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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemefr32snb | Structured version Visualization version GIF version |
Description: Show closure of β¦π / π β¦π. (Contributed by NM, 28-Mar-2013.) |
Ref | Expression |
---|---|
cdlemefr27.b | β’ π΅ = (BaseβπΎ) |
cdlemefr27.l | β’ β€ = (leβπΎ) |
cdlemefr27.j | β’ β¨ = (joinβπΎ) |
cdlemefr27.m | β’ β§ = (meetβπΎ) |
cdlemefr27.a | β’ π΄ = (AtomsβπΎ) |
cdlemefr27.h | β’ π» = (LHypβπΎ) |
cdlemefr27.u | β’ π = ((π β¨ π) β§ π) |
cdlemefr27.c | β’ πΆ = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) |
cdlemefr27.n | β’ π = if(π β€ (π β¨ π), πΌ, πΆ) |
Ref | Expression |
---|---|
cdlemefr32snb | 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΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemefr27.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | cdlemefr27.l | . . . 4 β’ β€ = (leβπΎ) | |
3 | cdlemefr27.j | . . . 4 β’ β¨ = (joinβπΎ) | |
4 | cdlemefr27.m | . . . 4 β’ β§ = (meetβπΎ) | |
5 | cdlemefr27.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
6 | cdlemefr27.h | . . . 4 β’ π» = (LHypβπΎ) | |
7 | cdlemefr27.u | . . . 4 β’ π = ((π β¨ π) β§ π) | |
8 | cdlemefr27.c | . . . 4 β’ πΆ = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) | |
9 | cdlemefr27.n | . . . 4 β’ π = if(π β€ (π β¨ π), πΌ, πΆ) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | cdlemefr32sn2aw 39801 | . . 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 β π΄ β§ Β¬ β¦π / π β¦π β€ π)) |
11 | 10 | simpld 494 | . 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΄) |
12 | 1, 5 | atbase 38685 | . 2 β’ (β¦π / π β¦π β π΄ β β¦π / π β¦π β π΅) |
13 | 11, 12 | syl 17 | 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¦π / π β¦π β π΅) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β wne 2935 β¦csb 3889 ifcif 4524 class class class wbr 5142 βcfv 6542 (class class class)co 7414 Basecbs 17165 lecple 17225 joincjn 18288 meetcmee 18289 Atomscatm 38659 HLchlt 38746 LHypclh 39381 |
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 7732 |
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-iin 4994 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-mpo 7419 df-1st 7985 df-2nd 7986 df-proset 18272 df-poset 18290 df-plt 18307 df-lub 18323 df-glb 18324 df-join 18325 df-meet 18326 df-p0 18402 df-p1 18403 df-lat 18409 df-clat 18476 df-oposet 38572 df-ol 38574 df-oml 38575 df-covers 38662 df-ats 38663 df-atl 38694 df-cvlat 38718 df-hlat 38747 df-lines 38898 df-psubsp 38900 df-pmap 38901 df-padd 39193 df-lhyp 39385 |
This theorem is referenced by: cdlemefr29clN 39804 cdlemefr32fvaN 39806 cdlemefr32fva1 39807 |
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