Metamath Proof Explorer

Theorem mapdpglem2

Description: Lemma for mapdpg . Baer p. 45, lines 1 and 2: "we have (F(x-y))* = Gt where t necessarily belongs to (Fx)*+(Fy)*." (We scope $d t ph locally to avoid clashes with later substitutions into ph .) (Contributed by NM, 15-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 )
Assertion mapdpglem2 ( 𝜑 → ∃ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) )

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 eqid ⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
14 1 3 8 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
15 4 5 lmodvsubcl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 − 𝑌 ) ∈ 𝑉 )
16 14 9 10 15 syl3anc ⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ∈ 𝑉 )
17 1 2 3 4 6 7 13 12 8 16 mapdspex ⊢ ( 𝜑 → ∃ 𝑡 ∈ ( Base ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) )
18 1 7 8 lcdlmod ⊢ ( 𝜑 → 𝐶 ∈ LMod )
19 13 12 lspsnid ⊢ ( ( 𝐶 ∈ LMod ∧ 𝑡 ∈ ( Base ‘ 𝐶 ) ) → 𝑡 ∈ ( 𝐽 ‘ { 𝑡 } ) )
20 18 19 sylan ⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝐶 ) ) → 𝑡 ∈ ( 𝐽 ‘ { 𝑡 } ) )
21 20 adantrr ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( Base ‘ 𝐶 ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ) → 𝑡 ∈ ( 𝐽 ‘ { 𝑡 } ) )
22 simprr ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( Base ‘ 𝐶 ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) )
23 21 22 eleqtrrd ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( Base ‘ 𝐶 ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ) → 𝑡 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) )
24 17 23 22 reximssdv ⊢ ( 𝜑 → ∃ 𝑡 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) )
25 1 2 3 4 5 6 7 8 9 10 11 mapdpglem1 ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) ⊆ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
26 25 sseld ⊢ ( 𝜑 → ( 𝑡 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) )
27 26 anim1d ⊢ ( 𝜑 → ( ( 𝑡 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → ( 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ) )
28 27 reximdv2 ⊢ ( 𝜑 → ( ∃ 𝑡 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) → ∃ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) )
29 24 28 mpd ⊢ ( 𝜑 → ∃ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) )