Description: Lemma for mapdpg . Baer p. 45, line 6: "Hence Fx=Fy, an impossibility." (Contributed by NM, 20-Mar-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | mapdpglem.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | |
mapdpglem.m | ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) | ||
mapdpglem.u | ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) | ||
mapdpglem.v | ⊢ 𝑉 = ( Base ‘ 𝑈 ) | ||
mapdpglem.s | ⊢ − = ( -g ‘ 𝑈 ) | ||
mapdpglem.n | ⊢ 𝑁 = ( LSpan ‘ 𝑈 ) | ||
mapdpglem.c | ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) | ||
mapdpglem.k | ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) | ||
mapdpglem.x | ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) | ||
mapdpglem.y | ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) | ||
mapdpglem1.p | ⊢ ⊕ = ( LSSum ‘ 𝐶 ) | ||
mapdpglem2.j | ⊢ 𝐽 = ( LSpan ‘ 𝐶 ) | ||
mapdpglem3.f | ⊢ 𝐹 = ( Base ‘ 𝐶 ) | ||
mapdpglem3.te | ⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) | ||
mapdpglem3.a | ⊢ 𝐴 = ( Scalar ‘ 𝑈 ) | ||
mapdpglem3.b | ⊢ 𝐵 = ( Base ‘ 𝐴 ) | ||
mapdpglem3.t | ⊢ · = ( ·𝑠 ‘ 𝐶 ) | ||
mapdpglem3.r | ⊢ 𝑅 = ( -g ‘ 𝐶 ) | ||
mapdpglem3.g | ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) | ||
mapdpglem3.e | ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ) | ||
mapdpglem4.q | ⊢ 𝑄 = ( 0g ‘ 𝑈 ) | ||
mapdpglem.ne | ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) | ||
mapdpglem4.jt | ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) | ||
mapdpglem4.z | ⊢ 0 = ( 0g ‘ 𝐴 ) | ||
mapdpglem4.g4 | ⊢ ( 𝜑 → 𝑔 ∈ 𝐵 ) | ||
mapdpglem4.z4 | ⊢ ( 𝜑 → 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) | ||
mapdpglem4.t4 | ⊢ ( 𝜑 → 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) | ||
mapdpglem4.xn | ⊢ ( 𝜑 → 𝑋 ≠ 𝑄 ) | ||
mapdpglem4.g0 | ⊢ ( 𝜑 → 𝑔 = 0 ) | ||
Assertion | mapdpglem10 | ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpglem.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | |
2 | mapdpglem.m | ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) | |
3 | mapdpglem.u | ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) | |
4 | mapdpglem.v | ⊢ 𝑉 = ( Base ‘ 𝑈 ) | |
5 | mapdpglem.s | ⊢ − = ( -g ‘ 𝑈 ) | |
6 | mapdpglem.n | ⊢ 𝑁 = ( LSpan ‘ 𝑈 ) | |
7 | mapdpglem.c | ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) | |
8 | mapdpglem.k | ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) | |
9 | mapdpglem.x | ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) | |
10 | mapdpglem.y | ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) | |
11 | mapdpglem1.p | ⊢ ⊕ = ( LSSum ‘ 𝐶 ) | |
12 | mapdpglem2.j | ⊢ 𝐽 = ( LSpan ‘ 𝐶 ) | |
13 | mapdpglem3.f | ⊢ 𝐹 = ( Base ‘ 𝐶 ) | |
14 | mapdpglem3.te | ⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) | |
15 | mapdpglem3.a | ⊢ 𝐴 = ( Scalar ‘ 𝑈 ) | |
16 | mapdpglem3.b | ⊢ 𝐵 = ( Base ‘ 𝐴 ) | |
17 | mapdpglem3.t | ⊢ · = ( ·𝑠 ‘ 𝐶 ) | |
18 | mapdpglem3.r | ⊢ 𝑅 = ( -g ‘ 𝐶 ) | |
19 | mapdpglem3.g | ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) | |
20 | mapdpglem3.e | ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ) | |
21 | mapdpglem4.q | ⊢ 𝑄 = ( 0g ‘ 𝑈 ) | |
22 | mapdpglem.ne | ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) | |
23 | mapdpglem4.jt | ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) | |
24 | mapdpglem4.z | ⊢ 0 = ( 0g ‘ 𝐴 ) | |
25 | mapdpglem4.g4 | ⊢ ( 𝜑 → 𝑔 ∈ 𝐵 ) | |
26 | mapdpglem4.z4 | ⊢ ( 𝜑 → 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) | |
27 | mapdpglem4.t4 | ⊢ ( 𝜑 → 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) | |
28 | mapdpglem4.xn | ⊢ ( 𝜑 → 𝑋 ≠ 𝑄 ) | |
29 | mapdpglem4.g0 | ⊢ ( 𝜑 → 𝑔 = 0 ) | |
30 | 1 3 8 | dvhlvec | ⊢ ( 𝜑 → 𝑈 ∈ LVec ) |
31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 | mapdpglem9 | ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 } ) ) |
32 | 4 21 6 30 10 31 28 | lspsneleq | ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) = ( 𝑁 ‘ { 𝑌 } ) ) |