Metamath Proof Explorer

Theorem mapdpglem1

Description: Lemma for mapdpg . Baer p. 44, last line: "(F(x-y))* <= (Fx)*+(Fy)*." (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 )
Assertion mapdpglem1 ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) ⊆ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )

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 1 3 8 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
13 eqid ⊢ ( LSSum ‘ 𝑈 ) = ( LSSum ‘ 𝑈 )
14 4 5 13 6 lspsntrim ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) )
15 12 9 10 14 syl3anc ⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) )
16 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
17 4 5 lmodvsubcl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 − 𝑌 ) ∈ 𝑉 )
18 12 9 10 17 syl3anc ⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ∈ 𝑉 )
19 4 16 6 lspsncl ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑋 − 𝑌 ) ∈ 𝑉 ) → ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ∈ ( LSubSp ‘ 𝑈 ) )
20 12 18 19 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ∈ ( LSubSp ‘ 𝑈 ) )
21 4 16 6 lspsncl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
22 12 9 21 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
23 4 16 6 lspsncl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
24 12 10 23 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
25 16 13 lsmcl ⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝑈 ) )
26 12 22 24 25 syl3anc ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝑈 ) )
27 1 3 16 2 8 20 26 mapdord ⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) ⊆ ( 𝑀 ‘ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) ) ↔ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ⊆ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) ) )
28 15 27 mpbird ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) ⊆ ( 𝑀 ‘ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) ) )
29 1 2 3 16 13 7 11 8 22 24 mapdlsm ⊢ ( 𝜑 → ( 𝑀 ‘ ( ( 𝑁 ‘ { 𝑋 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑌 } ) ) ) = ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
30 28 29 sseqtrd ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) ⊆ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )