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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemefrs32fva1 | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. 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΅) |
cdleme29frs.o | β’ π = (β©π§ β π΅ βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π₯ β§ π)) = π₯) β π§ = (π β¨ (π₯ β§ π)))) |
cdleme29frs.f | β’ πΉ = (π₯ β π΅ β¦ if((π β π β§ Β¬ π₯ β€ π), π, π₯)) |
Ref | Expression |
---|---|
cdlemefrs32fva1 | 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 (πΉβπ ) = β¦π / π β¦π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2rl 1243 | . . . 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 π β π΄) | |
2 | cdlemefrs27.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
3 | cdlemefrs27.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
4 | 2, 3 | atbase 37780 | . . . 4 β’ (π β π΄ β π β π΅) |
5 | 1, 4 | syl 17 | . . 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΅) |
6 | simp2l 1200 | . . 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) | |
7 | simp2rr 1244 | . . 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 Β¬ π β€ π) | |
8 | cdleme29frs.o | . . . 4 β’ π = (β©π§ β π΅ βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π₯ β§ π)) = π₯) β π§ = (π β¨ (π₯ β§ π)))) | |
9 | cdleme29frs.f | . . . 4 β’ πΉ = (π₯ β π΅ β¦ if((π β π β§ Β¬ π₯ β€ π), π, π₯)) | |
10 | 8, 9 | cdleme31fv1s 38884 | . . 3 β’ ((π β π΅ β§ (π β π β§ Β¬ π β€ π)) β (πΉβπ ) = β¦π / π₯β¦π) |
11 | 5, 6, 7, 10 | syl12anc 836 | . 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 (πΉβπ ) = β¦π / π₯β¦π) |
12 | cdlemefrs27.l | . . 3 β’ β€ = (leβπΎ) | |
13 | cdlemefrs27.j | . . 3 β’ β¨ = (joinβπΎ) | |
14 | cdlemefrs27.m | . . 3 β’ β§ = (meetβπΎ) | |
15 | cdlemefrs27.h | . . 3 β’ π» = (LHypβπΎ) | |
16 | cdlemefrs27.eq | . . 3 β’ (π = π β (π β π)) | |
17 | cdlemefrs27.nb | . . 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΅) | |
18 | cdlemefrs27.rnb | . . 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 β¦π / π β¦π β π΅) | |
19 | 2, 12, 13, 14, 3, 15, 16, 17, 18, 8 | cdlemefrs32fva 38892 | . 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 / π β¦π) |
20 | 11, 19 | eqtrd 2777 | 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 (πΉβπ ) = β¦π / π β¦π) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2944 βwral 3065 β¦csb 3860 ifcif 4491 class class class wbr 5110 β¦ cmpt 5193 βcfv 6501 β©crio 7317 (class class class)co 7362 Basecbs 17090 lecple 17147 joincjn 18207 meetcmee 18208 Atomscatm 37754 HLchlt 37841 LHypclh 38476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-proset 18191 df-poset 18209 df-plt 18226 df-lub 18242 df-glb 18243 df-join 18244 df-meet 18245 df-p0 18321 df-p1 18322 df-lat 18328 df-clat 18395 df-oposet 37667 df-ol 37669 df-oml 37670 df-covers 37757 df-ats 37758 df-atl 37789 df-cvlat 37813 df-hlat 37842 df-lhyp 38480 |
This theorem is referenced by: cdlemefr32fva1 38902 cdlemefs32fva1 38915 |
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