Metamath Proof Explorer

Theorem mapdpglem23

Description: Lemma for mapdpg . Baer p. 45, line 10: "and so y' meets all our requirements." Our h is Baer's y'. (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 )
mapdpglem12.yn ⊢ ( 𝜑 → 𝑌 ≠ 𝑄 )
mapdpglem17.ep ⊢ 𝐸 = ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · 𝑧 )
Assertion mapdpglem23 ( 𝜑 → ∃ ℎ ∈ 𝐹 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) )

Proof

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 mapdpglem12.yn ⊢ ( 𝜑 → 𝑌 ≠ 𝑄 )
30 mapdpglem17.ep ⊢ 𝐸 = ( ( ( invr ‘ 𝐴 ) ‘ 𝑔 ) · 𝑧 )
31 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
32 eqid ⊢ ( LSubSp ‘ 𝐶 ) = ( LSubSp ‘ 𝐶 )
33 1 3 8 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
34 4 31 6 lspsncl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
35 33 10 34 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
36 1 2 3 31 7 32 8 35 mapdcl2 ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝐶 ) )
37 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 30 mapdpglem19 ⊢ ( 𝜑 → 𝐸 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) )
38 13 32 lssel ⊢ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝐶 ) ∧ 𝐸 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) → 𝐸 ∈ 𝐹 )
39 36 37 38 syl2anc ⊢ ( 𝜑 → 𝐸 ∈ 𝐹 )
40 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 30 mapdpglem20 ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐸 } ) )
41 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 30 mapdpglem22 ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝐸 ) } ) )
42 sneq ⊢ ( ℎ = 𝐸 → { ℎ } = { 𝐸 } )
43 42 fveq2d ⊢ ( ℎ = 𝐸 → ( 𝐽 ‘ { ℎ } ) = ( 𝐽 ‘ { 𝐸 } ) )
44 43 eqeq2d ⊢ ( ℎ = 𝐸 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐸 } ) ) )
45 oveq2 ⊢ ( ℎ = 𝐸 → ( 𝐺 𝑅 ℎ ) = ( 𝐺 𝑅 𝐸 ) )
46 45 sneqd ⊢ ( ℎ = 𝐸 → { ( 𝐺 𝑅 ℎ ) } = { ( 𝐺 𝑅 𝐸 ) } )
47 46 fveq2d ⊢ ( ℎ = 𝐸 → ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝐸 ) } ) )
48 47 eqeq2d ⊢ ( ℎ = 𝐸 → ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝐸 ) } ) ) )
49 44 48 anbi12d ⊢ ( ℎ = 𝐸 → ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐸 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝐸 ) } ) ) ) )
50 49 rspcev ⊢ ( ( 𝐸 ∈ 𝐹 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐸 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝐸 ) } ) ) ) → ∃ ℎ ∈ 𝐹 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) )
51 39 40 41 50 syl12anc ⊢ ( 𝜑 → ∃ ℎ ∈ 𝐹 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) )