Metamath Proof Explorer


Theorem hdmap14lem1

Description: Prior to part 14 in Baer p. 49, line 25. (Contributed by NM, 31-May-2015)

Ref Expression
Hypotheses hdmap14lem1.h 𝐻 = ( LHyp ‘ 𝐾 )
hdmap14lem1.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hdmap14lem1.v 𝑉 = ( Base ‘ 𝑈 )
hdmap14lem1.t · = ( ·𝑠𝑈 )
hdmap14lem3.o 0 = ( 0g𝑈 )
hdmap14lem1.r 𝑅 = ( Scalar ‘ 𝑈 )
hdmap14lem1.b 𝐵 = ( Base ‘ 𝑅 )
hdmap14lem1.z 𝑍 = ( 0g𝑅 )
hdmap14lem1.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
hdmap14lem2.e = ( ·𝑠𝐶 )
hdmap14lem1.l 𝐿 = ( LSpan ‘ 𝐶 )
hdmap14lem2.p 𝑃 = ( Scalar ‘ 𝐶 )
hdmap14lem2.a 𝐴 = ( Base ‘ 𝑃 )
hdmap14lem2.q 𝑄 = ( 0g𝑃 )
hdmap14lem1.s 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
hdmap14lem1.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
hdmap14lem3.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap14lem1.f ( 𝜑𝐹 ∈ ( 𝐵 ∖ { 𝑍 } ) )
Assertion hdmap14lem1 ( 𝜑 → ( 𝐿 ‘ { ( 𝑆𝑋 ) } ) = ( 𝐿 ‘ { ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) } ) )

Proof

Step Hyp Ref Expression
1 hdmap14lem1.h 𝐻 = ( LHyp ‘ 𝐾 )
2 hdmap14lem1.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 hdmap14lem1.v 𝑉 = ( Base ‘ 𝑈 )
4 hdmap14lem1.t · = ( ·𝑠𝑈 )
5 hdmap14lem3.o 0 = ( 0g𝑈 )
6 hdmap14lem1.r 𝑅 = ( Scalar ‘ 𝑈 )
7 hdmap14lem1.b 𝐵 = ( Base ‘ 𝑅 )
8 hdmap14lem1.z 𝑍 = ( 0g𝑅 )
9 hdmap14lem1.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
10 hdmap14lem2.e = ( ·𝑠𝐶 )
11 hdmap14lem1.l 𝐿 = ( LSpan ‘ 𝐶 )
12 hdmap14lem2.p 𝑃 = ( Scalar ‘ 𝐶 )
13 hdmap14lem2.a 𝐴 = ( Base ‘ 𝑃 )
14 hdmap14lem2.q 𝑄 = ( 0g𝑃 )
15 hdmap14lem1.s 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
16 hdmap14lem1.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
17 hdmap14lem3.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
18 hdmap14lem1.f ( 𝜑𝐹 ∈ ( 𝐵 ∖ { 𝑍 } ) )
19 17 eldifad ( 𝜑𝑋𝑉 )
20 18 eldifad ( 𝜑𝐹𝐵 )
21 eldifsni ( 𝐹 ∈ ( 𝐵 ∖ { 𝑍 } ) → 𝐹𝑍 )
22 18 21 syl ( 𝜑𝐹𝑍 )
23 1 2 3 4 6 7 9 10 11 12 13 15 16 19 20 8 22 hdmap14lem1a ( 𝜑 → ( 𝐿 ‘ { ( 𝑆𝑋 ) } ) = ( 𝐿 ‘ { ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) } ) )