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Type | Label | Description |
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Statement | ||
Theorem | hdmapffval 41001* | Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.) |
β’ π» = (LHypβπΎ) β β’ (πΎ β π β (HDMapβπΎ) = (π€ β π» β¦ {π β£ [β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ€))β© / π][((DVecHβπΎ)βπ€) / π’][(Baseβπ’) / π£][((HDMap1βπΎ)βπ€) / π]π β (π‘ β π£ β¦ (β©π¦ β (Baseβ((LCDualβπΎ)βπ€))βπ§ β π£ (Β¬ π§ β (((LSpanβπ’)β{π}) βͺ ((LSpanβπ’)β{π‘})) β π¦ = (πββ¨π§, (πββ¨π, (((HVMapβπΎ)βπ€)βπ), π§β©), π‘β©))))})) | ||
Theorem | hdmapfval 41002* | Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π½ = ((HVMapβπΎ)βπ) & β’ πΌ = ((HDMap1βπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β π΄ β§ π β π»)) β β’ (π β π = (π‘ β π β¦ (β©π¦ β π· βπ§ β π (Β¬ π§ β ((πβ{πΈ}) βͺ (πβ{π‘})) β π¦ = (πΌββ¨π§, (πΌββ¨πΈ, (π½βπΈ), π§β©), π‘β©))))) | ||
Theorem | hdmapval 41003* | Value of map from vectors to functionals in the closed kernel dual space. This is the function sigma on line 27 above part 9 in [Baer] p. 48. We select a convenient fixed reference vector πΈ to be β¨0, 1β© (corresponding to vector u on p. 48 line 7) whose span is the lattice isomorphism map of the fiducial atom π = ((ocβπΎ)βπ) (see dvheveccl 40287). (π½βπΈ) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 40944 shows in Baer's notation (Fu)* = Gu'). Baer's independent vectors v and w on line 7 correspond to our π§ that the βπ§ β π ranges over. The middle term (πΌββ¨πΈ, (π½βπΈ), π§β©) provides isolation to allow πΈ and π to assume the same value without conflict. Closure is shown by hdmapcl 41005. If a separate auxiliary vector is known, hdmapval2 41007 provides a version without quantification. (Contributed by NM, 15-May-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π½ = ((HVMapβπΎ)βπ) & β’ πΌ = ((HDMap1βπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β π΄ β§ π β π»)) & β’ (π β π β π) β β’ (π β (πβπ) = (β©π¦ β π· βπ§ β π (Β¬ π§ β ((πβ{πΈ}) βͺ (πβ{π})) β π¦ = (πΌββ¨π§, (πΌββ¨πΈ, (π½βπΈ), π§β©), πβ©)))) | ||
Theorem | hdmapfnN 41004 | Functionality of map from vectors to functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) β β’ (π β π Fn π) | ||
Theorem | hdmapcl 41005 | Closure of map from vectors to functionals with closed kernels. (Contributed by NM, 15-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) β β’ (π β (πβπ) β π·) | ||
Theorem | hdmapval2lem 41006* | Lemma for hdmapval2 41007. (Contributed by NM, 15-May-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π½ = ((HVMapβπΎ)βπ) & β’ πΌ = ((HDMap1βπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β πΉ β π·) β β’ (π β ((πβπ) = πΉ β βπ§ β π (Β¬ π§ β ((πβ{πΈ}) βͺ (πβ{π})) β πΉ = (πΌββ¨π§, (πΌββ¨πΈ, (π½βπΈ), π§β©), πβ©)))) | ||
Theorem | hdmapval2 41007 | Value of map from vectors to functionals with a specific auxiliary vector. TODO: Would shorter proofs result if the .ne hypothesis were changed to two β hypothesis? Consider hdmaplem1 40946 through hdmaplem4 40949, which would become obsolete. (Contributed by NM, 15-May-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π½ = ((HVMapβπΎ)βπ) & β’ πΌ = ((HDMap1βπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β Β¬ π β ((πβ{πΈ}) βͺ (πβ{π}))) β β’ (π β (πβπ) = (πΌββ¨π, (πΌββ¨πΈ, (π½βπΈ), πβ©), πβ©)) | ||
Theorem | hdmapval0 41008 | Value of map from vectors to functionals at zero. Note: we use dvh3dim 40621 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 41019 could be derived from the other to shorten proof. (Contributed by NM, 17-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ 0 = (0gβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π = (0gβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) β β’ (π β (πβ 0 ) = π) | ||
Theorem | hdmapeveclem 41009 | Lemma for hdmapevec 41010. TODO: combine with hdmapevec 41010 if it shortens overall. (Contributed by NM, 16-May-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π½ = ((HVMapβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ πΌ = ((HDMap1βπΎ)βπ) & β’ (π β π β π) & β’ (π β Β¬ π β ((πβ{πΈ}) βͺ (πβ{πΈ}))) β β’ (π β (πβπΈ) = (π½βπΈ)) | ||
Theorem | hdmapevec 41010 | Value of map from vectors to functionals at the reference vector πΈ. (Contributed by NM, 16-May-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π½ = ((HVMapβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) β β’ (π β (πβπΈ) = (π½βπΈ)) | ||
Theorem | hdmapevec2 41011 | The inner product of the reference vector πΈ with itself is nonzero. This shows the inner product condition in the proof of Theorem 3.6 of [Holland95] p. 14 line 32, [ e , e ] β 0 is satisfied. TODO: remove redundant hypothesis hdmapevec.j. (Contributed by NM, 1-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π½ = ((HVMapβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Scalarβπ) & β’ 1 = (1rβπ ) β β’ (π β ((πβπΈ)βπΈ) = 1 ) | ||
Theorem | hdmapval3lemN 41012 | Value of map from vectors to functionals at arguments not colinear with the reference vector πΈ. (Contributed by NM, 17-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π½ = ((HVMapβπΎ)βπ) & β’ πΌ = ((HDMap1βπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β (πβ{π}) β (πβ{πΈ})) & β’ (π β π β (π β {(0gβπ)})) & β’ (π β π₯ β π) & β’ (π β Β¬ π₯ β (πβ{πΈ, π})) β β’ (π β (πβπ) = (πΌββ¨πΈ, (π½βπΈ), πβ©)) | ||
Theorem | hdmapval3N 41013 | Value of map from vectors to functionals at arguments not colinear with the reference vector πΈ. (Contributed by NM, 17-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π½ = ((HVMapβπΎ)βπ) & β’ πΌ = ((HDMap1βπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β (πβ{π}) β (πβ{πΈ})) & β’ (π β π β π) β β’ (π β (πβπ) = (πΌββ¨πΈ, (π½βπΈ), πβ©)) | ||
Theorem | hdmap10lem 41014 | Lemma for hdmap10 41015. (Contributed by NM, 17-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ 0 = (0gβπ) & β’ π· = (BaseβπΆ) & β’ π½ = ((HVMapβπΎ)βπ) & β’ πΌ = ((HDMap1βπΎ)βπ) & β’ (π β π β (π β { 0 })) β β’ (π β (πβ(πβ{π})) = (πΏβ{(πβπ)})) | ||
Theorem | hdmap10 41015 | Part 10 in [Baer] p. 48 line 33, (Ft)* = G(tS) in their notation (S = sigma). (Contributed by NM, 17-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) β β’ (π β (πβ(πβ{π})) = (πΏβ{(πβπ)})) | ||
Theorem | hdmap11lem1 41016 | Lemma for hdmapadd 41018. (Contributed by NM, 26-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = (+gβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ π· = (BaseβπΆ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π½ = ((HVMapβπΎ)βπ) & β’ πΌ = ((HDMap1βπΎ)βπ) & β’ (π β π§ β π) & β’ (π β Β¬ π§ β (πβ{π, π})) & β’ (π β (πβ{π§}) β (πβ{πΈ})) β β’ (π β (πβ(π + π)) = ((πβπ) β (πβπ))) | ||
Theorem | hdmap11lem2 41017 | Lemma for hdmapadd 41018. (Contributed by NM, 26-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = (+gβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ π· = (BaseβπΆ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π½ = ((HVMapβπΎ)βπ) & β’ πΌ = ((HDMap1βπΎ)βπ) β β’ (π β (πβ(π + π)) = ((πβπ) β (πβπ))) | ||
Theorem | hdmapadd 41018 | Part 11 in [Baer] p. 48 line 35, (a+b)S = aS+bS in their notation (S = sigma). (Contributed by NM, 22-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = (+gβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β (πβ(π + π)) = ((πβπ) β (πβπ))) | ||
Theorem | hdmapeq0 41019 | Part of proof of part 12 in [Baer] p. 49 line 3. (Contributed by NM, 22-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π = (0gβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) β β’ (π β ((πβπ) = π β π = 0 )) | ||
Theorem | hdmapnzcl 41020 | Nonzero vector closure of map from vectors to functionals with closed kernels. (Contributed by NM, 27-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π = (0gβπΆ) & β’ π· = (BaseβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π β { 0 })) β β’ (π β (πβπ) β (π· β {π})) | ||
Theorem | hdmapneg 41021 | Part of proof of part 12 in [Baer] p. 49 line 4. The sigma map of a negative is the negative of the sigma map. (Contributed by NM, 24-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (invgβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΌ = (invgβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) β β’ (π β (πβ(πβπ)) = (πΌβ(πβπ))) | ||
Theorem | hdmapsub 41022 | Part of proof of part 12 in [Baer] p. 49 line 5, (a-b)S = aS-bS in their notation (S = sigma). (Contributed by NM, 26-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π = (-gβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β (πβ(π β π)) = ((πβπ)π(πβπ))) | ||
Theorem | hdmap11 41023 | Part of proof of part 12 in [Baer] p. 49 line 4, aS=bS iff a=b in their notation (S = sigma). The sigma map is one-to-one. (Contributed by NM, 26-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β ((πβπ) = (πβπ) β π = π)) | ||
Theorem | hdmaprnlem1N 41024 | Part of proof of part 12 in [Baer] p. 49 line 10, Gu' β Gs. Our (πβ{π£}) is Baer's T. (Contributed by NM, 26-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β {π})) & β’ (π β π£ β π) & β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) & β’ (π β π’ β π) & β’ (π β Β¬ π’ β (πβ{π£})) β β’ (π β (πΏβ{(πβπ’)}) β (πΏβ{π })) | ||
Theorem | hdmaprnlem3N 41025 | Part of proof of part 12 in [Baer] p. 49 line 15, T β P. Our (β‘πβ(πΏβ{((πβπ’) β π )})) is Baer's P, where P* = G(u'+s). (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β {π})) & β’ (π β π£ β π) & β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) & β’ (π β π’ β π) & β’ (π β Β¬ π’ β (πβ{π£})) & β’ π· = (BaseβπΆ) & β’ π = (0gβπΆ) & β’ 0 = (0gβπ) & β’ β = (+gβπΆ) β β’ (π β (πβ{π£}) β (β‘πβ(πΏβ{((πβπ’) β π )}))) | ||
Theorem | hdmaprnlem3uN 41026 | Part of proof of part 12 in [Baer] p. 49. (Contributed by NM, 29-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β {π})) & β’ (π β π£ β π) & β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) & β’ (π β π’ β π) & β’ (π β Β¬ π’ β (πβ{π£})) & β’ π· = (BaseβπΆ) & β’ π = (0gβπΆ) & β’ 0 = (0gβπ) & β’ β = (+gβπΆ) β β’ (π β (πβ{π’}) β (β‘πβ(πΏβ{((πβπ’) β π )}))) | ||
Theorem | hdmaprnlem4tN 41027 | Lemma for hdmaprnN 41039. TODO: This lemma doesn't quite pay for itself even though used six times. Maybe prove this directly instead. (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β {π})) & β’ (π β π£ β π) & β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) & β’ (π β π’ β π) & β’ (π β Β¬ π’ β (πβ{π£})) & β’ π· = (BaseβπΆ) & β’ π = (0gβπΆ) & β’ 0 = (0gβπ) & β’ β = (+gβπΆ) & β’ (π β π‘ β ((πβ{π£}) β { 0 })) β β’ (π β π‘ β π) | ||
Theorem | hdmaprnlem4N 41028 | Part of proof of part 12 in [Baer] p. 49 line 19. (T* =) (Ft)* = Gs. (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β {π})) & β’ (π β π£ β π) & β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) & β’ (π β π’ β π) & β’ (π β Β¬ π’ β (πβ{π£})) & β’ π· = (BaseβπΆ) & β’ π = (0gβπΆ) & β’ 0 = (0gβπ) & β’ β = (+gβπΆ) & β’ (π β π‘ β ((πβ{π£}) β { 0 })) β β’ (π β (πβ(πβ{π‘})) = (πΏβ{π })) | ||
Theorem | hdmaprnlem6N 41029 | Part of proof of part 12 in [Baer] p. 49 line 18, G(u'+s) = G(u'+t). (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β {π})) & β’ (π β π£ β π) & β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) & β’ (π β π’ β π) & β’ (π β Β¬ π’ β (πβ{π£})) & β’ π· = (BaseβπΆ) & β’ π = (0gβπΆ) & β’ 0 = (0gβπ) & β’ β = (+gβπΆ) & β’ (π β π‘ β ((πβ{π£}) β { 0 })) & β’ + = (+gβπ) & β’ (π β (πΏβ{((πβπ’) β π )}) = (πβ(πβ{(π’ + π‘)}))) β β’ (π β (πΏβ{((πβπ’) β π )}) = (πΏβ{((πβπ’) β (πβπ‘))})) | ||
Theorem | hdmaprnlem7N 41030 | Part of proof of part 12 in [Baer] p. 49 line 19, s-St β G(u'+s) = P*. (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β {π})) & β’ (π β π£ β π) & β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) & β’ (π β π’ β π) & β’ (π β Β¬ π’ β (πβ{π£})) & β’ π· = (BaseβπΆ) & β’ π = (0gβπΆ) & β’ 0 = (0gβπ) & β’ β = (+gβπΆ) & β’ (π β π‘ β ((πβ{π£}) β { 0 })) & β’ + = (+gβπ) & β’ (π β (πΏβ{((πβπ’) β π )}) = (πβ(πβ{(π’ + π‘)}))) β β’ (π β (π (-gβπΆ)(πβπ‘)) β (πΏβ{((πβπ’) β π )})) | ||
Theorem | hdmaprnlem8N 41031 | Part of proof of part 12 in [Baer] p. 49 line 19, s-St β (Ft)* = T*. (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β {π})) & β’ (π β π£ β π) & β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) & β’ (π β π’ β π) & β’ (π β Β¬ π’ β (πβ{π£})) & β’ π· = (BaseβπΆ) & β’ π = (0gβπΆ) & β’ 0 = (0gβπ) & β’ β = (+gβπΆ) & β’ (π β π‘ β ((πβ{π£}) β { 0 })) & β’ + = (+gβπ) & β’ (π β (πΏβ{((πβπ’) β π )}) = (πβ(πβ{(π’ + π‘)}))) β β’ (π β (π (-gβπΆ)(πβπ‘)) β (πβ(πβ{π‘}))) | ||
Theorem | hdmaprnlem9N 41032 | Part of proof of part 12 in [Baer] p. 49 line 21, s=S(t). TODO: we seem to be going back and forth with mapd11 40814 and mapdcnv11N 40834. Use better hypotheses and/or theorems? (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β {π})) & β’ (π β π£ β π) & β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) & β’ (π β π’ β π) & β’ (π β Β¬ π’ β (πβ{π£})) & β’ π· = (BaseβπΆ) & β’ π = (0gβπΆ) & β’ 0 = (0gβπ) & β’ β = (+gβπΆ) & β’ (π β π‘ β ((πβ{π£}) β { 0 })) & β’ + = (+gβπ) & β’ (π β (πΏβ{((πβπ’) β π )}) = (πβ(πβ{(π’ + π‘)}))) β β’ (π β π = (πβπ‘)) | ||
Theorem | hdmaprnlem3eN 41033* | Lemma for hdmaprnN 41039. (Contributed by NM, 29-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β {π})) & β’ (π β π£ β π) & β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) & β’ (π β π’ β π) & β’ (π β Β¬ π’ β (πβ{π£})) & β’ π· = (BaseβπΆ) & β’ π = (0gβπΆ) & β’ 0 = (0gβπ) & β’ β = (+gβπΆ) & β’ + = (+gβπ) β β’ (π β βπ‘ β ((πβ{π£}) β { 0 })(πΏβ{((πβπ’) β π )}) = (πβ(πβ{(π’ + π‘)}))) | ||
Theorem | hdmaprnlem10N 41034* | Lemma for hdmaprnN 41039. Show π is in the range of π. (Contributed by NM, 29-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β {π})) & β’ (π β π£ β π) & β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) & β’ (π β π’ β π) & β’ (π β Β¬ π’ β (πβ{π£})) & β’ π· = (BaseβπΆ) & β’ π = (0gβπΆ) & β’ 0 = (0gβπ) & β’ β = (+gβπΆ) & β’ + = (+gβπ) β β’ (π β βπ‘ β π (πβπ‘) = π ) | ||
Theorem | hdmaprnlem11N 41035* | Lemma for hdmaprnN 41039. Show π is in the range of π. (Contributed by NM, 29-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β {π})) & β’ (π β π£ β π) & β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) & β’ (π β π’ β π) & β’ (π β Β¬ π’ β (πβ{π£})) & β’ π· = (BaseβπΆ) & β’ π = (0gβπΆ) & β’ 0 = (0gβπ) & β’ β = (+gβπΆ) & β’ + = (+gβπ) β β’ (π β π β ran π) | ||
Theorem | hdmaprnlem15N 41036* | Lemma for hdmaprnN 41039. Eliminate π’. (Contributed by NM, 30-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ 0 = (0gβπΆ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β { 0 })) & β’ (π β π£ β π) & β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) β β’ (π β π β ran π) | ||
Theorem | hdmaprnlem16N 41037 | Lemma for hdmaprnN 41039. Eliminate π£. (Contributed by NM, 30-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ 0 = (0gβπΆ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π· β { 0 })) β β’ (π β π β ran π) | ||
Theorem | hdmaprnlem17N 41038 | Lemma for hdmaprnN 41039. Include zero. (Contributed by NM, 30-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (LSpanβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ 0 = (0gβπΆ) & β’ πΏ = (LSpanβπΆ) & β’ π = ((mapdβπΎ)βπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π·) β β’ (π β π β ran π) | ||
Theorem | hdmaprnN 41039 | Part of proof of part 12 in [Baer] p. 49 line 21, As=B. (Contributed by NM, 30-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) β β’ (π β ran π = π·) | ||
Theorem | hdmapf1oN 41040 | Part 12 in [Baer] p. 49. The map from vectors to functionals with closed kernels maps one-to-one onto. Combined with hdmapadd 41018, this shows the map is an automorphism from the additive group of vectors to the additive group of functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) β β’ (π β π:πβ1-1-ontoβπ·) | ||
Theorem | hdmap14lem1a 41041 | Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ πΏ = (LSpanβπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β πΉ β π΅) & β’ 0 = (0gβπ ) & β’ (π β πΉ β 0 ) β β’ (π β (πΏβ{(πβπ)}) = (πΏβ{(πβ(πΉ Β· π))})) | ||
Theorem | hdmap14lem2a 41042* | Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include πΉ = 0 so it can be used in hdmap14lem10 41052. (Contributed by NM, 31-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ πΏ = (LSpanβπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β πΉ β π΅) β β’ (π β βπ β π΄ (πβ(πΉ Β· π)) = (π β (πβπ))) | ||
Theorem | hdmap14lem1 41043 | Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ π = (0gβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ πΏ = (LSpanβπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ π = (0gβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π β { 0 })) & β’ (π β πΉ β (π΅ β {π})) β β’ (π β (πΏβ{(πβπ)}) = (πΏβ{(πβ(πΉ Β· π))})) | ||
Theorem | hdmap14lem2N 41044* | Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include πΉ = π so it can be used in hdmap14lem10 41052. (Contributed by NM, 31-May-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ π = (0gβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ πΏ = (LSpanβπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ π = (0gβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π β { 0 })) & β’ (π β πΉ β (π΅ β {π})) β β’ (π β βπ β (π΄ β {π})(πβ(πΉ Β· π)) = (π β (πβπ))) | ||
Theorem | hdmap14lem3 41045* | Prior to part 14 in [Baer] p. 49, line 26. (Contributed by NM, 31-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ π = (0gβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ πΏ = (LSpanβπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ π = (0gβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π β { 0 })) & β’ (π β πΉ β (π΅ β {π})) β β’ (π β β!π β (π΄ β {π})(πβ(πΉ Β· π)) = (π β (πβπ))) | ||
Theorem | hdmap14lem4a 41046* | Simplify (π΄ β {π}) in hdmap14lem3 41045 to provide a slightly simpler definition later. (Contributed by NM, 31-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ π = (0gβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ πΏ = (LSpanβπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ π = (0gβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π β { 0 })) & β’ (π β πΉ β (π΅ β {π})) β β’ (π β (β!π β (π΄ β {π})(πβ(πΉ Β· π)) = (π β (πβπ)) β β!π β π΄ (πβ(πΉ Β· π)) = (π β (πβπ)))) | ||
Theorem | hdmap14lem4 41047* | Simplify (π΄ β {π}) in hdmap14lem3 41045 to provide a slightly simpler definition later. TODO: Use hdmap14lem4a 41046 if that one is also used directly elsewhere. Otherwise, merge hdmap14lem4a 41046 into this one. (Contributed by NM, 31-May-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ π = (0gβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ πΏ = (LSpanβπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ π = (0gβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π β { 0 })) & β’ (π β πΉ β (π΅ β {π})) β β’ (π β β!π β π΄ (πβ(πΉ Β· π)) = (π β (πβπ))) | ||
Theorem | hdmap14lem6 41048* | Case where πΉ is zero. (Contributed by NM, 1-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ π = (0gβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ πΏ = (LSpanβπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ π = (0gβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π β { 0 })) & β’ (π β πΉ = π) β β’ (π β β!π β π΄ (πβ(πΉ Β· π)) = (π β (πβπ))) | ||
Theorem | hdmap14lem7 41049* | Combine cases of πΉ. TODO: Can this be done at once in hdmap14lem3 41045, in order to get rid of hdmap14lem6 41048? Perhaps modify lspsneu 20882 to become β!π β πΎ instead of β!π β (πΎ β { 0 })? (Contributed by NM, 1-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π β { 0 })) & β’ (π β πΉ β π΅) β β’ (π β β!π β π΄ (πβ(πΉ Β· π)) = (π β (πβπ))) | ||
Theorem | hdmap14lem8 41050 | Part of proof of part 14 in [Baer] p. 49 lines 33-35. (Contributed by NM, 1-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = (+gβπΆ) & β’ β = ( Β·π βπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π β { 0 })) & β’ (π β π β (π β { 0 })) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΄) & β’ (π β πΌ β π΄) & β’ (π β (πβ(πΉ Β· π)) = (πΊ β (πβπ))) & β’ (π β (πβ(πΉ Β· π)) = (πΌ β (πβπ))) & β’ (π β (πβ{π}) β (πβ{π})) & β’ (π β π½ β π΄) & β’ (π β (πβ(πΉ Β· (π + π))) = (π½ β (πβ(π + π)))) β β’ (π β ((π½ β (πβπ)) β (π½ β (πβπ))) = ((πΊ β (πβπ)) β (πΌ β (πβπ)))) | ||
Theorem | hdmap14lem9 41051 | Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 1-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = (+gβπΆ) & β’ β = ( Β·π βπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π β { 0 })) & β’ (π β π β (π β { 0 })) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΄) & β’ (π β πΌ β π΄) & β’ (π β (πβ(πΉ Β· π)) = (πΊ β (πβπ))) & β’ (π β (πβ(πΉ Β· π)) = (πΌ β (πβπ))) & β’ (π β (πβ{π}) β (πβ{π})) & β’ (π β π½ β π΄) & β’ (π β (πβ(πΉ Β· (π + π))) = (π½ β (πβ(π + π)))) β β’ (π β πΊ = πΌ) | ||
Theorem | hdmap14lem10 41052 | Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 3-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = (+gβπΆ) & β’ β = ( Β·π βπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π β { 0 })) & β’ (π β π β (π β { 0 })) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΄) & β’ (π β πΌ β π΄) & β’ (π β (πβ(πΉ Β· π)) = (πΊ β (πβπ))) & β’ (π β (πβ(πΉ Β· π)) = (πΌ β (πβπ))) & β’ (π β (πβ{π}) β (πβ{π})) β β’ (π β πΊ = πΌ) | ||
Theorem | hdmap14lem11 41053 | Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 3-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (LSpanβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = (+gβπΆ) & β’ β = ( Β·π βπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π β { 0 })) & β’ (π β π β (π β { 0 })) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΄) & β’ (π β πΌ β π΄) & β’ (π β (πβ(πΉ Β· π)) = (πΊ β (πβπ))) & β’ (π β (πβ(πΉ Β· π)) = (πΌ β (πβπ))) β β’ (π β πΊ = πΌ) | ||
Theorem | hdmap14lem12 41054* | Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β πΉ β π΅) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ 0 = (0gβπ) & β’ (π β π β (π β { 0 })) & β’ (π β πΊ β π΄) β β’ (π β ((πβ(πΉ Β· π)) = (πΊ β (πβπ)) β βπ¦ β (π β { 0 })(πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)))) | ||
Theorem | hdmap14lem13 41055* | Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β πΉ β π΅) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ 0 = (0gβπ) & β’ (π β π β (π β { 0 })) & β’ (π β πΊ β π΄) β β’ (π β ((πβ(πΉ Β· π)) = (πΊ β (πβπ)) β βπ¦ β π (πβ(πΉ Β· π¦)) = (πΊ β (πβπ¦)))) | ||
Theorem | hdmap14lem14 41056* | Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 6-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β πΉ β π΅) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) β β’ (π β β!π β π΄ βπ₯ β π (πβ(πΉ Β· π₯)) = (π β (πβπ₯))) | ||
Theorem | hdmap14lem15 41057* | Part of proof of part 14 in [Baer] p. 50 line 3. Convert scalar base of dual to scalar base of vector space. (Contributed by NM, 6-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β πΉ β π΅) β β’ (π β β!π β π΅ βπ₯ β π (πβ(πΉ Β· π₯)) = (π β (πβπ₯))) | ||
Syntax | chg 41058 | Extend class notation with g-map. |
class HGMap | ||
Definition | df-hgmap 41059* | Define map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.) |
β’ HGMap = (π β V β¦ (π€ β (LHypβπ) β¦ {π β£ [((DVecHβπ)βπ€) / π’][(Baseβ(Scalarβπ’)) / π][((HDMapβπ)βπ€) / π]π β (π₯ β π β¦ (β©π¦ β π βπ£ β (Baseβπ’)(πβ(π₯( Β·π βπ’)π£)) = (π¦( Β·π β((LCDualβπ)βπ€))(πβπ£))))})) | ||
Theorem | hgmapffval 41060* | Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.) |
β’ π» = (LHypβπΎ) β β’ (πΎ β π β (HGMapβπΎ) = (π€ β π» β¦ {π β£ [((DVecHβπΎ)βπ€) / π’][(Baseβ(Scalarβπ’)) / π][((HDMapβπΎ)βπ€) / π]π β (π₯ β π β¦ (β©π¦ β π βπ£ β (Baseβπ’)(πβ(π₯( Β·π βπ’)π£)) = (π¦( Β·π β((LCDualβπΎ)βπ€))(πβπ£))))})) | ||
Theorem | hgmapfval 41061* | Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΌ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β π β§ π β π»)) β β’ (π β πΌ = (π₯ β π΅ β¦ (β©π¦ β π΅ βπ£ β π (πβ(π₯ Β· π£)) = (π¦ β (πβπ£))))) | ||
Theorem | hgmapval 41062* | Value of map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. Function sigma of scalar f in part 14 of [Baer] p. 50 line 4. TODO: variable names are inherited from older version. Maybe make more consistent with hdmap14lem15 41057. (Contributed by NM, 25-Mar-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΌ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β π β§ π β π»)) & β’ (π β π β π΅) β β’ (π β (πΌβπ) = (β©π¦ β π΅ βπ£ β π (πβ(π Β· π£)) = (π¦ β (πβπ£)))) | ||
Theorem | hgmapfnN 41063 | Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) β β’ (π β πΊ Fn π΅) | ||
Theorem | hgmapcl 41064 | Closure of scalar sigma map i.e. the map from the vector space scalar base to the dual space scalar base. (Contributed by NM, 6-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β πΉ β π΅) β β’ (π β (πΊβπΉ) β π΅) | ||
Theorem | hgmapdcl 41065 | Closure of the vector space to dual space scalar map, in the scalar sigma map. (Contributed by NM, 6-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β πΉ β π΅) β β’ (π β (πΊβπΉ) β π΄) | ||
Theorem | hgmapvs 41066 | Part 15 of [Baer] p. 50 line 6. Also line 15 in [Holland95] p. 14. (Contributed by NM, 6-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ β = ( Β·π βπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β πΉ β π΅) β β’ (π β (πβ(πΉ Β· π)) = ((πΊβπΉ) β (πβπ))) | ||
Theorem | hgmapval0 41067 | Value of the scalar sigma map at zero. (Contributed by NM, 12-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Scalarβπ) & β’ 0 = (0gβπ ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) β β’ (π β (πΊβ 0 ) = 0 ) | ||
Theorem | hgmapval1 41068 | Value of the scalar sigma map at one. (Contributed by NM, 12-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Scalarβπ) & β’ 1 = (1rβπ ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) β β’ (π β (πΊβ 1 ) = 1 ) | ||
Theorem | hgmapadd 41069 | Part 15 of [Baer] p. 50 line 13. (Contributed by NM, 6-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ + = (+gβπ ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (πΊβ(π + π)) = ((πΊβπ) + (πΊβπ))) | ||
Theorem | hgmapmul 41070 | Part 15 of [Baer] p. 50 line 16. The multiplication is reversed after converting to the dual space scalar to the vector space scalar. (Contributed by NM, 7-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Β· = (.rβπ ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (πΊβ(π Β· π)) = ((πΊβπ) Β· (πΊβπ))) | ||
Theorem | hgmaprnlem1N 41071 | Lemma for hgmaprnN 41076. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ β = ( Β·π βπΆ) & β’ π = (0gβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π§ β π΄) & β’ (π β π‘ β (π β { 0 })) & β’ (π β π β π) & β’ (π β (πβπ ) = (π§ β (πβπ‘))) & β’ (π β π β π΅) & β’ (π β π = (π Β· π‘)) β β’ (π β π§ β ran πΊ) | ||
Theorem | hgmaprnlem2N 41072 | Lemma for hgmaprnN 41076. Part 15 of [Baer] p. 50 line 20. We only require a subset relation, rather than equality, so that the case of zero π§ is taken care of automatically. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ β = ( Β·π βπΆ) & β’ π = (0gβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π§ β π΄) & β’ (π β π‘ β (π β { 0 })) & β’ (π β π β π) & β’ (π β (πβπ ) = (π§ β (πβπ‘))) & β’ π = ((mapdβπΎ)βπ) & β’ π = (LSpanβπ) & β’ πΏ = (LSpanβπΆ) β β’ (π β (πβ{π }) β (πβ{π‘})) | ||
Theorem | hgmaprnlem3N 41073* | Lemma for hgmaprnN 41076. Eliminate π. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ β = ( Β·π βπΆ) & β’ π = (0gβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π§ β π΄) & β’ (π β π‘ β (π β { 0 })) & β’ (π β π β π) & β’ (π β (πβπ ) = (π§ β (πβπ‘))) & β’ π = ((mapdβπΎ)βπ) & β’ π = (LSpanβπ) & β’ πΏ = (LSpanβπΆ) β β’ (π β π§ β ran πΊ) | ||
Theorem | hgmaprnlem4N 41074* | Lemma for hgmaprnN 41076. Eliminate π . (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ β = ( Β·π βπΆ) & β’ π = (0gβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π§ β π΄) & β’ (π β π‘ β (π β { 0 })) β β’ (π β π§ β ran πΊ) | ||
Theorem | hgmaprnlem5N 41075 | Lemma for hgmaprnN 41076. Eliminate π‘. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ πΆ = ((LCDualβπΎ)βπ) & β’ π· = (BaseβπΆ) & β’ π = (ScalarβπΆ) & β’ π΄ = (Baseβπ) & β’ β = ( Β·π βπΆ) & β’ π = (0gβπΆ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π§ β π΄) β β’ (π β π§ β ran πΊ) | ||
Theorem | hgmaprnN 41076 | Part of proof of part 16 in [Baer] p. 50 line 23, Fs=G, except that we use the original vector space scalars for the range. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) β β’ (π β ran πΊ = π΅) | ||
Theorem | hgmap11 41077 | The scalar sigma map is one-to-one. (Contributed by NM, 7-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β ((πΊβπ) = (πΊβπ) β π = π)) | ||
Theorem | hgmapf1oN 41078 | The scalar sigma map is a one-to-one onto function. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) β β’ (π β πΊ:π΅β1-1-ontoβπ΅) | ||
Theorem | hgmapeq0 41079 | The scalar sigma map is zero iff its argument is zero. (Contributed by NM, 12-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π΅) β β’ (π β ((πΊβπ) = 0 β π = 0 )) | ||
Theorem | hdmapipcl 41080 | The inner product (Hermitian form) (π, π) will be defined as ((πβπ)βπ). Show closure. (Contributed by NM, 7-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β ((πβπ)βπ) β π΅) | ||
Theorem | hdmapln1 41081 | Linearity property that will be used for inner product. TODO: try to combine hypotheses in hdmap*ln* series. (Contributed by NM, 7-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & ⒠⨣ = (+gβπ ) & β’ Γ = (.rβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π΄ β π΅) β β’ (π β ((πβπ)β((π΄ Β· π) + π)) = ((π΄ Γ ((πβπ)βπ)) ⨣ ((πβπ)βπ))) | ||
Theorem | hdmaplna1 41082 | Additive property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ π = (Scalarβπ) & ⒠⨣ = (+gβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β ((πβπ)β(π + π)) = (((πβπ)βπ) ⨣ ((πβπ)βπ))) | ||
Theorem | hdmaplns1 41083 | Subtraction property of first (inner product) argument. (Contributed by NM, 12-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (Scalarβπ) & β’ π = (-gβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β ((πβπ)β(π β π)) = (((πβπ)βπ)π((πβπ)βπ))) | ||
Theorem | hdmaplnm1 41084 | Multiplicative property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Γ = (.rβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π΄ β π΅) β β’ (π β ((πβπ)β(π΄ Β· π)) = (π΄ Γ ((πβπ)βπ))) | ||
Theorem | hdmaplna2 41085 | Additive property of second (inner product) argument. (Contributed by NM, 10-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ π = (Scalarβπ) & ⒠⨣ = (+gβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β ((πβ(π + π))βπ) = (((πβπ)βπ) ⨣ ((πβπ)βπ))) | ||
Theorem | hdmapglnm2 41086 | g-linear property of second (inner product) argument. Line 19 in [Holland95] p. 14. (Contributed by NM, 10-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Γ = (.rβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π΄ β π΅) β β’ (π β ((πβ(π΄ Β· π))βπ) = (((πβπ)βπ) Γ (πΊβπ΄))) | ||
Theorem | hdmapgln2 41087 | g-linear property that will be used for inner product. (Contributed by NM, 14-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & ⒠⨣ = (+gβπ ) & β’ Γ = (.rβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π΄ β π΅) β β’ (π β ((πβ((π΄ Β· π) + π))βπ) = ((((πβπ)βπ) Γ (πΊβπ΄)) ⨣ ((πβπ)βπ))) | ||
Theorem | hdmaplkr 41088 | Kernel of the vector to dual map. Line 16 in [Holland95] p. 14. TODO: eliminate πΉ hypothesis. (Contributed by NM, 9-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((ocHβπΎ)βπ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ πΉ = (LFnlβπ) & β’ π = (LKerβπ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) β β’ (π β (πβ(πβπ)) = (πβ{π})) | ||
Theorem | hdmapellkr 41089 | Membership in the kernel (as shown by hdmaplkr 41088) of the vector to dual map. Line 17 in [Holland95] p. 14. (Contributed by NM, 16-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((ocHβπΎ)βπ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (Scalarβπ) & β’ 0 = (0gβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β (((πβπ)βπ) = 0 β π β (πβ{π}))) | ||
Theorem | hdmapip0 41090 | Zero property that will be used for inner product. (Contributed by NM, 9-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ 0 = (0gβπ) & β’ π = (Scalarβπ) & β’ π = (0gβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) β β’ (π β (((πβπ)βπ) = π β π = 0 )) | ||
Theorem | hdmapip1 41091 | Construct a proportional vector π whose inner product with the original π equals one. (Contributed by NM, 13-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ 0 = (0gβπ) & β’ π = (Scalarβπ) & β’ 1 = (1rβπ ) & β’ π = (invrβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π β { 0 })) & β’ π = ((πβ((πβπ)βπ)) Β· π) β β’ (π β ((πβπ)βπ) = 1 ) | ||
Theorem | hdmapip0com 41092 | Commutation property of Baer's sigma map (Holland's A map). Line 20 of [Holland95] p. 14. Also part of Lemma 1 of [Baer] p. 110 line 7. (Contributed by NM, 9-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (Scalarβπ) & β’ 0 = (0gβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β (((πβπ)βπ) = 0 β ((πβπ)βπ) = 0 )) | ||
Theorem | hdmapinvlem1 41093 | Line 27 in [Baer] p. 110. We use πΆ for Baer's u. Our unit vector πΈ has the required properties for his w by hdmapevec2 41011. Our ((πβπΈ)βπΆ) means the inner product β¨πΆ, πΈβ© i.e. his f(u,w) (note argument reversal). (Contributed by NM, 11-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((ocHβπΎ)βπ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Β· = (.rβπ ) & β’ 0 = (0gβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β πΆ β (πβ{πΈ})) β β’ (π β ((πβπΈ)βπΆ) = 0 ) | ||
Theorem | hdmapinvlem2 41094 | Line 28 in [Baer] p. 110, 0 = f(w,u). (Contributed by NM, 11-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((ocHβπΎ)βπ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Β· = (.rβπ ) & β’ 0 = (0gβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β πΆ β (πβ{πΈ})) β β’ (π β ((πβπΆ)βπΈ) = 0 ) | ||
Theorem | hdmapinvlem3 41095 | Line 30 in [Baer] p. 110, f(sw + u, tw - v) = 0. (Contributed by NM, 12-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((ocHβπΎ)βπ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ β = (-gβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Γ = (.rβπ ) & β’ 0 = (0gβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β πΆ β (πβ{πΈ})) & β’ (π β π· β (πβ{πΈ})) & β’ (π β πΌ β π΅) & β’ (π β π½ β π΅) & β’ (π β (πΌ Γ (πΊβπ½)) = ((πβπ·)βπΆ)) β β’ (π β ((πβ((π½ Β· πΈ) β π·))β((πΌ Β· πΈ) + πΆ)) = 0 ) | ||
Theorem | hdmapinvlem4 41096 | Part 1.1 of Proposition 1 of [Baer] p. 110. We use πΆ, π·, πΌ, and π½ for Baer's u, v, s, and t. Our unit vector πΈ has the required properties for his w by hdmapevec2 41011. Our ((πβπ·)βπΆ) means his f(u,v) (note argument reversal). (Contributed by NM, 12-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((ocHβπΎ)βπ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ β = (-gβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Γ = (.rβπ ) & β’ 0 = (0gβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β πΆ β (πβ{πΈ})) & β’ (π β π· β (πβ{πΈ})) & β’ (π β πΌ β π΅) & β’ (π β π½ β π΅) & β’ (π β (πΌ Γ (πΊβπ½)) = ((πβπ·)βπΆ)) β β’ (π β (π½ Γ (πΊβπΌ)) = ((πβπΆ)βπ·)) | ||
Theorem | hdmapglem5 41097 | Part 1.2 in [Baer] p. 110 line 34, f(u,v) alpha = f(v,u). (Contributed by NM, 12-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((ocHβπΎ)βπ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & β’ β = (-gβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Γ = (.rβπ ) & β’ 0 = (0gβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β πΆ β (πβ{πΈ})) & β’ (π β π· β (πβ{πΈ})) & β’ (π β πΌ β π΅) & β’ (π β π½ β π΅) β β’ (π β (πΊβ((πβπ·)βπΆ)) = ((πβπΆ)βπ·)) | ||
Theorem | hgmapvvlem1 41098 | Involution property of scalar sigma map. Line 10 in [Baer] p. 111, t sigma squared = t. Our πΈ, πΆ, π·, π, π correspond to Baer's w, h, k, s, t. (Contributed by NM, 13-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((ocHβπΎ)βπ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Γ = (.rβπ ) & β’ 0 = (0gβπ ) & β’ 1 = (1rβπ ) & β’ π = (invrβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π΅ β { 0 })) & β’ (π β πΆ β (πβ{πΈ})) & β’ (π β π· β (πβ{πΈ})) & β’ (π β ((πβπ·)βπΆ) = 1 ) & β’ (π β π β (π΅ β { 0 })) & β’ (π β (π Γ (πΊβπ)) = 1 ) β β’ (π β (πΊβ(πΊβπ)) = π) | ||
Theorem | hgmapvvlem2 41099 | Lemma for hgmapvv 41101. Eliminate π (Baer's s). (Contributed by NM, 13-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((ocHβπΎ)βπ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Γ = (.rβπ ) & β’ 0 = (0gβπ ) & β’ 1 = (1rβπ ) & β’ π = (invrβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π΅ β { 0 })) & β’ (π β πΆ β (πβ{πΈ})) & β’ (π β π· β (πβ{πΈ})) & β’ (π β ((πβπ·)βπΆ) = 1 ) β β’ (π β (πΊβ(πΊβπ)) = π) | ||
Theorem | hgmapvvlem3 41100 | Lemma for hgmapvv 41101. Eliminate ((πβπ·)βπΆ) = 1 (Baer's f(h,k)=1). (Contributed by NM, 13-Jun-2015.) |
β’ π» = (LHypβπΎ) & β’ πΈ = β¨( I βΎ (BaseβπΎ)), ( I βΎ ((LTrnβπΎ)βπ))β© & β’ π = ((ocHβπΎ)βπ) & β’ π = ((DVecHβπΎ)βπ) & β’ π = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ π = (Scalarβπ) & β’ π΅ = (Baseβπ ) & β’ Γ = (.rβπ ) & β’ 0 = (0gβπ ) & β’ 1 = (1rβπ ) & β’ π = (invrβπ ) & β’ π = ((HDMapβπΎ)βπ) & β’ πΊ = ((HGMapβπΎ)βπ) & β’ (π β (πΎ β HL β§ π β π»)) & β’ (π β π β (π΅ β { 0 })) β β’ (π β (πΊβ(πΊβπ)) = π) |
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