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 . (Convert mapdh6N to use the function HDMap1 .) (Contributed by NM, 17-May-2015)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | hdmap1-6.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | |
| hdmap1-6.u | ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) | ||
| hdmap1-6.v | ⊢ 𝑉 = ( Base ‘ 𝑈 ) | ||
| hdmap1-6.p | ⊢ + = ( +g ‘ 𝑈 ) | ||
| hdmap1-6.o | ⊢ 0 = ( 0g ‘ 𝑈 ) | ||
| hdmap1-6.n | ⊢ 𝑁 = ( LSpan ‘ 𝑈 ) | ||
| hdmap1-6.c | ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) | ||
| hdmap1-6.d | ⊢ 𝐷 = ( Base ‘ 𝐶 ) | ||
| hdmap1-6.a | ⊢ ✚ = ( +g ‘ 𝐶 ) | ||
| hdmap1-6.l | ⊢ 𝐿 = ( LSpan ‘ 𝐶 ) | ||
| hdmap1-6.m | ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) | ||
| hdmap1-6.i | ⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) | ||
| hdmap1-6.k | ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) | ||
| hdmap1-6.f | ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) | ||
| hdmap1-6.x | ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) | ||
| hdmap1-6.y | ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) | ||
| hdmap1-6.z | ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) | ||
| hdmap1-6.xn | ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) | ||
| hdmap1-6.mn | ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) ) | ||
| Assertion | hdmap1l6 | ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hdmap1-6.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | |
| 2 | hdmap1-6.u | ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) | |
| 3 | hdmap1-6.v | ⊢ 𝑉 = ( Base ‘ 𝑈 ) | |
| 4 | hdmap1-6.p | ⊢ + = ( +g ‘ 𝑈 ) | |
| 5 | hdmap1-6.o | ⊢ 0 = ( 0g ‘ 𝑈 ) | |
| 6 | hdmap1-6.n | ⊢ 𝑁 = ( LSpan ‘ 𝑈 ) | |
| 7 | hdmap1-6.c | ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) | |
| 8 | hdmap1-6.d | ⊢ 𝐷 = ( Base ‘ 𝐶 ) | |
| 9 | hdmap1-6.a | ⊢ ✚ = ( +g ‘ 𝐶 ) | |
| 10 | hdmap1-6.l | ⊢ 𝐿 = ( LSpan ‘ 𝐶 ) | |
| 11 | hdmap1-6.m | ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) | |
| 12 | hdmap1-6.i | ⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) | |
| 13 | hdmap1-6.k | ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) | |
| 14 | hdmap1-6.f | ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) | |
| 15 | hdmap1-6.x | ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) | |
| 16 | hdmap1-6.y | ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) | |
| 17 | hdmap1-6.z | ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) | |
| 18 | hdmap1-6.xn | ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) | |
| 19 | hdmap1-6.mn | ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) ) | |
| 20 | eqid | ⊢ ( -g ‘ 𝑈 ) = ( -g ‘ 𝑈 ) | |
| 21 | eqid | ⊢ ( -g ‘ 𝐶 ) = ( -g ‘ 𝐶 ) | |
| 22 | eqid | ⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 ) | |
| 23 | 1 2 3 4 20 5 6 7 8 9 21 22 10 11 12 13 14 15 19 16 17 18 | hdmap1l6k | ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) |