| Metamath
Proof Explorer Theorem List (p. 419 of 494) | < Previous Next > | |
| Bad symbols? Try the
GIF version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | (1-30937) |
(30938-32460) |
(32461-49324) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | hdmap1val0 41801 | Value of preliminary map from vectors to functionals at zero. (Restated mapdhval0 41727.) (Contributed by NM, 17-May-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 0 〉) = 𝑄) | ||
| Theorem | hdmap1val2 41802* | Value of preliminary map from vectors to functionals in the closed kernel dual space, for nonzero 𝑌. (Contributed by NM, 16-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) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)})))) | ||
| Theorem | hdmap1eq 41803 | The defining equation for h(x,x',y)=y' in part (2) in [Baer] p. 45 line 24. (Contributed by NM, 16-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) & ⊢ 𝐿 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐷) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) ⇒ ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅𝐺)})))) | ||
| Theorem | hdmap1cbv 41804* | Frequently used lemma to change bound variables in 𝐿 hypothesis. (Contributed by NM, 15-May-2015.) |
| ⊢ 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)}))))) ⇒ ⊢ 𝐿 = (𝑦 ∈ V ↦ if((2nd ‘𝑦) = 0 , 𝑄, (℩𝑖 ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑦)})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑦)) − (2nd ‘𝑦))})) = (𝐽‘{((2nd ‘(1st ‘𝑦))𝑅𝑖)}))))) | ||
| Theorem | hdmap1valc 41805* | Connect the value of the preliminary map from vectors to functionals 𝐼 to the hypothesis 𝐿 used by earlier theorems. Note: the 𝑋 ∈ (𝑉 ∖ { 0 }) hypothesis could be the more general 𝑋 ∈ 𝑉 but the former will be easier to use. TODO: use the 𝐼 function directly in those theorems, so this theorem becomes unnecessary? TODO: The hdmap1cbv 41804 is probably unnecessary, but it would mean different $d's later on. (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 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ 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 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (𝐿‘〈𝑋, 𝐹, 𝑌〉)) | ||
| Theorem | hdmap1cl 41806 | Convert closure theorem mapdhcl 41729 to use HDMap1 function. (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 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷) | ||
| Theorem | hdmap1eq2 41807 | Convert mapdheq2 41731 to use HDMap1 function. (Contributed by NM, 16-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 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑋〉) = 𝐹) | ||
| Theorem | hdmap1eq4N 41808 | Convert mapdheq4 41734 to use HDMap1 function. (Contributed by NM, 17-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 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐵) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑍〉) = 𝐵) | ||
| Theorem | hdmap1l6lem1 41809 | Lemma for hdmap1l6 41823. Part (6) in [Baer] p. 47, lines 16-18. (Contributed by NM, 13-Apr-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 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅(𝐺 ✚ 𝐸))})) | ||
| Theorem | hdmap1l6lem2 41810 | Lemma for hdmap1l6 41823. Part (6) in [Baer] p. 47, lines 20-22. (Contributed by NM, 13-Apr-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 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{(𝐺 ✚ 𝐸)})) | ||
| Theorem | hdmap1l6a 41811 | Lemma for hdmap1l6 41823. Part (6) in [Baer] p. 47, case 1. (Contributed by NM, 23-Apr-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 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) | ||
| Theorem | hdmap1l6b0N 41812 | Lemmma for hdmap1l6 41823. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.) |
| ⊢ 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 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌, 𝑍})) = { 0 }) ⇒ ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | ||
| Theorem | hdmap1l6b 41813 | Lemmma for hdmap1l6 41823. (Contributed by NM, 24-Apr-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 = 0 ) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) | ||
| Theorem | hdmap1l6c 41814 | Lemmma for hdmap1l6 41823. (Contributed by NM, 24-Apr-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 = 0 ) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) | ||
| Theorem | hdmap1l6d 41815 | Lemmma for hdmap1l6 41823. (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 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))〉) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉))) | ||
| Theorem | hdmap1l6e 41816 | Lemmma for hdmap1l6 41823. Part (6) in [Baer] p. 47 line 38. (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 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, (𝑤 + 𝑌)〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) | ||
| Theorem | hdmap1l6f 41817 | Lemmma for hdmap1l6 41823. Part (6) in [Baer] p. 47 line 38. (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 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑤 + 𝑌)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑌〉))) | ||
| Theorem | hdmap1l6g 41818 | Lemmma for hdmap1l6 41823. Part (6) of [Baer] p. 47 line 39. (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 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉)) = (((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑌〉)) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) | ||
| Theorem | hdmap1l6h 41819 | Lemmma for hdmap1l6 41823. Part (6) of [Baer] p. 48 line 2. (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 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) | ||
| Theorem | hdmap1l6i 41820 | Lemmma for hdmap1l6 41823. Eliminate auxiliary vector 𝑤. (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 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) | ||
| Theorem | hdmap1l6j 41821 | Lemmma for hdmap1l6 41823. 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 41822 | Lemmma for hdmap1l6 41823. 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 41823 | 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 21128. (Convert mapdh6N 41749 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 41824* | Lemma for hdmap1eu 41826. TODO: combine with hdmap1eu 41826 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 41825* | Lemma for hdmap1euOLDN 41827. TODO: combine with hdmap1euOLDN 41827 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 41826* | Convert mapdh9a 41791 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 41827* | Convert mapdh9aOLDN 41792 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 41828* | 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 41829* | 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 41830* | 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 41114). (𝐽‘𝐸) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 41771 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 41832. If a separate auxiliary vector is known, hdmapval2 41834 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 41831 | 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 41832 | 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 41833* | Lemma for hdmapval2 41834. (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 41834 | 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 41773 through hdmaplem4 41776, 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 41835 | Value of map from vectors to functionals at zero. Note: we use dvh3dim 41448 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 41846 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 41836 | Lemma for hdmapevec 41837. TODO: combine with hdmapevec 41837 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 41837 | 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 41838 | 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 41839 | 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 41840 | 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 41841 | Lemma for hdmap10 41842. (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 41842 | 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 41843 | Lemma for hdmapadd 41845. (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 41844 | Lemma for hdmapadd 41845. (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 41845 | 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 41846 | 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 41847 | 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 41848 | 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 41849 | 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 41850 | 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 41851 | 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 41852 | 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 41853 | 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 41854 | Lemma for hdmaprnN 41866. 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 41855 | 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 41856 | 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 41857 | 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 41858 | 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 41859 | 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 41641 and mapdcnv11N 41661. 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 41860* | Lemma for hdmaprnN 41866. (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 41861* | Lemma for hdmaprnN 41866. 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 41862* | Lemma for hdmaprnN 41866. 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 41863* | Lemma for hdmaprnN 41866. 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 41864 | Lemma for hdmaprnN 41866. 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 41865 | Lemma for hdmaprnN 41866. 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 41866 | 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 41867 | Part 12 in [Baer] p. 49. The map from vectors to functionals with closed kernels maps one-to-one onto. Combined with hdmapadd 41845, 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 41868 | 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 41869* | Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 0 so it can be used in hdmap14lem10 41879. (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 41870 | 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 41871* | Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 𝑍 so it can be used in hdmap14lem10 41879. (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 41872* | 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 41873* | Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 41872 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 41874* | Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 41872 to provide a slightly simpler definition later. TODO: Use hdmap14lem4a 41873 if that one is also used directly elsewhere. Otherwise, merge hdmap14lem4a 41873 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 41875* | 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 41876* | Combine cases of 𝐹. TODO: Can this be done at once in hdmap14lem3 41872, in order to get rid of hdmap14lem6 41875? Perhaps modify lspsneu 21125 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 41877 | 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 41878 | 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 41879 | 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 41880 | 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 41881* | 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 41882* | 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 41883* | 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 41884* | 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 41885 | Extend class notation with g-map. |
| class HGMap | ||
| Definition | df-hgmap 41886* | 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 41887* | 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 41888* | 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 41889* | 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 41884. (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 41890 | 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 41891 | 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 41892 | 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 41893 | 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 41894 | 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 41895 | 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 41896 | 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 41897 | 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 41898 | Lemma for hgmaprnN 41903. (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 41899 | Lemma for hgmaprnN 41903. 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 41900* | Lemma for hgmaprnN 41903. 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) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |