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Type | Label | Description |
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Statement | ||
Theorem | hdmap1l6j 41801 | Lemmma for hdmap1l6 41803. Eliminate (𝑁 { Y } ) = ( N {𝑍}) hypothesis. (Contributed by NM, 1-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+g‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) | ||
Theorem | hdmap1l6k 41802 | Lemmma for hdmap1l6 41803. Eliminate nonzero vector requirement. (Contributed by NM, 1-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+g‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) | ||
Theorem | hdmap1l6 41803 | Part (6) of [Baer] p. 47 line 6. Note that we use ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}) which is equivalent to Baer's "Fx ∩ (Fy + Fz)" by lspdisjb 21145. (Convert mapdh6N 41729 to use the function HDMap1.) (Contributed by NM, 17-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+g‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) | ||
Theorem | hdmap1eulem 41804* | Lemma for hdmap1eu 41806. TODO: combine with hdmap1eu 41806 or at least share some hypotheses. (Contributed by NM, 15-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) & ⊢ 𝐿 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) ⇒ ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) | ||
Theorem | hdmap1eulemOLDN 41805* | Lemma for hdmap1euOLDN 41807. TODO: combine with hdmap1euOLDN 41807 or at least share some hypotheses. (Contributed by NM, 15-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) & ⊢ 𝐿 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) ⇒ ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) | ||
Theorem | hdmap1eu 41806* | Convert mapdh9a 41771 to use the HDMap1 notation. (Contributed by NM, 15-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) | ||
Theorem | hdmap1euOLDN 41807* | Convert mapdh9aOLDN 41772 to use the HDMap1 notation. (Contributed by NM, 15-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝑋, 𝐹, 𝑧〉), 𝑇〉))) | ||
Theorem | hdmapffval 41808* | Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝑋 → (HDMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑎 ∣ [〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))〉 / 𝑒][((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡 ∈ 𝑣 ↦ (℩𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧 ∈ 𝑣 (¬ 𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘〈𝑧, (𝑖‘〈𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧〉), 𝑡〉))))})) | ||
Theorem | hdmapfval 41809* | Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑆 = (𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑡〉))))) | ||
Theorem | hdmapval 41810* | 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 𝐸 to be 〈0, 1〉 (corresponding to vector u on p. 48 line 7) whose span is the lattice isomorphism map of the fiducial atom 𝑃 = ((oc‘𝐾)‘𝑊) (see dvheveccl 41094). (𝐽‘𝐸) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 41751 shows in Baer's notation (Fu)* = Gu'). Baer's independent vectors v and w on line 7 correspond to our 𝑧 that the ∀𝑧 ∈ 𝑉 ranges over. The middle term (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉) provides isolation to allow 𝐸 and 𝑇 to assume the same value without conflict. Closure is shown by hdmapcl 41812. If a separate auxiliary vector is known, hdmapval2 41814 provides a version without quantification. (Contributed by NM, 15-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑆‘𝑇) = (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)))) | ||
Theorem | hdmapfnN 41811 | Functionality of map from vectors to functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑆 Fn 𝑉) | ||
Theorem | hdmapcl 41812 | Closure of map from vectors to functionals with closed kernels. (Contributed by NM, 15-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑆‘𝑇) ∈ 𝐷) | ||
Theorem | hdmapval2lem 41813* | Lemma for hdmapval2 41814. (Contributed by NM, 15-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) ⇒ ⊢ (𝜑 → ((𝑆‘𝑇) = 𝐹 ↔ ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝐹 = (𝐼‘〈𝑧, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑧〉), 𝑇〉)))) | ||
Theorem | hdmapval2 41814 | 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 ≠ hypothesis? Consider hdmaplem1 41753 through hdmaplem4 41756, which would become obsolete. (Contributed by NM, 15-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇}))) ⇒ ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝑋, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑋〉), 𝑇〉)) | ||
Theorem | hdmapval0 41815 | Value of map from vectors to functionals at zero. Note: we use dvh3dim 41428 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 41826 could be derived from the other to shorten proof. (Contributed by NM, 17-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → (𝑆‘ 0 ) = 𝑄) | ||
Theorem | hdmapeveclem 41816 | Lemma for hdmapevec 41817. TODO: combine with hdmapevec 41817 if it shortens overall. (Contributed by NM, 16-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸}))) ⇒ ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸)) | ||
Theorem | hdmapevec 41817 | Value of map from vectors to functionals at the reference vector 𝐸. (Contributed by NM, 16-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸)) | ||
Theorem | hdmapevec2 41818 | The inner product of the reference vector 𝐸 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 ] ≠ 0 is satisfied. TODO: remove redundant hypothesis hdmapevec.j. (Contributed by NM, 1-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = 1 ) | ||
Theorem | hdmapval3lemN 41819 | Value of map from vectors to functionals at arguments not colinear with the reference vector 𝐸. (Contributed by NM, 17-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑁‘{𝑇}) ≠ (𝑁‘{𝐸})) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ {(0g‘𝑈)})) & ⊢ (𝜑 → 𝑥 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘{𝐸, 𝑇})) ⇒ ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑇〉)) | ||
Theorem | hdmapval3N 41820 | Value of map from vectors to functionals at arguments not colinear with the reference vector 𝐸. (Contributed by NM, 17-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑁‘{𝑇}) ≠ (𝑁‘{𝐸})) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑇〉)) | ||
Theorem | hdmap10lem 41821 | Lemma for hdmap10 41822. (Contributed by NM, 17-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)})) | ||
Theorem | hdmap10 41822 | Part 10 in [Baer] p. 48 line 33, (Ft)* = G(tS) in their notation (S = sigma). (Contributed by NM, 17-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)})) | ||
Theorem | hdmap11lem1 41823 | Lemma for hdmapadd 41825. (Contributed by NM, 26-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ✚ = (+g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → 𝑧 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) & ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸})) ⇒ ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) | ||
Theorem | hdmap11lem2 41824 | Lemma for hdmapadd 41825. (Contributed by NM, 26-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ✚ = (+g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) ⇒ ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) | ||
Theorem | hdmapadd 41825 | Part 11 in [Baer] p. 48 line 35, (a+b)S = aS+bS in their notation (S = sigma). (Contributed by NM, 22-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ✚ = (+g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) | ||
Theorem | hdmapeq0 41826 | Part of proof of part 12 in [Baer] p. 49 line 3. (Contributed by NM, 22-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑆‘𝑇) = 𝑄 ↔ 𝑇 = 0 )) | ||
Theorem | hdmapnzcl 41827 | Nonzero vector closure of map from vectors to functionals with closed kernels. (Contributed by NM, 27-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑆‘𝑇) ∈ (𝐷 ∖ {𝑄})) | ||
Theorem | hdmapneg 41828 | 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.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑀 = (invg‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐼 = (invg‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑆‘(𝑀‘𝑇)) = (𝐼‘(𝑆‘𝑇))) | ||
Theorem | hdmapsub 41829 | 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.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝑁 = (-g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑆‘(𝑋 − 𝑌)) = ((𝑆‘𝑋)𝑁(𝑆‘𝑌))) | ||
Theorem | hdmap11 41830 | 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.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑆‘𝑋) = (𝑆‘𝑌) ↔ 𝑋 = 𝑌)) | ||
Theorem | hdmaprnlem1N 41831 | Part of proof of part 12 in [Baer] p. 49 line 10, Gu' ≠ Gs. Our (𝑁‘{𝑣}) is Baer's T. (Contributed by NM, 26-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) & ⊢ (𝜑 → 𝑣 ∈ 𝑉) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) & ⊢ (𝜑 → 𝑢 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) ⇒ ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠})) | ||
Theorem | hdmaprnlem3N 41832 | Part of proof of part 12 in [Baer] p. 49 line 15, T ≠ P. Our (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) is Baer's P, where P* = G(u'+s). (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) & ⊢ (𝜑 → 𝑣 ∈ 𝑉) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) & ⊢ (𝜑 → 𝑢 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 0 = (0g‘𝑈) & ⊢ ✚ = (+g‘𝐶) ⇒ ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) | ||
Theorem | hdmaprnlem3uN 41833 | Part of proof of part 12 in [Baer] p. 49. (Contributed by NM, 29-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) & ⊢ (𝜑 → 𝑣 ∈ 𝑉) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) & ⊢ (𝜑 → 𝑢 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 0 = (0g‘𝑈) & ⊢ ✚ = (+g‘𝐶) ⇒ ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))) | ||
Theorem | hdmaprnlem4tN 41834 | Lemma for hdmaprnN 41846. 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.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) & ⊢ (𝜑 → 𝑣 ∈ 𝑉) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) & ⊢ (𝜑 → 𝑢 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 0 = (0g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ⇒ ⊢ (𝜑 → 𝑡 ∈ 𝑉) | ||
Theorem | hdmaprnlem4N 41835 | 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.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) & ⊢ (𝜑 → 𝑣 ∈ 𝑉) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) & ⊢ (𝜑 → 𝑢 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 0 = (0g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠})) | ||
Theorem | hdmaprnlem6N 41836 | 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.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) & ⊢ (𝜑 → 𝑣 ∈ 𝑉) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) & ⊢ (𝜑 → 𝑢 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 0 = (0g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)}))) ⇒ ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝐿‘{((𝑆‘𝑢) ✚ (𝑆‘𝑡))})) | ||
Theorem | hdmaprnlem7N 41837 | Part of proof of part 12 in [Baer] p. 49 line 19, s-St ∈ G(u'+s) = P*. (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) & ⊢ (𝜑 → 𝑣 ∈ 𝑉) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) & ⊢ (𝜑 → 𝑢 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 0 = (0g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)}))) ⇒ ⊢ (𝜑 → (𝑠(-g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) | ||
Theorem | hdmaprnlem8N 41838 | Part of proof of part 12 in [Baer] p. 49 line 19, s-St ∈ (Ft)* = T*. (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) & ⊢ (𝜑 → 𝑣 ∈ 𝑉) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) & ⊢ (𝜑 → 𝑢 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 0 = (0g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)}))) ⇒ ⊢ (𝜑 → (𝑠(-g‘𝐶)(𝑆‘𝑡)) ∈ (𝑀‘(𝑁‘{𝑡}))) | ||
Theorem | hdmaprnlem9N 41839 | 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 41621 and mapdcnv11N 41641. Use better hypotheses and/or theorems? (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) & ⊢ (𝜑 → 𝑣 ∈ 𝑉) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) & ⊢ (𝜑 → 𝑢 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 0 = (0g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)}))) ⇒ ⊢ (𝜑 → 𝑠 = (𝑆‘𝑡)) | ||
Theorem | hdmaprnlem3eN 41840* | Lemma for hdmaprnN 41846. (Contributed by NM, 29-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) & ⊢ (𝜑 → 𝑣 ∈ 𝑉) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) & ⊢ (𝜑 → 𝑢 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 0 = (0g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ + = (+g‘𝑈) ⇒ ⊢ (𝜑 → ∃𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)}))) | ||
Theorem | hdmaprnlem10N 41841* | Lemma for hdmaprnN 41846. Show 𝑠 is in the range of 𝑆. (Contributed by NM, 29-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) & ⊢ (𝜑 → 𝑣 ∈ 𝑉) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) & ⊢ (𝜑 → 𝑢 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 0 = (0g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ + = (+g‘𝑈) ⇒ ⊢ (𝜑 → ∃𝑡 ∈ 𝑉 (𝑆‘𝑡) = 𝑠) | ||
Theorem | hdmaprnlem11N 41842* | Lemma for hdmaprnN 41846. Show 𝑠 is in the range of 𝑆. (Contributed by NM, 29-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄})) & ⊢ (𝜑 → 𝑣 ∈ 𝑉) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) & ⊢ (𝜑 → 𝑢 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣})) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 0 = (0g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ + = (+g‘𝑈) ⇒ ⊢ (𝜑 → 𝑠 ∈ ran 𝑆) | ||
Theorem | hdmaprnlem15N 41843* | Lemma for hdmaprnN 41846. Eliminate 𝑢. (Contributed by NM, 30-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 0 = (0g‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ { 0 })) & ⊢ (𝜑 → 𝑣 ∈ 𝑉) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠})) ⇒ ⊢ (𝜑 → 𝑠 ∈ ran 𝑆) | ||
Theorem | hdmaprnlem16N 41844 | Lemma for hdmaprnN 41846. Eliminate 𝑣. (Contributed by NM, 30-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 0 = (0g‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ { 0 })) ⇒ ⊢ (𝜑 → 𝑠 ∈ ran 𝑆) | ||
Theorem | hdmaprnlem17N 41845 | Lemma for hdmaprnN 41846. Include zero. (Contributed by NM, 30-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 0 = (0g‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑠 ∈ 𝐷) ⇒ ⊢ (𝜑 → 𝑠 ∈ ran 𝑆) | ||
Theorem | hdmaprnN 41846 | Part of proof of part 12 in [Baer] p. 49 line 21, As=B. (Contributed by NM, 30-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ran 𝑆 = 𝐷) | ||
Theorem | hdmapf1oN 41847 | Part 12 in [Baer] p. 49. The map from vectors to functionals with closed kernels maps one-to-one onto. Combined with hdmapadd 41825, 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.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑆:𝑉–1-1-onto→𝐷) | ||
Theorem | hdmap14lem1a 41848 | Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐹 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑋)}) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))})) | ||
Theorem | hdmap14lem2a 41849* | Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 0 so it can be used in hdmap14lem10 41859. (Contributed by NM, 31-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))) | ||
Theorem | hdmap14lem1 41850 | Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑍 = (0g‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝑄 = (0g‘𝑃) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍})) ⇒ ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑋)}) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))})) | ||
Theorem | hdmap14lem2N 41851* | Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 𝑍 so it can be used in hdmap14lem10 41859. (Contributed by NM, 31-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑍 = (0g‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝑄 = (0g‘𝑃) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍})) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))) | ||
Theorem | hdmap14lem3 41852* | Prior to part 14 in [Baer] p. 49, line 26. (Contributed by NM, 31-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑍 = (0g‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝑄 = (0g‘𝑃) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍})) ⇒ ⊢ (𝜑 → ∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))) | ||
Theorem | hdmap14lem4a 41853* | Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 41852 to provide a slightly simpler definition later. (Contributed by NM, 31-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑍 = (0g‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝑄 = (0g‘𝑃) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍})) ⇒ ⊢ (𝜑 → (∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))) | ||
Theorem | hdmap14lem4 41854* | Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 41852 to provide a slightly simpler definition later. TODO: Use hdmap14lem4a 41853 if that one is also used directly elsewhere. Otherwise, merge hdmap14lem4a 41853 into this one. (Contributed by NM, 31-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑍 = (0g‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝑄 = (0g‘𝑃) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍})) ⇒ ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))) | ||
Theorem | hdmap14lem6 41855* | Case where 𝐹 is zero. (Contributed by NM, 1-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑍 = (0g‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝑄 = (0g‘𝑃) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 = 𝑍) ⇒ ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))) | ||
Theorem | hdmap14lem7 41856* | Combine cases of 𝐹. TODO: Can this be done at once in hdmap14lem3 41852, in order to get rid of hdmap14lem6 41855? Perhaps modify lspsneu 21142 to become ∃!𝑘 ∈ 𝐾 instead of ∃!𝑘 ∈ (𝐾 ∖ { 0 })? (Contributed by NM, 1-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))) | ||
Theorem | hdmap14lem8 41857 | Part of proof of part 14 in [Baer] p. 49 lines 33-35. (Contributed by NM, 1-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ✚ = (+g‘𝐶) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐴) & ⊢ (𝜑 → 𝐼 ∈ 𝐴) & ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) & ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌))) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → 𝐽 ∈ 𝐴) & ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 ∙ (𝑆‘(𝑋 + 𝑌)))) ⇒ ⊢ (𝜑 → ((𝐽 ∙ (𝑆‘𝑋)) ✚ (𝐽 ∙ (𝑆‘𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌)))) | ||
Theorem | hdmap14lem9 41858 | Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 1-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ✚ = (+g‘𝐶) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐴) & ⊢ (𝜑 → 𝐼 ∈ 𝐴) & ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) & ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌))) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → 𝐽 ∈ 𝐴) & ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 ∙ (𝑆‘(𝑋 + 𝑌)))) ⇒ ⊢ (𝜑 → 𝐺 = 𝐼) | ||
Theorem | hdmap14lem10 41859 | Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 3-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ✚ = (+g‘𝐶) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐴) & ⊢ (𝜑 → 𝐼 ∈ 𝐴) & ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) & ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌))) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → 𝐺 = 𝐼) | ||
Theorem | hdmap14lem11 41860 | Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 3-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ✚ = (+g‘𝐶) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐴) & ⊢ (𝜑 → 𝐼 ∈ 𝐴) & ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) & ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌))) ⇒ ⊢ (𝜑 → 𝐺 = 𝐼) | ||
Theorem | hdmap14lem12 41861* | Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐴) ⇒ ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) | ||
Theorem | hdmap14lem13 41862* | Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐴) ⇒ ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ 𝑉 (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦)))) | ||
Theorem | hdmap14lem14 41863* | Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 6-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) ⇒ ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥))) | ||
Theorem | hdmap14lem15 41864* | 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.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃!𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥))) | ||
Syntax | chg 41865 | Extend class notation with g-map. |
class HGMap | ||
Definition | df-hgmap 41866* | 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.) |
⊢ HGMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎 ∣ [((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠 ‘𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝑘)‘𝑤))(𝑚‘𝑣))))})) | ||
Theorem | hgmapffval 41867* | 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.) |
⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝑋 → (HGMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑎 ∣ [((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠 ‘𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣))))})) | ||
Theorem | hgmapfval 41868* | 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.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) | ||
Theorem | hgmapval 41869* | 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 41864. (Contributed by NM, 25-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼‘𝑋) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) | ||
Theorem | hgmapfnN 41870 | Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝐺 Fn 𝐵) | ||
Theorem | hgmapcl 41871 | 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.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐵) | ||
Theorem | hgmapdcl 41872 | Closure of the vector space to dual space scalar map, in the scalar sigma map. (Contributed by NM, 6-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝑄 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑄) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐴) | ||
Theorem | hgmapvs 41873 | Part 15 of [Baer] p. 50 line 6. Also line 15 in [Holland95] p. 14. (Contributed by NM, 6-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋))) | ||
Theorem | hgmapval0 41874 | Value of the scalar sigma map at zero. (Contributed by NM, 12-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → (𝐺‘ 0 ) = 0 ) | ||
Theorem | hgmapval1 41875 | Value of the scalar sigma map at one. (Contributed by NM, 12-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → (𝐺‘ 1 ) = 1 ) | ||
Theorem | hgmapadd 41876 | Part 15 of [Baer] p. 50 line 13. (Contributed by NM, 6-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺‘(𝑋 + 𝑌)) = ((𝐺‘𝑋) + (𝐺‘𝑌))) | ||
Theorem | hgmapmul 41877 | 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.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺‘(𝑋 · 𝑌)) = ((𝐺‘𝑌) · (𝐺‘𝑋))) | ||
Theorem | hgmaprnlem1N 41878 | Lemma for hgmaprnN 41883. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑧 ∈ 𝐴) & ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑠 ∈ 𝑉) & ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) & ⊢ (𝜑 → 𝑘 ∈ 𝐵) & ⊢ (𝜑 → 𝑠 = (𝑘 · 𝑡)) ⇒ ⊢ (𝜑 → 𝑧 ∈ ran 𝐺) | ||
Theorem | hgmaprnlem2N 41879 | Lemma for hgmaprnN 41883. Part 15 of [Baer] p. 50 line 20. We only require a subset relation, rather than equality, so that the case of zero 𝑧 is taken care of automatically. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑧 ∈ 𝐴) & ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑠 ∈ 𝑉) & ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐿 = (LSpan‘𝐶) ⇒ ⊢ (𝜑 → (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡})) | ||
Theorem | hgmaprnlem3N 41880* | Lemma for hgmaprnN 41883. Eliminate 𝑘. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑧 ∈ 𝐴) & ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑠 ∈ 𝑉) & ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐿 = (LSpan‘𝐶) ⇒ ⊢ (𝜑 → 𝑧 ∈ ran 𝐺) | ||
Theorem | hgmaprnlem4N 41881* | Lemma for hgmaprnN 41883. Eliminate 𝑠. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑧 ∈ 𝐴) & ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → 𝑧 ∈ ran 𝐺) | ||
Theorem | hgmaprnlem5N 41882 | Lemma for hgmaprnN 41883. Eliminate 𝑡. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑧 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝑧 ∈ ran 𝐺) | ||
Theorem | hgmaprnN 41883 | 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.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ran 𝐺 = 𝐵) | ||
Theorem | hgmap11 41884 | The scalar sigma map is one-to-one. (Contributed by NM, 7-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐺‘𝑋) = (𝐺‘𝑌) ↔ 𝑋 = 𝑌)) | ||
Theorem | hgmapf1oN 41885 | The scalar sigma map is a one-to-one onto function. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐵) | ||
Theorem | hgmapeq0 41886 | The scalar sigma map is zero iff its argument is zero. (Contributed by NM, 12-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐺‘𝑋) = 0 ↔ 𝑋 = 0 )) | ||
Theorem | hdmapipcl 41887 | The inner product (Hermitian form) (𝑋, 𝑌) will be defined as ((𝑆‘𝑌)‘𝑋). Show closure. (Contributed by NM, 7-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑆‘𝑌)‘𝑋) ∈ 𝐵) | ||
Theorem | hdmapln1 41888 | Linearity property that will be used for inner product. TODO: try to combine hypotheses in hdmap*ln* series. (Contributed by NM, 7-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑆‘𝑍)‘((𝐴 · 𝑋) + 𝑌)) = ((𝐴 × ((𝑆‘𝑍)‘𝑋)) ⨣ ((𝑆‘𝑍)‘𝑌))) | ||
Theorem | hdmaplna1 41889 | Additive property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑆‘𝑍)‘(𝑋 + 𝑌)) = (((𝑆‘𝑍)‘𝑋) ⨣ ((𝑆‘𝑍)‘𝑌))) | ||
Theorem | hdmaplns1 41890 | Subtraction property of first (inner product) argument. (Contributed by NM, 12-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝑁 = (-g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑆‘𝑍)‘(𝑋 − 𝑌)) = (((𝑆‘𝑍)‘𝑋)𝑁((𝑆‘𝑍)‘𝑌))) | ||
Theorem | hdmaplnm1 41891 | Multiplicative property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑆‘𝑌)‘(𝐴 · 𝑋)) = (𝐴 × ((𝑆‘𝑌)‘𝑋))) | ||
Theorem | hdmaplna2 41892 | Additive property of second (inner product) argument. (Contributed by NM, 10-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑆‘(𝑌 + 𝑍))‘𝑋) = (((𝑆‘𝑌)‘𝑋) ⨣ ((𝑆‘𝑍)‘𝑋))) | ||
Theorem | hdmapglnm2 41893 | g-linear property of second (inner product) argument. Line 19 in [Holland95] p. 14. (Contributed by NM, 10-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑆‘(𝐴 · 𝑌))‘𝑋) = (((𝑆‘𝑌)‘𝑋) × (𝐺‘𝐴))) | ||
Theorem | hdmapgln2 41894 | g-linear property that will be used for inner product. (Contributed by NM, 14-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑆‘((𝐴 · 𝑌) + 𝑍))‘𝑋) = ((((𝑆‘𝑌)‘𝑋) × (𝐺‘𝐴)) ⨣ ((𝑆‘𝑍)‘𝑋))) | ||
Theorem | hdmaplkr 41895 | Kernel of the vector to dual map. Line 16 in [Holland95] p. 14. TODO: eliminate 𝐹 hypothesis. (Contributed by NM, 9-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝑌 = (LKer‘𝑈) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑌‘(𝑆‘𝑋)) = (𝑂‘{𝑋})) | ||
Theorem | hdmapellkr 41896 | Membership in the kernel (as shown by hdmaplkr 41895) of the vector to dual map. Line 17 in [Holland95] p. 14. (Contributed by NM, 16-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑌) = 0 ↔ 𝑌 ∈ (𝑂‘{𝑋}))) | ||
Theorem | hdmapip0 41897 | Zero property that will be used for inner product. (Contributed by NM, 9-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝑍 = (0g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 ↔ 𝑋 = 0 )) | ||
Theorem | hdmapip1 41898 | Construct a proportional vector 𝑌 whose inner product with the original 𝑋 equals one. (Contributed by NM, 13-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (invr‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ 𝑌 = ((𝑁‘((𝑆‘𝑋)‘𝑋)) · 𝑋) ⇒ ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 1 ) | ||
Theorem | hdmapip0com 41899 | 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.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑌) = 0 ↔ ((𝑆‘𝑌)‘𝑋) = 0 )) | ||
Theorem | hdmapinvlem1 41900 | Line 27 in [Baer] p. 110. We use 𝐶 for Baer's u. Our unit vector 𝐸 has the required properties for his w by hdmapevec2 41818. Our ((𝑆‘𝐸)‘𝐶) means the inner product 〈𝐶, 𝐸〉 i.e. his f(u,w) (note argument reversal). (Contributed by NM, 11-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) ⇒ ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐶) = 0 ) |
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