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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | mapdpglem24 42001* | Lemma for mapdpg 42003. Existence part - consolidate hypotheses in mapdpglem23 41991. (Contributed by NM, 21-Mar-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) ⇒ ⊢ (𝜑 → ∃ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))) | ||
| Theorem | mapdpglem32 42002* | Lemma for mapdpg 42003. Uniqueness part - consolidate hypotheses in mapdpglem31 42000. (Contributed by NM, 23-Mar-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) ⇒ ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ℎ = 𝑖) | ||
| Theorem | mapdpg 42003* | Part 1 of proof of the first fundamental theorem of projective geometry. Part (1) in [Baer] p. 44. Our notation corresponds to Baer's as follows: 𝑀 for *, 𝑁‘{} for F(), 𝐽‘{} for G(), 𝑋 for x, 𝐺 for x', 𝑌 for y, ℎ for y'. TODO: Rename variables per mapdhval 42021. (Contributed by NM, 22-Mar-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) ⇒ ⊢ (𝜑 → ∃!ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))) | ||
| Theorem | baerlem3lem1 42004 | Lemma for baerlem3 42010. (Contributed by NM, 9-Apr-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝐿 = (-g‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) & ⊢ (𝜑 → 𝑎 ∈ 𝐵) & ⊢ (𝜑 → 𝑏 ∈ 𝐵) & ⊢ (𝜑 → 𝑑 ∈ 𝐵) & ⊢ (𝜑 → 𝑒 ∈ 𝐵) & ⊢ (𝜑 → 𝑗 = ((𝑎 · 𝑌) + (𝑏 · 𝑍))) & ⊢ (𝜑 → 𝑗 = ((𝑑 · (𝑋 − 𝑌)) + (𝑒 · (𝑋 − 𝑍)))) ⇒ ⊢ (𝜑 → 𝑗 = (𝑎 · (𝑌 − 𝑍))) | ||
| Theorem | baerlem5alem1 42005 | Lemma for baerlem5a 42011. (Contributed by NM, 13-Apr-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝐿 = (-g‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) & ⊢ (𝜑 → 𝑎 ∈ 𝐵) & ⊢ (𝜑 → 𝑏 ∈ 𝐵) & ⊢ (𝜑 → 𝑑 ∈ 𝐵) & ⊢ (𝜑 → 𝑒 ∈ 𝐵) & ⊢ (𝜑 → 𝑗 = ((𝑎 · (𝑋 − 𝑌)) + (𝑏 · 𝑍))) & ⊢ (𝜑 → 𝑗 = ((𝑑 · (𝑋 − 𝑍)) + (𝑒 · 𝑌))) ⇒ ⊢ (𝜑 → 𝑗 = (𝑎 · (𝑋 − (𝑌 + 𝑍)))) | ||
| Theorem | baerlem5blem1 42006 | Lemma for baerlem5b 42012. (Contributed by NM, 9-Apr-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝐿 = (-g‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) & ⊢ (𝜑 → 𝑎 ∈ 𝐵) & ⊢ (𝜑 → 𝑏 ∈ 𝐵) & ⊢ (𝜑 → 𝑑 ∈ 𝐵) & ⊢ (𝜑 → 𝑒 ∈ 𝐵) & ⊢ (𝜑 → 𝑗 = ((𝑎 · 𝑌) + (𝑏 · 𝑍))) & ⊢ (𝜑 → 𝑗 = ((𝑑 · (𝑋 − (𝑌 + 𝑍))) + (𝑒 · 𝑋))) ⇒ ⊢ (𝜑 → 𝑗 = ((𝐼‘𝑑) · (𝑌 + 𝑍))) | ||
| Theorem | baerlem3lem2 42007 | Lemma for baerlem3 42010. (Contributed by NM, 9-Apr-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝐿 = (-g‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑌 − 𝑍)}) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{(𝑋 − 𝑍)})))) | ||
| Theorem | baerlem5alem2 42008 | Lemma for baerlem5a 42011. (Contributed by NM, 9-Apr-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝐿 = (-g‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)}) ⊕ (𝑁‘{𝑌})))) | ||
| Theorem | baerlem5blem2 42009 | Lemma for baerlem5b 42012. (Contributed by NM, 13-Apr-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝐿 = (-g‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 + 𝑍))}) ⊕ (𝑁‘{𝑋})))) | ||
| Theorem | baerlem3 42010 | An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Part (3) in [Baer] p. 45. TODO fix ref. (Contributed by NM, 9-Apr-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑌 − 𝑍)}) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{(𝑋 − 𝑍)})))) | ||
| Theorem | baerlem5a 42011 | An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. First equation of part (5) in [Baer] p. 46. (Contributed by NM, 10-Apr-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)}) ⊕ (𝑁‘{𝑌})))) | ||
| Theorem | baerlem5b 42012 | An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Second equation of part (5) in [Baer] p. 46. (Contributed by NM, 13-Apr-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 + 𝑍))}) ⊕ (𝑁‘{𝑋})))) | ||
| Theorem | baerlem5amN 42013 | An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Subtraction version of first equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 42015 is used. (Contributed by NM, 24-May-2015.) (New usage is discouraged.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 − 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 + 𝑍)}) ⊕ (𝑁‘{𝑌})))) | ||
| Theorem | baerlem5bmN 42014 | An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Subtraction version of second equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 42015 is used. (Contributed by NM, 24-May-2015.) (New usage is discouraged.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑌 − 𝑍)}) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 − 𝑍))}) ⊕ (𝑁‘{𝑋})))) | ||
| Theorem | baerlem5abmN 42015 | An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Subtraction versions of first and second equations of part (5) in [Baer] p. 46, conjoined to share commonality in their proofs. TODO: Delete if not needed. (Contributed by NM, 24-May-2015.) (New usage is discouraged.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) ⇒ ⊢ (𝜑 → ((𝑁‘{(𝑋 − (𝑌 − 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 + 𝑍)}) ⊕ (𝑁‘{𝑌}))) ∧ (𝑁‘{(𝑌 − 𝑍)}) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 − 𝑍))}) ⊕ (𝑁‘{𝑋}))))) | ||
| Theorem | mapdindp0 42016 | Vector independence lemma. (Contributed by NM, 29-Apr-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) & ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌})) | ||
| Theorem | mapdindp1 42017 | Vector independence lemma. (Contributed by NM, 1-May-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) | ||
| Theorem | mapdindp2 42018 | Vector independence lemma. (Contributed by NM, 1-May-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)})) | ||
| Theorem | mapdindp3 42019 | Vector independence lemma. (Contributed by NM, 29-Apr-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)})) | ||
| Theorem | mapdindp4 42020 | Vector independence lemma. (Contributed by NM, 29-Apr-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)})) | ||
| Theorem | mapdhval 42021* | Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐸) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))) | ||
| Theorem | mapdhval0 42022* | Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄) | ||
| Theorem | mapdhval2 42023* | Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))) | ||
| Theorem | mapdhcl 42024* | Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷) | ||
| Theorem | mapdheq 42025* | Lemmma for ~? mapdh . The defining equation for h(x,x',y)=y' in part (2) in [Baer] p. 45 line 24. (Contributed by NM, 4-Apr-2015.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐷) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})))) | ||
| Theorem | mapdheq2 42026* | Lemmma for ~? mapdh . One direction of part (2) in [Baer] p. 45. (Contributed by NM, 4-Apr-2015.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐷) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 → (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹)) | ||
| Theorem | mapdheq2biN 42027* | Lemmma for ~? mapdh . Part (2) in [Baer] p. 45. The bidirectional version of mapdheq2 42026 seems to require an additional hypothesis not mentioned in Baer. TODO fix ref. TODO: We probably don't need this; delete if never used. (Contributed by NM, 4-Apr-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐷) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})) ⇒ ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹)) | ||
| Theorem | mapdheq4lem 42028* | Lemma for mapdheq4 42029. Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (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)})) = (𝐽‘{(𝐺𝑅𝐸)})) | ||
| Theorem | mapdheq4 42029* | Lemma for ~? mapdh . Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (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‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐸) | ||
| Theorem | mapdh6lem1N 42030* | Lemma for mapdh6N 42044. Part (6) in [Baer] p. 47, lines 16-18. (Contributed by NM, 13-Apr-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (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))})) = (𝐽‘{(𝐹𝑅(𝐺 ✚ 𝐸))})) | ||
| Theorem | mapdh6lem2N 42031* | Lemma for mapdh6N 42044. Part (6) in [Baer] p. 47, lines 20-22. (Contributed by NM, 13-Apr-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (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)})) = (𝐽‘{(𝐺 ✚ 𝐸)})) | ||
| Theorem | mapdh6aN 42032* | Lemma for mapdh6N 42044. Part (6) in [Baer] p. 47, case 1. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (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 | mapdh6b0N 42033* | Lemmma for mapdh6N 42044. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌, 𝑍})) = { 0 }) ⇒ ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | ||
| Theorem | mapdh6bN 42034* | Lemmma for mapdh6N 42044. (Contributed by NM, 24-Apr-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (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 | mapdh6cN 42035* | Lemmma for mapdh6N 42044. (Contributed by NM, 24-Apr-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (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 | mapdh6dN 42036* | Lemmma for mapdh6N 42044. (Contributed by NM, 1-May-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (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 | mapdh6eN 42037* | Lemmma for mapdh6N 42044. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (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 | mapdh6fN 42038* | Lemmma for mapdh6N 42044. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (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 | mapdh6gN 42039* | Lemmma for mapdh6N 42044. Part (6) of [Baer] p. 47 line 39. (Contributed by NM, 1-May-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (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 | mapdh6hN 42040* | Lemmma for mapdh6N 42044. Part (6) of [Baer] p. 48 line 2. (Contributed by NM, 1-May-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (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 | mapdh6iN 42041* | Lemmma for mapdh6N 42044. Eliminate auxiliary vector 𝑤. (Contributed by NM, 1-May-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (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 | mapdh6jN 42042* | Lemmma for mapdh6N 42044. Eliminate (𝑁‘{𝑌}) = (𝑁‘{𝑍}) hypothesis. (Contributed by NM, 1-May-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (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 | mapdh6kN 42043* | Lemmma for mapdh6N 42044. Eliminate nonzero vector requirement. (Contributed by NM, 1-May-2015.) (New usage is discouraged.) |
| ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (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)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (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 | mapdh6N 42044* | 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 21085. TODO: If disjoint variable conditions with 𝐼 and 𝜑 become a problem later, use cbv* theorems on 𝐼 variables here to get rid of them. Maybe reorder hypotheses in lemmas to the more consistent order of this theorem, so they can be shared with this theorem. TODO: may be deleted (with its lemmas), if not needed, in view of hdmap1l6 42118. (Contributed by NM, 1-May-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) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (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 → (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})) = (𝐽‘{𝐹})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩))) | ||
| Theorem | mapdh7eN 42045* | Part (7) of [Baer] p. 48 line 10 (5 of 6 cases). (Note: 1 of 6 and 2 of 6 are hypotheses a and b.) (Contributed by NM, 2-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣})) & ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑤⟩) = 𝐸) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑢⟩) = 𝐹) | ||
| Theorem | mapdh7cN 42046* | Part (7) of [Baer] p. 48 line 10 (3 of 6 cases). (Contributed by NM, 2-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣})) & ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑣⟩) = 𝐺) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑣, 𝐺, 𝑢⟩) = 𝐹) | ||
| Theorem | mapdh7dN 42047* | Part (7) of [Baer] p. 48 line 10 (4 of 6 cases). (Contributed by NM, 2-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣})) & ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑣⟩) = 𝐺) & ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑤⟩) = 𝐸) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑣, 𝐺, 𝑤⟩) = 𝐸) | ||
| Theorem | mapdh7fN 42048* | Part (7) of [Baer] p. 48 line 10 (6 of 6 cases). (Contributed by NM, 2-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣})) & ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑣⟩) = 𝐺) & ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑤⟩) = 𝐸) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑣⟩) = 𝐺) | ||
| Theorem | mapdh75e 42049* | Part (7) of [Baer] p. 48 line 10 (5 of 6 cases). 𝑋, 𝑌, 𝑍 are Baer's u, v, w. (Note: Cases 1 of 6 and 2 of 6 are hypotheses mapdh75b here and mapdh75a in mapdh75cN 42050.) (Contributed by NM, 2-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑍, 𝐸, 𝑋⟩) = 𝐹) | ||
| Theorem | mapdh75cN 42050* | Part (7) of [Baer] p. 48 line 10 (3 of 6 cases). (Contributed by NM, 2-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹) | ||
| Theorem | mapdh75d 42051* | Part (7) of [Baer] p. 48 line 10 (4 of 6 cases). (Contributed by NM, 2-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐸) | ||
| Theorem | mapdh75fN 42052* | Part (7) of [Baer] p. 48 line 10 (6 of 6 cases). (Contributed by NM, 2-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑍, 𝐸, 𝑌⟩) = 𝐺) | ||
| Syntax | chvm 42053 | Extend class notation with vector to dual map. |
| class HVMap | ||
| Definition | df-hvmap 42054* | Extend class notation with a map from each nonzero vector 𝑥 to a unique nonzero functional in the closed kernel dual space. (We could extend it to include the zero vector, but that is unnecessary for our purposes.) TODO: This pattern is used several times earlier, e.g., lcf1o 41848, dochfl1 41773- should we update those to use this definition? (Contributed by NM, 23-Mar-2015.) |
| ⊢ HVMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ ((Base‘((DVecH‘𝑘)‘𝑤)) ∖ {(0g‘((DVecH‘𝑘)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝑘)‘𝑤))𝑥))))))) | ||
| Theorem | hvmapffval 42055* | Map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝑋 → (HVMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))))) | ||
| Theorem | hvmapfval 42056* | Map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑆 = (Scalar‘𝑈) & ⊢ 𝑅 = (Base‘𝑆) & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))))) | ||
| Theorem | hvmapval 42057* | Value of map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑆 = (Scalar‘𝑈) & ⊢ 𝑅 = (Base‘𝑆) & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑀‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋))))) | ||
| Theorem | hvmapvalvalN 42058* | Value of value of map (i.e. functional value) from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.) (New usage is discouraged.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑆 = (Scalar‘𝑈) & ⊢ 𝑅 = (Base‘𝑆) & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑀‘𝑋)‘𝑌) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋)))) | ||
| Theorem | hvmapidN 42059 | The value of the vector to functional map, at the vector, is one. (Contributed by NM, 23-Mar-2015.) (New usage is discouraged.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑆 = (Scalar‘𝑈) & ⊢ 1 = (1r‘𝑆) & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → ((𝑀‘𝑋)‘𝑋) = 1 ) | ||
| Theorem | hvmap1o 42060* | The vector to functional map provides a bijection from nonzero vectors 𝑉 to nonzero functionals with closed kernels 𝐶. (Contributed by NM, 27-Mar-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝐷 = (LDual‘𝑈) & ⊢ 𝑄 = (0g‘𝐷) & ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)} & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄})) | ||
| Theorem | hvmapclN 42061* | Closure of the vector to functional map. (Contributed by NM, 27-Mar-2015.) (New usage is discouraged.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝐷 = (LDual‘𝑈) & ⊢ 𝑄 = (0g‘𝐷) & ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)} & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑀‘𝑋) ∈ (𝐶 ∖ {𝑄})) | ||
| Theorem | hvmap1o2 42062 | The vector to functional map provides a bijection from nonzero vectors 𝑉 to nonzero functionals with closed kernels 𝐶. (Contributed by NM, 27-Mar-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑂 = (0g‘𝐶) & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐹 ∖ {𝑂})) | ||
| Theorem | hvmapcl2 42063 | Closure of the vector to functional map. (Contributed by NM, 27-Mar-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑂 = (0g‘𝐶) & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑀‘𝑋) ∈ (𝐹 ∖ {𝑂})) | ||
| Theorem | hvmaplfl 42064 | The vector to functional map value is a functional. (Contributed by NM, 28-Mar-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑀‘𝑋) ∈ 𝐹) | ||
| Theorem | hvmaplkr 42065 | Kernel of the vector to functional map. TODO: make this become lcfrlem11 41850. (Contributed by NM, 29-Mar-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐿‘(𝑀‘𝑋)) = (𝑂‘{𝑋})) | ||
| Theorem | mapdhvmap 42066 | Relationship between mapd and HVMap, which can be used to satisfy the last hypothesis of mapdpg 42003. Equation 10 of [Baer] p. 48. (Contributed by NM, 29-Mar-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑃 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{(𝑃‘𝑋)})) | ||
| Theorem | lspindp5 42067 | Obtain an independent vector set 𝑈, 𝑋, 𝑌 from a vector 𝑈 dependent on 𝑋 and 𝑍 and another independent set 𝑍, 𝑋, 𝑌. (Here we don't show the (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) part of the independence, which passes straight through. We also don't show nonzero vector requirements that are redundant for this theorem. Different orderings can be obtained using lspexch 21088 and prcom 4690.) (Contributed by NM, 4-May-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ (𝑁‘{𝑋, 𝑈})) & ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → ¬ 𝑈 ∈ (𝑁‘{𝑋, 𝑌})) | ||
| Theorem | hdmaplem1 42068 | Lemma to convert a frequently-used union condition. TODO: see if this can be applied to other hdmap* theorems. (Contributed by NM, 17-May-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑍 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑋})) | ||
| Theorem | hdmaplem2N 42069 | Lemma to convert a frequently-used union condition. TODO: see if this can be applied to other hdmap* theorems. (Contributed by NM, 17-May-2015.) (New usage is discouraged.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑍 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑌})) | ||
| Theorem | hdmaplem3 42070 | Lemma to convert a frequently-used union condition. TODO: see if this can be applied to other hdmap* theorems. (Contributed by NM, 17-May-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑍 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | ||
| Theorem | hdmaplem4 42071 | Lemma to convert a frequently-used union condition. TODO: see if this can be applied to other hdmap* theorems. (Contributed by NM, 17-May-2015.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑋})) & ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → ¬ 𝑍 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌}))) | ||
| Theorem | mapdh8a 42072* | Part of Part (8) in [Baer] p. 48. (Contributed by NM, 5-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩)) | ||
| Theorem | mapdh8aa 42073* | Part of Part (8) in [Baer] p. 48. (Contributed by NM, 12-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩)) | ||
| Theorem | mapdh8ab 42074* | Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩)) | ||
| Theorem | mapdh8ac 42075* | Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) = 𝐵) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) & ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑍})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩)) | ||
| Theorem | mapdh8ad 42076* | Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩)) | ||
| Theorem | mapdh8b 42077* | Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})) & ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑤⟩) = 𝐸) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) & ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑌, 𝐺, 𝑇⟩)) | ||
| Theorem | mapdh8c 42078* | Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) = 𝐸) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) & ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩)) | ||
| Theorem | mapdh8d0N 42079* | Part of Part (8) in [Baer] p. 48. (Contributed by NM, 10-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) & ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩)) | ||
| Theorem | mapdh8d 42080* | Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩)) | ||
| Theorem | mapdh8e 42081* | Part of Part (8) in [Baer] p. 48. Eliminate 𝑤. (Contributed by NM, 10-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩)) | ||
| Theorem | mapdh8g 42082* | Part of Part (8) in [Baer] p. 48. Eliminate 𝑋 ∈ (𝑁‘{𝑌, 𝑇}). TODO: break out 𝑇 ≠ 0 in mapdh8e 42081 so we can share hypotheses. Also, look at hypothesis sharing for earlier mapdh8* and mapdh75* stuff. (Contributed by NM, 10-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩)) | ||
| Theorem | mapdh8i 42083* | Part of Part (8) in [Baer] p. 48. (Contributed by NM, 11-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇})) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩)) | ||
| Theorem | mapdh8j 42084* | Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩)) | ||
| Theorem | mapdh8 42085* | Part (8) in [Baer] p. 48. Given a reference vector 𝑋, the value of function 𝐼 at a vector 𝑇 is independent of the choice of auxiliary vectors 𝑌 and 𝑍. Unlike Baer's, our version does not require 𝑋, 𝑌, and 𝑍 to be independent, and also is defined for all 𝑌 and 𝑍 that are not colinear with 𝑋 or 𝑇. We do this to make the definition of Baer's sigma function more straightforward. (This part eliminates 𝑇 ≠ 0.) (Contributed by NM, 13-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇})) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩)) | ||
| Theorem | mapdh9a 42086* | Lemma for part (9) in [Baer] p. 48. TODO: why is this 50% larger than mapdh9aOLDN 42087? (Contributed by NM, 14-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩))) | ||
| Theorem | mapdh9aOLDN 42087* | Lemma for part (9) in [Baer] p. 48. (Contributed by NM, 14-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 = (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 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩))) | ||
| Syntax | chdma1 42088 | Extend class notation with preliminary map from vectors to functionals in the closed kernel dual space. |
| class HDMap1 | ||
| Syntax | chdma 42089 | Extend class notation with map from vectors to functionals in the closed kernel dual space. |
| class HDMap | ||
| Definition | df-hdmap1 42090* | Define preliminary map from vectors to functionals in the closed kernel dual space. See hdmap1fval 42093 description for more details. (Contributed by NM, 14-May-2015.) |
| ⊢ HDMap1 = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎 ∣ [((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝑘)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd ‘𝑥) = (0g‘𝑢), (0g‘𝑐), (℩ℎ ∈ 𝑑 ((𝑚‘(𝑛‘{(2nd ‘𝑥)})) = (𝑗‘{ℎ}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st ‘𝑥))(-g‘𝑢)(2nd ‘𝑥))})) = (𝑗‘{((2nd ‘(1st ‘𝑥))(-g‘𝑐)ℎ)})))))})) | ||
| Definition | df-hdmap 42091* | Define map from vectors to functionals in the closed kernel dual space. See hdmapfval 42124 description for more details. (Contributed by NM, 15-May-2015.) |
| ⊢ HDMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {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 | hdmap1ffval 42092* | Preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 14-May-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝑋 → (HDMap1‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑎 ∣ [((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝐾)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd ‘𝑥) = (0g‘𝑢), (0g‘𝑐), (℩ℎ ∈ 𝑑 ((𝑚‘(𝑛‘{(2nd ‘𝑥)})) = (𝑗‘{ℎ}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st ‘𝑥))(-g‘𝑢)(2nd ‘𝑥))})) = (𝑗‘{((2nd ‘(1st ‘𝑥))(-g‘𝑐)ℎ)})))))})) | ||
| Theorem | hdmap1fval 42093* | Preliminary map from vectors to functionals in the closed kernel dual space. TODO: change span 𝐽 to the convention 𝐿 for this section. (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 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝐼 = (𝑥 ∈ ((𝑉 × 𝐷) × 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)})))))) | ||
| Theorem | hdmap1vallem 42094* | Value of preliminary map from vectors to functionals in the closed kernel dual space. (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 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑇 ∈ ((𝑉 × 𝐷) × 𝑉)) ⇒ ⊢ (𝜑 → (𝐼‘𝑇) = if((2nd ‘𝑇) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑇)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑇)) − (2nd ‘𝑇))})) = (𝐽‘{((2nd ‘(1st ‘𝑇))𝑅ℎ)}))))) | ||
| Theorem | hdmap1val 42095* | Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 42021.) TODO: change 𝐼 = (𝑥 ∈ V ↦... to (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌 > ) =... in e.g. mapdh8 42085 to shorten proofs with no $d on 𝑥. (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 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))) | ||
| Theorem | hdmap1val0 42096 | Value of preliminary map from vectors to functionals at zero. (Restated mapdhval0 42022.) (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 42097* | 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 42098 | 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 42099* | 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 42100* | 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 42099 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⟩)) | ||
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