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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemefs32snb | Structured version Visualization version GIF version |
Description: Show closure of β¦π / π β¦π. (Contributed by NM, 24-Mar-2013.) |
Ref | Expression |
---|---|
cdlemefs32.b | β’ π΅ = (BaseβπΎ) |
cdlemefs32.l | β’ β€ = (leβπΎ) |
cdlemefs32.j | β’ β¨ = (joinβπΎ) |
cdlemefs32.m | β’ β§ = (meetβπΎ) |
cdlemefs32.a | β’ π΄ = (AtomsβπΎ) |
cdlemefs32.h | β’ π» = (LHypβπΎ) |
cdlemefs32.u | β’ π = ((π β¨ π) β§ π) |
cdlemefs32.d | β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) |
cdlemefs32.e | β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) |
cdlemefs32.i | β’ πΌ = (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)) |
cdlemefs32.n | β’ π = if(π β€ (π β¨ π), πΌ, πΆ) |
Ref | Expression |
---|---|
cdlemefs32snb | 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)) β β¦π / π β¦π β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemefs32.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | cdlemefs32.l | . . . 4 β’ β€ = (leβπΎ) | |
3 | cdlemefs32.j | . . . 4 β’ β¨ = (joinβπΎ) | |
4 | cdlemefs32.m | . . . 4 β’ β§ = (meetβπΎ) | |
5 | cdlemefs32.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
6 | cdlemefs32.h | . . . 4 β’ π» = (LHypβπΎ) | |
7 | cdlemefs32.u | . . . 4 β’ π = ((π β¨ π) β§ π) | |
8 | cdlemefs32.d | . . . 4 β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) | |
9 | cdlemefs32.e | . . . 4 β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) | |
10 | cdlemefs32.i | . . . 4 β’ πΌ = (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)) | |
11 | cdlemefs32.n | . . . 4 β’ π = if(π β€ (π β¨ π), πΌ, πΆ) | |
12 | eqid 2731 | . . . 4 β’ ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) | |
13 | eqid 2731 | . . . 4 β’ (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))))) = (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))))) | |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | cdlemefs32sn1aw 39589 | . . 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€ (π β¨ π)) β (β¦π / π β¦π β π΄ β§ Β¬ β¦π / π β¦π β€ π)) |
15 | 14 | 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§ π β€ (π β¨ π)) β β¦π / π β¦π β π΄) |
16 | 1, 5 | atbase 38463 | . 2 β’ (β¦π / π β¦π β π΄ β β¦π / π β¦π β π΅) |
17 | 15, 16 | 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§ 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 1086 = wceq 1540 β wcel 2105 β wne 2939 βwral 3060 β¦csb 3894 ifcif 4529 class class class wbr 5149 βcfv 6544 β©crio 7367 (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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-riotaBAD 38127 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 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 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-iin 5001 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7978 df-2nd 7979 df-undef 8261 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-llines 38673 df-lplanes 38674 df-lvols 38675 df-lines 38676 df-psubsp 38678 df-pmap 38679 df-padd 38971 df-lhyp 39163 |
This theorem is referenced by: cdlemefs29bpre1N 39592 cdlemefs29cpre1N 39593 cdlemefs29clN 39594 cdlemefs32fvaN 39597 cdlemefs32fva1 39598 |
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