ) |
| cdleme19.j | β’ β¨ = (joinβπΎ) |
| cdleme19.m | β’ β§ = (meetβπΎ) |
| cdleme19.a | β’ π΄ = (AtomsβπΎ) |
| cdleme19.h | β’ π» = (LHypβπΎ) |
| cdleme19.u | β’ π = ((π β¨ π) β§ π) |
| cdleme19.f | β’ πΉ = ((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) |
| cdleme19.g | β’ πΊ = ((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) |
| cdleme19.d | β’ π· = ((π β¨ π) β§ π) |
| cdleme19.y | β’ π = ((π β¨ π) β§ π) |
| cdleme19.n | β’ π = ((π β¨ π) β§ (πΉ β¨ π·)) |
| cdleme19.o | β’ π = ((π β¨ π) β§ (πΊ β¨ π)) |
| Ref | Expression |
|---|---|
| cdleme19f | 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§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π) β§ π β π΄) β§ ((π β π 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 β¨ π)))) β π = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdleme19.l | . . . 4 β’ β€ = (leβπΎ) | |
| 2 | cdleme19.j | . . . 4 β’ β¨ = (joinβπΎ) | |
| 3 | cdleme19.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 4 | cdleme19.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
| 5 | cdleme19.h | . . . 4 β’ π» = (LHypβπΎ) | |
| 6 | cdleme19.u | . . . 4 β’ π = ((π β¨ π) β§ π) | |
| 7 | cdleme19.f | . . . 4 β’ πΉ = ((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) | |
| 8 | cdleme19.g | . . . 4 β’ πΊ = ((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) | |
| 9 | cdleme19.d | . . . 4 β’ π· = ((π β¨ π) β§ π) | |
| 10 | cdleme19.y | . . . 4 β’ π = ((π β¨ π) β§ π) | |
| 11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme19e 39820 | . . 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§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π) β§ π β π΄) β§ ((π β π 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 β¨ π)))) β (πΉ β¨ π·) = (πΊ β¨ π)) |
| 12 | 11 | oveq2d 7442 | . 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§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π) β§ π β π΄) β§ ((π β π 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 β¨ π)))) β ((π β¨ π) β§ (πΉ β¨ π·)) = ((π β¨ π) β§ (πΊ β¨ π))) |
| 13 | cdleme19.n | . 2 β’ π = ((π β¨ π) β§ (πΉ β¨ π·)) | |
| 14 | cdleme19.o | . 2 β’ π = ((π β¨ π) β§ (πΊ β¨ π)) | |
| 15 | 12, 13, 14 | 3eqtr4g 2793 | 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§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π) β§ π β π΄) β§ ((π β π 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 β¨ π)))) β π = π) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2937 class class class wbr 5152 βcfv 6553 (class class class)co 7426 lecple 17249 joincjn 18312 meetcmee 18313 Atomscatm 38775 HLchlt 38862 LHypclh 39497 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-proset 18296 df-poset 18314 df-plt 18331 df-lub 18347 df-glb 18348 df-join 18349 df-meet 18350 df-p0 18426 df-p1 18427 df-lat 18433 df-clat 18500 df-oposet 38688 df-ol 38690 df-oml 38691 df-covers 38778 df-ats 38779 df-atl 38810 df-cvlat 38834 df-hlat 38863 df-llines 39011 df-lplanes 39012 df-lvols 39013 df-lines 39014 df-psubsp 39016 df-pmap 39017 df-padd 39309 df-lhyp 39501 |
| This theorem is referenced by: cdleme20 39837 |
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