Description: Part (7) of Baer p. 48 line 10 (3 of 6 cases). (Contributed by NM, 2-May-2015) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | mapdh75.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | |
| mapdh75.u | ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) | ||
| mapdh75.v | ⊢ 𝑉 = ( Base ‘ 𝑈 ) | ||
| mapdh75.s | ⊢ − = ( -g ‘ 𝑈 ) | ||
| mapdh75.o | ⊢ 0 = ( 0g ‘ 𝑈 ) | ||
| mapdh75.n | ⊢ 𝑁 = ( LSpan ‘ 𝑈 ) | ||
| mapdh75.c | ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) | ||
| mapdh75.d | ⊢ 𝐷 = ( Base ‘ 𝐶 ) | ||
| mapdh75.r | ⊢ 𝑅 = ( -g ‘ 𝐶 ) | ||
| mapdh75.q | ⊢ 𝑄 = ( 0g ‘ 𝐶 ) | ||
| mapdh75.j | ⊢ 𝐽 = ( LSpan ‘ 𝐶 ) | ||
| mapdh75.m | ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) | ||
| mapdh75.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 ) } ) ) ) ) ) | ||
| mapdh75.k | ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) | ||
| mapdh75.f | ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) | ||
| mapdh75.mn | ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) ) | ||
| mapdh75a | ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ) | ||
| mapdh75c.ne | ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) | ||
| mapdh75c.x | ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) | ||
| mapdh75c.y | ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) | ||
| Assertion | mapdh75cN | ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑋 ⟩ ) = 𝐹 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mapdh75.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | |
| 2 | mapdh75.u | ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) | |
| 3 | mapdh75.v | ⊢ 𝑉 = ( Base ‘ 𝑈 ) | |
| 4 | mapdh75.s | ⊢ − = ( -g ‘ 𝑈 ) | |
| 5 | mapdh75.o | ⊢ 0 = ( 0g ‘ 𝑈 ) | |
| 6 | mapdh75.n | ⊢ 𝑁 = ( LSpan ‘ 𝑈 ) | |
| 7 | mapdh75.c | ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) | |
| 8 | mapdh75.d | ⊢ 𝐷 = ( Base ‘ 𝐶 ) | |
| 9 | mapdh75.r | ⊢ 𝑅 = ( -g ‘ 𝐶 ) | |
| 10 | mapdh75.q | ⊢ 𝑄 = ( 0g ‘ 𝐶 ) | |
| 11 | mapdh75.j | ⊢ 𝐽 = ( LSpan ‘ 𝐶 ) | |
| 12 | mapdh75.m | ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) | |
| 13 | mapdh75.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 ) } ) ) ) ) ) | |
| 14 | mapdh75.k | ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) | |
| 15 | mapdh75.f | ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) | |
| 16 | mapdh75.mn | ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) ) | |
| 17 | mapdh75a | ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ) | |
| 18 | mapdh75c.ne | ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) | |
| 19 | mapdh75c.x | ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) | |
| 20 | mapdh75c.y | ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) | |
| 21 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | mapdh75e | ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑋 ⟩ ) = 𝐹 ) |