Metamath Proof Explorer

Theorem hdmapsub

Description: Part of proof of part 12 in Baer p. 49 line 5, (a-b)S = aS-bS in their notation (S = sigma). (Contributed by NM, 26-May-2015)

Ref Expression
Hypotheses hdmap12c.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
hdmap12c.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hdmap12c.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
hdmap12c.m ⊢ − = ( -g ‘ 𝑈 )
hdmap12c.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
hdmap12c.n ⊢ 𝑁 = ( -g ‘ 𝐶 )
hdmap12c.s ⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
hdmap12c.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
hdmap12c.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
hdmap12c.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion hdmapsub ( 𝜑 → ( 𝑆 ‘ ( 𝑋 − 𝑌 ) ) = ( ( 𝑆 ‘ 𝑋 ) 𝑁 ( 𝑆 ‘ 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 hdmap12c.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 hdmap12c.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 hdmap12c.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
4 hdmap12c.m ⊢ − = ( -g ‘ 𝑈 )
5 hdmap12c.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
6 hdmap12c.n ⊢ 𝑁 = ( -g ‘ 𝐶 )
7 hdmap12c.s ⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
8 hdmap12c.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 hdmap12c.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
10 hdmap12c.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
11 eqid ⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 )
12 eqid ⊢ ( invg ‘ 𝑈 ) = ( invg ‘ 𝑈 )
13 3 11 12 4 grpsubval ⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 ( +g ‘ 𝑈 ) ( ( invg ‘ 𝑈 ) ‘ 𝑌 ) ) )
14 9 10 13 syl2anc ⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) = ( 𝑋 ( +g ‘ 𝑈 ) ( ( invg ‘ 𝑈 ) ‘ 𝑌 ) ) )
15 14 fveq2d ⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝑋 − 𝑌 ) ) = ( 𝑆 ‘ ( 𝑋 ( +g ‘ 𝑈 ) ( ( invg ‘ 𝑈 ) ‘ 𝑌 ) ) ) )
16 eqid ⊢ ( +g ‘ 𝐶 ) = ( +g ‘ 𝐶 )
17 1 2 8 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
18 3 12 lmodvnegcl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( ( invg ‘ 𝑈 ) ‘ 𝑌 ) ∈ 𝑉 )
19 17 10 18 syl2anc ⊢ ( 𝜑 → ( ( invg ‘ 𝑈 ) ‘ 𝑌 ) ∈ 𝑉 )
20 1 2 3 11 5 16 7 8 9 19 hdmapadd ⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝑋 ( +g ‘ 𝑈 ) ( ( invg ‘ 𝑈 ) ‘ 𝑌 ) ) ) = ( ( 𝑆 ‘ 𝑋 ) ( +g ‘ 𝐶 ) ( 𝑆 ‘ ( ( invg ‘ 𝑈 ) ‘ 𝑌 ) ) ) )
21 eqid ⊢ ( invg ‘ 𝐶 ) = ( invg ‘ 𝐶 )
22 1 2 3 12 5 21 7 8 10 hdmapneg ⊢ ( 𝜑 → ( 𝑆 ‘ ( ( invg ‘ 𝑈 ) ‘ 𝑌 ) ) = ( ( invg ‘ 𝐶 ) ‘ ( 𝑆 ‘ 𝑌 ) ) )
23 22 oveq2d ⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑋 ) ( +g ‘ 𝐶 ) ( 𝑆 ‘ ( ( invg ‘ 𝑈 ) ‘ 𝑌 ) ) ) = ( ( 𝑆 ‘ 𝑋 ) ( +g ‘ 𝐶 ) ( ( invg ‘ 𝐶 ) ‘ ( 𝑆 ‘ 𝑌 ) ) ) )
24 15 20 23 3eqtrd ⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝑋 − 𝑌 ) ) = ( ( 𝑆 ‘ 𝑋 ) ( +g ‘ 𝐶 ) ( ( invg ‘ 𝐶 ) ‘ ( 𝑆 ‘ 𝑌 ) ) ) )
25 eqid ⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
26 1 2 3 5 25 7 8 9 hdmapcl ⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ∈ ( Base ‘ 𝐶 ) )
27 1 2 3 5 25 7 8 10 hdmapcl ⊢ ( 𝜑 → ( 𝑆 ‘ 𝑌 ) ∈ ( Base ‘ 𝐶 ) )
28 25 16 21 6 grpsubval ⊢ ( ( ( 𝑆 ‘ 𝑋 ) ∈ ( Base ‘ 𝐶 ) ∧ ( 𝑆 ‘ 𝑌 ) ∈ ( Base ‘ 𝐶 ) ) → ( ( 𝑆 ‘ 𝑋 ) 𝑁 ( 𝑆 ‘ 𝑌 ) ) = ( ( 𝑆 ‘ 𝑋 ) ( +g ‘ 𝐶 ) ( ( invg ‘ 𝐶 ) ‘ ( 𝑆 ‘ 𝑌 ) ) ) )
29 26 27 28 syl2anc ⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑋 ) 𝑁 ( 𝑆 ‘ 𝑌 ) ) = ( ( 𝑆 ‘ 𝑋 ) ( +g ‘ 𝐶 ) ( ( invg ‘ 𝐶 ) ‘ ( 𝑆 ‘ 𝑌 ) ) ) )
30 24 29 eqtr4d ⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝑋 − 𝑌 ) ) = ( ( 𝑆 ‘ 𝑋 ) 𝑁 ( 𝑆 ‘ 𝑌 ) ) )