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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemefrs29bpre1 | Structured version Visualization version GIF version |
Description: TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013.) |
Ref | Expression |
---|---|
cdlemefrs27.b | β’ π΅ = (BaseβπΎ) |
cdlemefrs27.l | β’ β€ = (leβπΎ) |
cdlemefrs27.j | β’ β¨ = (joinβπΎ) |
cdlemefrs27.m | β’ β§ = (meetβπΎ) |
cdlemefrs27.a | β’ π΄ = (AtomsβπΎ) |
cdlemefrs27.h | β’ π» = (LHypβπΎ) |
cdlemefrs27.eq | β’ (π = π β (π β π)) |
cdlemefrs27.nb | 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§ π))) β π β π΅) |
cdlemefrs27.rnb | 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§ π) β β¦π / π β¦π β π΅) |
Ref | Expression |
---|---|
cdlemefrs29bpre1 | 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§ π) β βπ§ β π΅ βπ β π΄ (((Β¬ π β€ π β§ π) β§ (π β¨ (π β§ π)) = π ) β π§ = (π β¨ (π β§ π)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemefrs27.rnb | . 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§ π) β β¦π / π β¦π β π΅) | |
2 | cdlemefrs27.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
3 | cdlemefrs27.l | . . . . 5 β’ β€ = (leβπΎ) | |
4 | cdlemefrs27.j | . . . . 5 β’ β¨ = (joinβπΎ) | |
5 | cdlemefrs27.m | . . . . 5 β’ β§ = (meetβπΎ) | |
6 | cdlemefrs27.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
7 | cdlemefrs27.h | . . . . 5 β’ π» = (LHypβπΎ) | |
8 | cdlemefrs27.eq | . . . . 5 β’ (π = π β (π β π)) | |
9 | cdlemefrs27.nb | . . . . 5 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§ π))) β π β π΅) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | cdlemefrs29bpre0 39806 | . . . 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§ π) β (βπ β π΄ (((Β¬ π β€ π β§ π) β§ (π β¨ (π β§ π)) = π ) β π§ = (π β¨ (π β§ π))) β π§ = β¦π / π β¦π)) |
11 | 10 | rexbidv 3173 | . . 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§ π) β (βπ§ β π΅ βπ β π΄ (((Β¬ π β€ π β§ π) β§ (π β¨ (π β§ π)) = π ) β π§ = (π β¨ (π β§ π))) β βπ§ β π΅ π§ = β¦π / π β¦π)) |
12 | risset 3225 | . . 3 β’ (β¦π / π β¦π β π΅ β βπ§ β π΅ π§ = β¦π / π β¦π) | |
13 | 11, 12 | bitr4di 289 | . 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§ π) β (βπ§ β π΅ βπ β π΄ (((Β¬ π β€ π β§ π) β§ (π β¨ (π β§ π)) = π ) β π§ = (π β¨ (π β§ π))) β β¦π / π β¦π β π΅)) |
14 | 1, 13 | mpbird 257 | 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§ π) β βπ§ β π΅ βπ β π΄ (((Β¬ π β€ π β§ π) β§ (π β¨ (π β§ π)) = π ) β π§ = (π β¨ (π β§ π)))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β wne 2935 βwral 3056 βwrex 3065 β¦csb 3889 class class class wbr 5142 βcfv 6542 (class class class)co 7414 Basecbs 17171 lecple 17231 joincjn 18294 meetcmee 18295 Atomscatm 38672 HLchlt 38759 LHypclh 39394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7370 df-ov 7417 df-oprab 7418 df-proset 18278 df-poset 18296 df-plt 18313 df-lub 18329 df-glb 18330 df-join 18331 df-meet 18332 df-p0 18408 df-lat 18415 df-oposet 38585 df-ol 38587 df-oml 38588 df-covers 38675 df-ats 38676 df-atl 38707 df-cvlat 38731 df-hlat 38760 df-lhyp 39398 |
This theorem is referenced by: cdlemefrs29cpre1 39808 cdlemefrs32fva 39810 cdlemefs29bpre1N 39827 |
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