Description: Part (6) of Baer p. 47 line 6. Note that we use -. X e. ( N{ Y , Z } ) which is equivalent to Baer's "Fx i^i (Fy + Fz)" by lspdisjb . TODO: If disjoint variable conditions with I and ph 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 . (Contributed by NM, 1-May-2015) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | mapdh6.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | |
| mapdh6.u | ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) | ||
| mapdh6.v | ⊢ 𝑉 = ( Base ‘ 𝑈 ) | ||
| mapdh6.p | ⊢ + = ( +g ‘ 𝑈 ) | ||
| mapdh6.s | ⊢ − = ( -g ‘ 𝑈 ) | ||
| mapdh6.o | ⊢ 0 = ( 0g ‘ 𝑈 ) | ||
| mapdh6.n | ⊢ 𝑁 = ( LSpan ‘ 𝑈 ) | ||
| mapdh6.c | ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) | ||
| mapdh6.d | ⊢ 𝐷 = ( Base ‘ 𝐶 ) | ||
| mapdh6.a | ⊢ ✚ = ( +g ‘ 𝐶 ) | ||
| mapdh6.r | ⊢ 𝑅 = ( -g ‘ 𝐶 ) | ||
| mapdh6.q | ⊢ 𝑄 = ( 0g ‘ 𝐶 ) | ||
| mapdh6.j | ⊢ 𝐽 = ( LSpan ‘ 𝐶 ) | ||
| mapdh6.m | ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) | ||
| mapdh6.i | ⊢ 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 ) ) 𝑅 ℎ ) } ) ) ) ) ) | ||
| mapdh6.k | ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) | ||
| mapdh6.f | ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) | ||
| mapdh6.x | ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) | ||
| mapdh6.y | ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) | ||
| mapdh6.z | ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) | ||
| mapdh6.xn | ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) | ||
| mapdh6.mn | ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) ) | ||
| Assertion | mapdh6N | ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mapdh6.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | |
| 2 | mapdh6.u | ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) | |
| 3 | mapdh6.v | ⊢ 𝑉 = ( Base ‘ 𝑈 ) | |
| 4 | mapdh6.p | ⊢ + = ( +g ‘ 𝑈 ) | |
| 5 | mapdh6.s | ⊢ − = ( -g ‘ 𝑈 ) | |
| 6 | mapdh6.o | ⊢ 0 = ( 0g ‘ 𝑈 ) | |
| 7 | mapdh6.n | ⊢ 𝑁 = ( LSpan ‘ 𝑈 ) | |
| 8 | mapdh6.c | ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) | |
| 9 | mapdh6.d | ⊢ 𝐷 = ( Base ‘ 𝐶 ) | |
| 10 | mapdh6.a | ⊢ ✚ = ( +g ‘ 𝐶 ) | |
| 11 | mapdh6.r | ⊢ 𝑅 = ( -g ‘ 𝐶 ) | |
| 12 | mapdh6.q | ⊢ 𝑄 = ( 0g ‘ 𝐶 ) | |
| 13 | mapdh6.j | ⊢ 𝐽 = ( LSpan ‘ 𝐶 ) | |
| 14 | mapdh6.m | ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) | |
| 15 | mapdh6.i | ⊢ 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 ) ) 𝑅 ℎ ) } ) ) ) ) ) | |
| 16 | mapdh6.k | ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) | |
| 17 | mapdh6.f | ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) | |
| 18 | mapdh6.x | ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) | |
| 19 | mapdh6.y | ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) | |
| 20 | mapdh6.z | ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) | |
| 21 | mapdh6.xn | ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) | |
| 22 | mapdh6.mn | ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) ) | |
| 23 | 12 15 1 14 2 3 5 6 7 8 9 11 13 16 17 22 18 4 10 19 20 21 | mapdh6kN | ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) |