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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | tendo0mul 40001* | Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 1-Aug-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ πΈ = ((TEndoβπΎ)βπ) & β’ π = (π β π β¦ ( I βΎ π΅)) β β’ (((πΎ β HL β§ π β π») β§ π β πΈ) β (π β π) = π) | ||
| Theorem | tendo0mulr 40002* | Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 13-Feb-2014.) |
| β’ π΅ = (BaseβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ πΈ = ((TEndoβπΎ)βπ) & β’ π = (π β π β¦ ( I βΎ π΅)) β β’ (((πΎ β HL β§ π β π») β§ π β πΈ) β (π β π) = π) | ||
| Theorem | tendo1ne0 40003* | The identity (unity) is not equal to the zero trace-preserving endomorphism. (Contributed by NM, 8-Aug-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ πΈ = ((TEndoβπΎ)βπ) & β’ π = (π β π β¦ ( I βΎ π΅)) β β’ ((πΎ β HL β§ π β π») β ( I βΎ π) β π) | ||
| Theorem | tendoconid 40004* | The composition (product) of trace-preserving endormorphisms is nonzero when each argument is nonzero. (Contributed by NM, 8-Aug-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ πΈ = ((TEndoβπΎ)βπ) & β’ π = (π β π β¦ ( I βΎ π΅)) β β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β π) β§ (π β πΈ β§ π β π)) β (π β π) β π) | ||
| Theorem | tendotr 40005* | The trace of the value of a nonzero trace-preserving endomorphism equals the trace of the argument. (Contributed by NM, 11-Aug-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ πΈ = ((TEndoβπΎ)βπ) & β’ π = (π β π β¦ ( I βΎ π΅)) β β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β π) β§ πΉ β π) β (π β(πβπΉ)) = (π βπΉ)) | ||
| Theorem | cdlemk1 40006 | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) β β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ π β π) β§ ((π βπΉ) = (π βπ) β§ (π β π΄ β§ Β¬ π β€ π))) β (π β¨ (πβπ)) = ((πΉβπ) β¨ (π βπΉ))) | ||
| Theorem | cdlemk2 40007 | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) β β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β ((πΊβπ) β¨ (π β(πΊ β β‘πΉ))) = ((πΉβπ) β¨ (π β(πΊ β β‘πΉ)))) | ||
| Theorem | cdlemk3 40008 | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ ((π βπΊ) β (π βπΉ) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (((πΉβπ) β¨ (π βπΉ)) β§ ((πΉβπ) β¨ (π β(πΊ β β‘πΉ)))) = (πΉβπ)) | ||
| Theorem | cdlemk4 40009 | Part of proof of Lemma K of [Crawley] p. 118, last line. We use π for their h, since π» is already used. (Contributed by NM, 24-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π)) β (πΉβπ) β€ ((πβπ) β¨ (π β(π β β‘πΉ)))) | ||
| Theorem | cdlemk5a 40010 | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π β§ π β π) β§ ((π βπΊ) β (π βπΉ) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (((πΉβπ) β¨ (π βπΉ)) β§ ((πΉβπ) β¨ (π β(πΊ β β‘πΉ)))) β€ ((πβπ) β¨ (π β(π β β‘πΉ)))) | ||
| Theorem | cdlemk5 40011 | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 25-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΊ β π) β§ ((π β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ (π βπΊ) β (π βπΉ))) β ((π β¨ (πβπ)) β§ ((πΊβπ) β¨ (π β(πΊ β β‘πΉ)))) β€ ((πβπ) β¨ (π β(π β β‘πΉ)))) | ||
| Theorem | cdlemk6 40012 | Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 39061. (Contributed by NM, 25-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΊ β π) β§ ((π β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ ((π βπΊ) β (π βπΉ) β§ (π βπ) β (π βπΉ)))) β ((π β¨ (πΊβπ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΉ)))) β€ ((((πΊβπ) β¨ (πβπ)) β§ ((π β(πΊ β β‘πΉ)) β¨ (π β(π β β‘πΉ)))) β¨ (((πβπ) β¨ π) β§ ((π β(π β β‘πΉ)) β¨ (πβπ))))) | ||
| Theorem | cdlemk8 40013 | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 26-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (πΊ β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π)) β ((πΊβπ) β¨ (πβπ)) = ((πΊβπ) β¨ (π β(π β β‘πΊ)))) | ||
| Theorem | cdlemk9 40014 | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 29-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (πΊ β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π)) β (((πΊβπ) β¨ (πβπ)) β§ π) = (π β(π β β‘πΊ))) | ||
| Theorem | cdlemk9bN 40015 | Part of proof of Lemma K of [Crawley] p. 118. TODO: is this needed? If so, shorten with cdlemk9 40014 if that one is also needed. (Contributed by NM, 28-Jun-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (πΊ β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π)) β (((πΊβπ) β¨ (πβπ)) β§ π) = (π β(πΊ β β‘π))) | ||
| Theorem | cdlemki 40016* | Part of proof of Lemma K of [Crawley] p. 118. TODO: Eliminate and put into cdlemksel 40020. (Contributed by NM, 25-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) & β’ πΌ = (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΉ))))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΊ β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ (π βπΊ) β (π βπΉ))) β πΌ β π) | ||
| Theorem | cdlemkvcl 40017 | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 27-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) & β’ π = (((πΊβπ) β¨ (πβπ)) β§ ((π β(πΊ β β‘πΉ)) β¨ (π β(π β β‘πΉ)))) β β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π β§ π β π) β§ π β π΄) β π β π΅) | ||
| Theorem | cdlemk10 40018 | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 29-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) & β’ π = (((πΊβπ) β¨ (πβπ)) β§ ((π β(πΊ β β‘πΉ)) β¨ (π β(π β β‘πΉ)))) β β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π)) β π β€ (π β(π β β‘πΊ))) | ||
| Theorem | cdlemksv 40019* | Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) β β’ (πΊ β π β (πβπΊ) = (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΉ)))))) | ||
| Theorem | cdlemksel 40020* | Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma(p) function to be a translation. TODO: combine cdlemki 40016? (Contributed by NM, 26-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΊ β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ (π βπΊ) β (π βπΉ))) β (πβπΊ) β π) | ||
| Theorem | cdlemksat 40021* | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 27-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΊ β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ (π βπΊ) β (π βπΉ))) β ((πβπΊ)βπ) β π΄) | ||
| Theorem | cdlemksv2 40022* | Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function π at the fixed π parameter. (Contributed by NM, 26-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΊ β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ (π βπΊ) β (π βπΉ))) β ((πβπΊ)βπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΉ))))) | ||
| Theorem | cdlemk7 40023* | Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119. (Contributed by NM, 27-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (((πΊβπ) β¨ (πβπ)) β§ ((π β(πΊ β β‘πΉ)) β¨ (π β(π β β‘πΉ)))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΊ β π) β§ ((π β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ ((πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π β ( I βΎ π΅)) β§ (π βπΊ) β (π βπΉ) β§ (π βπ) β (π βπΉ))) β ((πβπΊ)βπ) β€ (((πβπ)βπ) β¨ π)) | ||
| Theorem | cdlemk11 40024* | Part of proof of Lemma K of [Crawley] p. 118. Eq. 3, line 8, p. 119. (Contributed by NM, 29-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (((πΊβπ) β¨ (πβπ)) β§ ((π β(πΊ β β‘πΉ)) β¨ (π β(π β β‘πΉ)))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΊ β π) β§ ((π β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ ((πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π β ( I βΎ π΅)) β§ (π βπΊ) β (π βπΉ) β§ (π βπ) β (π βπΉ))) β ((πβπΊ)βπ) β€ (((πβπ)βπ) β¨ (π β(π β β‘πΊ)))) | ||
| Theorem | cdlemk12 40025* | Part of proof of Lemma K of [Crawley] p. 118. Eq. 4, line 10, p. 119. (Contributed by NM, 30-Jun-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ β§ = (meetβπΎ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ πΊ β π) β§ ((π β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ ((πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π β ( I βΎ π΅)) β§ ((π βπΊ) β (π βπΉ) β§ (π βπ) β (π βπΉ)) β§ (π βπΊ) β (π βπ))) β ((πβπΊ)βπ) = ((π β¨ (πΊβπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘πΊ))))) | ||
| Theorem | cdlemkoatnle 40026* | Utility lemma. (Contributed by NM, 2-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β ((πβπ) β π΄ β§ Β¬ (πβπ) β€ π)) | ||
| Theorem | cdlemk13 40027* | Part of proof of Lemma K of [Crawley] p. 118. Line 13 on p. 119. π, π· are k1, f1. (Contributed by NM, 1-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (πβπ) = ((π β¨ (π βπ·)) β§ ((πβπ) β¨ (π β(π· β β‘πΉ))))) | ||
| Theorem | cdlemkole 40028* | Utility lemma. (Contributed by NM, 2-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (πβπ) β€ (π β¨ (π βπ·))) | ||
| Theorem | cdlemk14 40029* | Part of proof of Lemma K of [Crawley] p. 118. Line 19 on p. 119. π, π· are k1, f1. (Contributed by NM, 1-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (πβπ) β€ ((πβπ) β¨ (π β(πΉ β β‘π·)))) | ||
| Theorem | cdlemk15 40030* | Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. π, π· are k1, f1. (Contributed by NM, 1-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (πβπ) β€ ((π β¨ (π βπΉ)) β§ ((πβπ) β¨ (π β(πΉ β β‘π·))))) | ||
| Theorem | cdlemk16a 40031* | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) β β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ π· β π β§ π β π) β§ (((π βπ·) β (π βπΉ) β§ (π βπ·) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘π·)))) β π΄ β§ Β¬ ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘π·)))) β€ π)) | ||
| Theorem | cdlemk16 40032* | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 1-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (((π β¨ (π βπΉ)) β§ ((πβπ) β¨ (π β(πΉ β β‘π·)))) β π΄ β§ Β¬ ((π β¨ (π βπΉ)) β§ ((πβπ) β¨ (π β(πΉ β β‘π·)))) β€ π)) | ||
| Theorem | cdlemk17 40033* | Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. π, π· are k1, f1. (Contributed by NM, 1-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (πβπ) = ((π β¨ (π βπΉ)) β§ ((πβπ) β¨ (π β(πΉ β β‘π·))))) | ||
| Theorem | cdlemk1u 40034* | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (π β¨ (πβπ)) β€ ((π·βπ) β¨ (π βπ·))) | ||
| Theorem | cdlemk5auN 40035* | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) β β’ (((πΎ β HL β§ π β π») β§ (π· β π β§ πΊ β π β§ π β π) β§ ((π βπΊ) β (π βπ·) β§ (π· β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (((π·βπ) β¨ (π βπ·)) β§ ((π·βπ) β¨ (π β(πΊ β β‘π·)))) β€ ((πβπ) β¨ (π β(π β β‘π·)))) | ||
| Theorem | cdlemk5u 40036* | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 4-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ ((π β π β§ πΊ β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ ((πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅)) β§ ((π βπ·) β (π βπΉ) β§ (π βπΊ) β (π βπ·) β§ (π βπ) β (π βπ·)))) β ((π β¨ (πβπ)) β§ ((πΊβπ) β¨ (π β(πΊ β β‘π·)))) β€ ((πβπ) β¨ (π β(π β β‘π·)))) | ||
| Theorem | cdlemk6u 40037* | Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 39061. (Contributed by NM, 4-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ ((π β π β§ πΊ β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ ((πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅)) β§ ((π βπ·) β (π βπΉ) β§ (π βπΊ) β (π βπ·) β§ (π βπ) β (π βπ·)))) β ((π β¨ (πΊβπ)) β§ ((πβπ) β¨ (π β(πΊ β β‘π·)))) β€ ((((πΊβπ) β¨ (πβπ)) β§ ((π β(πΊ β β‘π·)) β¨ (π β(π β β‘π·)))) β¨ (((πβπ) β¨ π) β§ ((π β(π β β‘π·)) β¨ (πβπ))))) | ||
| Theorem | cdlemkj 40038* | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 2-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) & β’ π = (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘π·))))) β β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ π· β π β§ π β π) β§ (((π βπ·) β (π βπΉ) β§ (π βπ·) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β π β π) | ||
| Theorem | cdlemkuvN 40039* | Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma1 (p) function π. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) β β’ (πΊ β π β (πβπΊ) = (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘π·)))))) | ||
| Theorem | cdlemkuel 40040* | Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma1 (p) function to be a translation. TODO: combine cdlemkj 40038? (Contributed by NM, 2-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) β β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ π· β π β§ π β π) β§ (((π βπ·) β (π βπΉ) β§ (π βπ·) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (πβπΊ) β π) | ||
| Theorem | cdlemkuat 40041* | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 4-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) β β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ π· β π β§ π β π) β§ (((π βπ·) β (π βπΉ) β§ (π βπ·) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πβπΊ)βπ) β π΄) | ||
| Theorem | cdlemkuv2 40042* | Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 1, where sigma1 (p) is π, f1 is π·, and k1 is π. (Contributed by NM, 2-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) β β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ π· β π β§ π β π) β§ (((π βπ·) β (π βπΉ) β§ (π βπ·) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πβπΊ)βπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘π·))))) | ||
| Theorem | cdlemk18 40043* | Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. π, π, π, π· are k, sigma1 (p), k1, f1. (Contributed by NM, 2-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (πβπ) = ((πβπΉ)βπ)) | ||
| Theorem | cdlemk19 40044* | Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. π, π, π, π· are k, sigma1 (p), k1, f1. (Contributed by NM, 2-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ (π βπ·) β (π βπΉ))) β (πβπΉ) = π) | ||
| Theorem | cdlemk7u 40045* | Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119 for the sigma1 case. (Contributed by NM, 3-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) & β’ π = (((πΊβπ) β¨ (πβπ)) β§ ((π β(πΊ β β‘π·)) β¨ (π β(π β β‘π·)))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ ((π β π β§ πΊ β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ ((πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅)) β§ π β ( I βΎ π΅) β§ ((π βπ·) β (π βπΉ) β§ (π βπΊ) β (π βπ·) β§ (π βπ) β (π βπ·)))) β ((πβπΊ)βπ) β€ (((πβπ)βπ) β¨ π)) | ||
| Theorem | cdlemk11u 40046* | Part of proof of Lemma K of [Crawley] p. 118. Line 17, p. 119, showing Eq. 3 (line 8, p. 119) for the sigma1 (π) case. (Contributed by NM, 4-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) & β’ π = (((πΊβπ) β¨ (πβπ)) β§ ((π β(πΊ β β‘π·)) β¨ (π β(π β β‘π·)))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ ((π β π β§ πΊ β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ ((πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅)) β§ π β ( I βΎ π΅) β§ ((π βπ·) β (π βπΉ) β§ (π βπΊ) β (π βπ·) β§ (π βπ) β (π βπ·)))) β ((πβπΊ)βπ) β€ (((πβπ)βπ) β¨ (π β(π β β‘πΊ)))) | ||
| Theorem | cdlemk12u 40047* | Part of proof of Lemma K of [Crawley] p. 118. Line 18, p. 119, showing Eq. 4 (line 10, p. 119) for the sigma1 (π) case. (Contributed by NM, 4-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ ((π β π β§ πΊ β π β§ π β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ ((πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅)) β§ (π β ( I βΎ π΅) β§ (π βπΊ) β (π βπ)) β§ ((π βπ·) β (π βπΉ) β§ (π βπΊ) β (π βπ·) β§ (π βπ) β (π βπ·)))) β ((πβπΊ)βπ) = ((π β¨ (πΊβπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘πΊ))))) | ||
| Theorem | cdlemk21N 40048* | Part of proof of Lemma K of [Crawley] p. 118. Lines 26-27, p. 119 for i=0 and j=1. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ ((π β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ ((πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅)) β§ ((π βπ·) β (π βπΉ) β§ (π βπΊ) β (π βπ·) β§ (π βπΊ) β (π βπΉ)))) β ((πβπΊ)βπ) = ((πβπΊ)βπ)) | ||
| Theorem | cdlemk20 40049* | Part of proof of Lemma K of [Crawley] p. 118. Line 22, p. 119 for the i=2, j=1 case. Note typo on line 22: f should be fi. Our π·, πΆ, π, π, π, π represent their f1, f2, k1, k2, sigma1, sigma2. (Contributed by NM, 5-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπ·) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) & β’ π = (πβπΆ) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ ((π β π β§ πΆ β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ ((πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ ((π βπ·) β (π βπΉ) β§ (π βπΆ) β (π βπΉ) β§ (π βπΆ) β (π βπ·)))) β ((πβπΆ)βπ) = (πβπ)) | ||
| Theorem | cdlemkoatnle-2N 40050* | Utility lemma. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ ((π βπΆ) β (π βπΉ) β§ (πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πβπ) β π΄ β§ Β¬ (πβπ) β€ π)) | ||
| Theorem | cdlemk13-2N 40051* | Part of proof of Lemma K of [Crawley] p. 118. Line 13 on p. 119. π, πΆ are k2, f2. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ ((π βπΆ) β (π βπΉ) β§ (πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (πβπ) = ((π β¨ (π βπΆ)) β§ ((πβπ) β¨ (π β(πΆ β β‘πΉ))))) | ||
| Theorem | cdlemkole-2N 40052* | Utility lemma. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ ((π βπΆ) β (π βπΉ) β§ (πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (πβπ) β€ (π β¨ (π βπΆ))) | ||
| Theorem | cdlemk14-2N 40053* | Part of proof of Lemma K of [Crawley] p. 118. Line 19 on p. 119. π, πΆ are k2, f2. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ ((π βπΆ) β (π βπΉ) β§ (πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (πβπ) β€ ((πβπ) β¨ (π β(πΉ β β‘πΆ)))) | ||
| Theorem | cdlemk15-2N 40054* | Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. π, πΆ are k2, f2. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ ((π βπΆ) β (π βπΉ) β§ (πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (πβπ) β€ ((π β¨ (π βπΉ)) β§ ((πβπ) β¨ (π β(πΉ β β‘πΆ))))) | ||
| Theorem | cdlemk16-2N 40055* | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ ((π βπΆ) β (π βπΉ) β§ (πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (((π β¨ (π βπΉ)) β§ ((πβπ) β¨ (π β(πΉ β β‘πΆ)))) β π΄ β§ Β¬ ((π β¨ (π βπΉ)) β§ ((πβπ) β¨ (π β(πΉ β β‘πΆ)))) β€ π)) | ||
| Theorem | cdlemk17-2N 40056* | Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. π, πΆ are k2, f2. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ ((π βπΆ) β (π βπΉ) β§ (πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (πβπ) = ((π β¨ (π βπΉ)) β§ ((πβπ) β¨ (π β(πΉ β β‘πΆ))))) | ||
| Theorem | cdlemkj-2N 40057* | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) & β’ π = (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΆ))))) β β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ (((π βπΆ) β (π βπΉ) β§ (π βπΆ) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β π β π) | ||
| Theorem | cdlemkuv-2N 40058* | Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma2 (p) function, given π. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΆ)))))) β β’ (πΊ β π β (πβπΊ) = (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΆ)))))) | ||
| Theorem | cdlemkuel-2N 40059* | Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma2 (p) function to be a translation. TODO: combine cdlemkj 40038? (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΆ)))))) β β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ (((π βπΆ) β (π βπΉ) β§ (π βπΆ) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (πβπΊ) β π) | ||
| Theorem | cdlemkuv2-2 40060* | Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 2, where sigma2 (p) is π, f2 is πΆ, and k2 is π. (Contributed by NM, 2-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΆ)))))) β β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ (((π βπΆ) β (π βπΉ) β§ (π βπΆ) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πβπΊ)βπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΆ))))) | ||
| Theorem | cdlemk18-2N 40061* | Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. π, π, π, πΆ are k, sigma2 (p), k2, f2. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΆ)))))) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ ((π βπΆ) β (π βπΉ) β§ (πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (πβπ) = ((πβπΉ)βπ)) | ||
| Theorem | cdlemk19-2N 40062* | Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. π, π, π, πΆ are k, sigma2 (p), k2, f2. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΆ)))))) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ (πΉ β π β§ πΆ β π β§ π β π) β§ ((π βπΆ) β (π βπΉ) β§ (πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (πβπΉ) = π) | ||
| Theorem | cdlemk7u-2N 40063* | Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119 for the sigma2 case. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΆ)))))) & β’ π = (((πΊβπ) β¨ (πβπ)) β§ ((π β(πΊ β β‘πΆ)) β¨ (π β(π β β‘πΆ)))) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ ((πΉ β π β§ πΆ β π β§ π β π) β§ (πΊ β π β§ πΊ β ( I βΎ π΅)) β§ (π β π β§ π β ( I βΎ π΅))) β§ (((π βπΆ) β (π βπΉ) β§ (π βπΊ) β (π βπΆ) β§ (π βπ) β (π βπΆ)) β§ (πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πβπΊ)βπ) β€ (((πβπ)βπ) β¨ π)) | ||
| Theorem | cdlemk11u-2N 40064* | Part of proof of Lemma K of [Crawley] p. 118. Line 17, p. 119, showing Eq. 3 (line 8, p. 119) for the sigma2 (π) case. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΆ)))))) & β’ π = (((πΊβπ) β¨ (πβπ)) β§ ((π β(πΊ β β‘πΆ)) β¨ (π β(π β β‘πΆ)))) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ ((πΉ β π β§ πΆ β π β§ π β π) β§ (πΊ β π β§ πΊ β ( I βΎ π΅)) β§ (π β π β§ π β ( I βΎ π΅))) β§ (((π βπΆ) β (π βπΉ) β§ (π βπΊ) β (π βπΆ) β§ (π βπ) β (π βπΆ)) β§ (πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πβπΊ)βπ) β€ (((πβπ)βπ) β¨ (π β(π β β‘πΊ)))) | ||
| Theorem | cdlemk12u-2N 40065* | Part of proof of Lemma K of [Crawley] p. 118. Line 18, p. 119, showing Eq. 4 (line 10, p. 119) for the sigma2 (π) case. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΆ)))))) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ ((πΉ β π β§ πΆ β π β§ π β π) β§ (πΊ β π β§ πΊ β ( I βΎ π΅)) β§ (π β π β§ π β ( I βΎ π΅))) β§ (((π βπΆ) β (π βπΉ) β§ (π βπΊ) β (π βπΆ) β§ (π βπ) β (π βπΆ)) β§ ((π βπΊ) β (π βπ) β§ πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πβπΊ)βπ) = ((π β¨ (πΊβπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘πΊ))))) | ||
| Theorem | cdlemk21-2N 40066* | Part of proof of Lemma K of [Crawley] p. 118. Lines 26-27, p. 119 for i=0 and j=2. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΆ)))))) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ ((πΉ β π β§ πΆ β π β§ π β π) β§ (πΊ β π β§ πΊ β ( I βΎ π΅))) β§ (((π βπΆ) β (π βπΉ) β§ (π βπΊ) β (π βπΆ)) β§ ((π βπΊ) β (π βπΉ) β§ πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πβπΊ)βπ) = ((πβπΊ)βπ)) | ||
| Theorem | cdlemk20-2N 40067* | Part of proof of Lemma K of [Crawley] p. 118. Line 22, p. 119 for the i=2, j=1 case. Note typo on line 22: f should be fi. Our π·, πΆ, π, π, π, π represent their f1, f2, k1, k2, sigma1, sigma2. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΆ)))))) & β’ π = (πβπ·) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ ((πΉ β π β§ πΆ β π β§ π β π) β§ (π· β π β§ π· β ( I βΎ π΅)) β§ (πΆ β π β§ πΆ β ( I βΎ π΅))) β§ (((π βπΆ) β (π βπΉ) β§ (π βπ·) β (π βπΉ) β§ (π βπ·) β (π βπΆ)) β§ (πΉ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πβπ·)βπ) = (πβπ)) | ||
| Theorem | cdlemk22 40068* | Part of proof of Lemma K of [Crawley] p. 118. Lines 26-27, p. 119 for i=1 and j=2. (Contributed by NM, 5-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (πβπΆ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΆ)))))) & β’ π = (πβπ·) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ ((π β π β§ πΊ β π β§ πΆ β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ ((πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅)) β§ (πΆ β ( I βΎ π΅) β§ (π βπΊ) β (π βπΆ) β§ (π βπΆ) β (π βπΉ)) β§ ((π βπ·) β (π βπΉ) β§ (π βπΊ) β (π βπ·) β§ (π βπΆ) β (π βπ·)))) β ((πβπΊ)βπ) = ((πβπΊ)βπ)) | ||
| Theorem | cdlemk30 40069* | Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 17-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) β β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ)) β§ (πΉ β π β§ π β π β§ π β π) β§ ((π βπ) β (π βπΉ) β§ (πΉ β ( I βΎ π΅) β§ π β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πβπ)βπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ))))) | ||
| Theorem | cdlemkuu 40070* | Convert between function and operation forms of π. TODO: Use operation form everywhere. (Contributed by NM, 6-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) & β’ π = (πβπ·) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘π·)))))) β β’ ((π· β π β§ πΊ β π) β (π·ππΊ) = (πβπΊ)) | ||
| Theorem | cdlemk31 40071* | Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 17-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) β β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ)) β§ ((πΉ β π β§ π β π β§ π β π) β§ πΊ β π) β§ (((π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ π β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πππΊ)βπ) = ((π β¨ (π βπΊ)) β§ (((πβπ)βπ) β¨ (π β(πΊ β β‘π))))) | ||
| Theorem | cdlemk32 40072* | Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 17-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) β β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ)) β§ ((πΉ β π β§ π β π β§ π β π) β§ πΊ β π) β§ (((π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ π β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πππΊ)βπ) = ((π β¨ (π βπΊ)) β§ (((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) β¨ (π β(πΊ β β‘π))))) | ||
| Theorem | cdlemkuel-3 40073* | Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma2 (p) function to be a translation. TODO: combine cdlemkj 40038? (Contributed by NM, 11-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) β β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ π· β π β§ π β π) β§ (((π βπ·) β (π βπΉ) β§ (π βπ·) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β (π·ππΊ) β π) | ||
| Theorem | cdlemkuv2-3N 40074* | Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 1, where sigma2 (p) is π, f1 is π·, and k1 is π. (Contributed by NM, 6-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) β β’ ((((πΎ β HL β§ π β π») β§ (π βπΉ) = (π βπ) β§ πΊ β π) β§ (πΉ β π β§ π· β π β§ π β π) β§ (((π βπ·) β (π βπΉ) β§ (π βπ·) β (π βπΊ)) β§ (πΉ β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((π·ππΊ)βπ) = ((π β¨ (π βπΊ)) β§ (((πβπ·)βπ) β¨ (π β(πΊ β β‘π·))))) | ||
| Theorem | cdlemk18-3N 40075* | Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. π, π, π, π· are k, sigma2 (p), k1, f1. (Contributed by NM, 7-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ (πΉ β π β§ π· β π β§ π β π) β§ ((π βπ·) β (π βπΉ) β§ (πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β ((π·ππΉ)βπ) = (πβπ)) | ||
| Theorem | cdlemk22-3 40076* | Part of proof of Lemma K of [Crawley] p. 118. Lines 26-27, p. 119 for i=1 and j=2. (Contributed by NM, 7-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π· β π) β§ ((π β π β§ πΊ β π β§ πΆ β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ ((πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅) β§ πΊ β ( I βΎ π΅)) β§ (πΆ β ( I βΎ π΅) β§ (π βπΊ) β (π βπΆ) β§ (π βπΆ) β (π βπΉ)) β§ ((π βπ·) β (π βπΉ) β§ (π βπΊ) β (π βπ·) β§ (π βπΆ) β (π βπ·)))) β ((π·ππΊ)βπ) = ((πΆππΊ)βπ)) | ||
| Theorem | cdlemk23-3 40077* | Part of proof of Lemma K of [Crawley] p. 118. Eliminate the (π βπΆ) β (π βπ·) requirement from cdlemk22-3 40076. (Contributed by NM, 7-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) β β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ π· β π β§ π β π) β§ (πΊ β π β§ πΆ β π β§ π₯ β π)) β§ ((π β π΄ β§ Β¬ π β€ π) β§ ((π βπΉ) = (π βπ) β§ πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (πΊ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅) β§ π₯ β ( I βΎ π΅))) β§ (((π βπΊ) β (π βπΆ) β§ (π βπΆ) β (π βπΉ) β§ (π βπ·) β (π βπΉ)) β§ ((π βπΊ) β (π βπ·) β§ (π βπ₯) β (π βπΆ)) β§ ((π βπ₯) β (π βπ·) β§ (π βπ₯) β (π βπΉ) β§ (π βπΊ) β (π βπ₯)))) β ((π·ππΊ)βπ) = ((πΆππΊ)βπ)) | ||
| Theorem | cdlemk24-3 40078* | Part of proof of Lemma K of [Crawley] p. 118. Eliminate the (π βπ₯) β (π βπΆ) requirement from cdlemk23-3 40077 using (π βπΆ) = (π βπ·). (Contributed by NM, 7-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) β β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ π· β π β§ π β π) β§ (πΊ β π β§ πΆ β π β§ π₯ β π)) β§ ((π β π΄ β§ Β¬ π β€ π) β§ ((π βπΉ) = (π βπ) β§ πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (πΊ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅) β§ π₯ β ( I βΎ π΅))) β§ (((π βπΊ) β (π βπΆ) β§ (π βπΆ) β (π βπΉ) β§ (π βπ·) β (π βπΉ)) β§ ((π βπΊ) β (π βπ·) β§ (π βπΆ) = (π βπ·)) β§ ((π βπ₯) β (π βπ·) β§ (π βπ₯) β (π βπΉ) β§ (π βπΊ) β (π βπ₯)))) β ((π·ππΊ)βπ) = ((πΆππΊ)βπ)) | ||
| Theorem | cdlemk25-3 40079* | Part of proof of Lemma K of [Crawley] p. 118. Eliminate the (π βπΆ) = (π βπ·) requirement from cdlemk24-3 40078. (Contributed by NM, 7-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) β β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ π· β π β§ π β π) β§ (πΊ β π β§ πΆ β π β§ π₯ β π)) β§ ((π β π΄ β§ Β¬ π β€ π) β§ ((π βπΉ) = (π βπ) β§ πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (πΊ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅) β§ π₯ β ( I βΎ π΅))) β§ (((π βπΊ) β (π βπΆ) β§ (π βπΆ) β (π βπΉ) β§ (π βπ·) β (π βπΉ)) β§ (π βπΊ) β (π βπ·) β§ ((π βπ₯) β (π βπ·) β§ (π βπ₯) β (π βπΉ) β§ (π βπΊ) β (π βπ₯)))) β ((π·ππΊ)βπ) = ((πΆππΊ)βπ)) | ||
| Theorem | cdlemk26b-3 40080* | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 14-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) β β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΉ β ( I βΎ π΅) β§ π β π) β§ (πΊ β π β§ πΊ β ( I βΎ π΅) β§ (π βπΉ) = (π βπ))) β§ (π β π΄ β§ Β¬ π β€ π)) β βπ₯ β π ((π₯ β ( I βΎ π΅) β§ (π βπ₯) β (π βπΉ) β§ (π βπ₯) β (π βπΊ)) β§ (π₯ππΊ) β π)) | ||
| Theorem | cdlemk26-3 40081* | Part of proof of Lemma K of [Crawley] p. 118. Eliminate the π₯ requirements from cdlemk25-3 40079. (Contributed by NM, 10-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) β β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ π· β π β§ π β π) β§ (πΊ β π β§ πΆ β π)) β§ ((π β π΄ β§ Β¬ π β€ π) β§ ((π βπΉ) = (π βπ) β§ πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (πΊ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅))) β§ (((π βπΊ) β (π βπΆ) β§ (π βπΆ) β (π βπΉ) β§ (π βπ·) β (π βπΉ)) β§ (π βπΊ) β (π βπ·))) β ((π·ππΊ)βπ) = ((πΆππΊ)βπ)) | ||
| Theorem | cdlemk27-3 40082* | Part of proof of Lemma K of [Crawley] p. 118. Eliminate the π from the conclusion of cdlemk25-3 40079. (Contributed by NM, 10-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) β β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ π· β π β§ π β π) β§ (πΊ β π β§ πΆ β π)) β§ ((π β π΄ β§ Β¬ π β€ π) β§ ((π βπΉ) = (π βπ) β§ πΉ β ( I βΎ π΅) β§ π· β ( I βΎ π΅)) β§ (πΊ β ( I βΎ π΅) β§ πΆ β ( I βΎ π΅))) β§ (((π βπΊ) β (π βπΆ) β§ (π βπΆ) β (π βπΉ) β§ (π βπ·) β (π βπΉ)) β§ (π βπΊ) β (π βπ·))) β (π·ππΊ) = (πΆππΊ)) | ||
| Theorem | cdlemk28-3 40083* | Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 14-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) β β’ (((πΎ β HL β§ π β π») β§ ((πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅)) β§ π β π) β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ))) β βπ§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)) β π§ = (πππΊ))) | ||
| Theorem | cdlemk33N 40084* | Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. TODO: not needed, is embodied in cdlemk34 40085. (Contributed by NM, 18-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) & β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)) β π§ = (πππΊ))) β β’ (((πΎ β HL β§ π β π» β§ (π βπΉ) = (π βπ)) β§ ((πΉ β π β§ πΉ β ( I βΎ π΅) β§ π β π) β§ (πΊ β π β§ πΊ β ( I βΎ π΅)) β§ (π β π΄ β§ Β¬ π β€ π))) β π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)) β (π§βπ) = ((πππΊ)βπ)))) | ||
| Theorem | cdlemk34 40085* | Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 18-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) & β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)) β π§ = (πππΊ))) β β’ (((πΎ β HL β§ π β π») β§ ((πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅)) β§ π β π) β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ))) β π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)) β (π§βπ) = ((π β¨ (π βπΊ)) β§ (((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) β¨ (π β(πΊ β β‘π))))))) | ||
| Theorem | cdlemk29-3 40086* | Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 14-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) & β’ π = (π β π, π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ (((πβπ)βπ) β¨ (π β(π β β‘π)))))) & β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)) β π§ = (πππΊ))) β β’ (((πΎ β HL β§ π β π») β§ ((πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅)) β§ π β π) β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ))) β π β π) | ||
| Theorem | cdlemk35 40087* | Part of proof of Lemma K of [Crawley] p. 118. cdlemk29-3 40086 with shorter hypotheses. (Contributed by NM, 18-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) & β’ π = ((π β¨ (π βπΊ)) β§ (π β¨ (π β(πΊ β β‘π)))) & β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)) β (π§βπ) = π)) β β’ (((πΎ β HL β§ π β π») β§ ((πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅)) β§ π β π) β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ))) β π β π) | ||
| Theorem | cdlemk36 40088* | Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 18-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) & β’ π = ((π β¨ (π βπΊ)) β§ (π β¨ (π β(πΊ β β‘π)))) & β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)) β (π§βπ) = π)) β β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅))) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (π β π β§ (π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)))) β (πβπ) = π) | ||
| Theorem | cdlemk37 40089* | Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 18-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) & β’ π = ((π β¨ (π βπΊ)) β§ (π β¨ (π β(πΊ β β‘π)))) & β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)) β (π§βπ) = π)) β β’ ((((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅))) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ)) β§ (π β π β§ (π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)))) β (πβπ) β€ (π β¨ (π βπΊ))) | ||
| Theorem | cdlemk38 40090* | Part of proof of Lemma K of [Crawley] p. 118. Line 31, p. 119. TODO: derive more directly with r19.23 3252? (Contributed by NM, 19-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) & β’ π = ((π β¨ (π βπΊ)) β§ (π β¨ (π β(πΊ β β‘π)))) & β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)) β (π§βπ) = π)) β β’ (((πΎ β HL β§ π β π») β§ ((πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅)) β§ π β π) β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ))) β (πβπ) β€ (π β¨ (π βπΊ))) | ||
| Theorem | cdlemk39 40091* | Part of proof of Lemma K of [Crawley] p. 118. Line 31, p. 119. Trace-preserving property of tau, represented by π. (Contributed by NM, 19-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) & β’ π = ((π β¨ (π βπΊ)) β§ (π β¨ (π β(πΊ β β‘π)))) & β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ)) β (π§βπ) = π)) β β’ (((πΎ β HL β§ π β π») β§ ((πΉ β π β§ πΉ β ( I βΎ π΅)) β§ (πΊ β π β§ πΊ β ( I βΎ π΅)) β§ π β π) β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π βπΉ) = (π βπ))) β (π βπ) β€ (π βπΊ)) | ||
| Theorem | cdlemk40 40092* | TODO: fix comment. (Contributed by NM, 31-Jul-2013.) |
| β’ π = (β©π§ β π π) & β’ π = (π β π β¦ if(πΉ = π, π, π)) β β’ (πΊ β π β (πβπΊ) = if(πΉ = π, πΊ, β¦πΊ / πβ¦π)) | ||
| Theorem | cdlemk40t 40093* | TODO: fix comment. (Contributed by NM, 31-Jul-2013.) |
| β’ π = (β©π§ β π π) & β’ π = (π β π β¦ if(πΉ = π, π, π)) β β’ ((πΉ = π β§ πΊ β π) β (πβπΊ) = πΊ) | ||
| Theorem | cdlemk40f 40094* | TODO: fix comment. (Contributed by NM, 31-Jul-2013.) |
| β’ π = (β©π§ β π π) & β’ π = (π β π β¦ if(πΉ = π, π, π)) β β’ ((πΉ β π β§ πΊ β π) β (πβπΊ) = β¦πΊ / πβ¦π) | ||
| Theorem | cdlemk41 40095* | Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 19-Jul-2013.) |
| β’ π = ((π β¨ (π βπ)) β§ (π β¨ (π β(π β β‘π)))) β β’ (πΊ β π β β¦πΊ / πβ¦π = ((π β¨ (π βπΊ)) β§ (π β¨ (π β(πΊ β β‘π))))) | ||
| Theorem | cdlemkfid1N 40096 | Lemma for cdlemkfid3N 40100. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) β β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΉ β ( I βΎ π΅) β§ πΊ β π) β§ ((π βπΊ) β (π βπΉ) β§ (π β π΄ β§ Β¬ π β€ π))) β ((π β¨ (π βπΊ)) β§ ((πΉβπ) β¨ (π β(πΊ β β‘πΉ)))) = (πΊβπ)) | ||
| Theorem | cdlemkid1 40097 | Lemma for cdlemkid 40111. (Contributed by NM, 24-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) β β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ π β π β§ (π βπΉ) = (π βπ)) β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π β§ π β ( I βΎ π΅)))) β (π β¨ (π βπ)) = (π β¨ (π βπ))) | ||
| Theorem | cdlemkfid2N 40098 | Lemma for cdlemkfid3N 40100. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ = π) β§ (πΉ β π β§ πΉ β ( I βΎ π΅) β§ π β π) β§ ((π βπ) β (π βπΉ) β§ (π β π΄ β§ Β¬ π β€ π))) β π = (πβπ)) | ||
| Theorem | cdlemkid2 40099* | Lemma for cdlemkid 40111. (Contributed by NM, 24-Jul-2013.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) & β’ π = ((π β¨ (π βπ)) β§ (π β¨ (π β(π β β‘π)))) β β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ π β π β§ (π βπΉ) = (π βπ)) β§ ((π β π΄ β§ Β¬ π β€ π) β§ πΊ = ( I βΎ π΅) β§ (π β π β§ π β ( I βΎ π΅)))) β β¦πΊ / πβ¦π = π) | ||
| Theorem | cdlemkfid3N 40100* | TODO: is this useful or should it be deleted? (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.) |
| β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) & β’ π = ((trLβπΎ)βπ) & β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) & β’ π = ((π β¨ (π βπ)) β§ (π β¨ (π β(π β β‘π)))) β β’ ((((πΎ β HL β§ π β π») β§ πΉ = π) β§ ((πΉ β π β§ πΉ β ( I βΎ π΅)) β§ πΊ β π β§ (π β π β§ π β ( I βΎ π΅))) β§ ((π βπ) β (π βπΉ) β§ (π βπ) β (π βπΊ) β§ (π β π΄ β§ Β¬ π β€ π))) β β¦πΊ / πβ¦π = (πΊβπ)) | ||
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