Theorem hdmap14lem8

Description: Part of proof of part 14 in Baer p. 49 lines 33-35. (Contributed by NM, 1-Jun-2015)

	Ref	Expression
Hypotheses	hdmap14lem8.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmap14lem8.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem8.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmap14lem8.q	⊢ + = ( +g ‘ 𝑈 )
	hdmap14lem8.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
	hdmap14lem8.o	⊢ 0 = ( 0g ‘ 𝑈 )
	hdmap14lem8.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	hdmap14lem8.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
	hdmap14lem8.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	hdmap14lem8.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem8.d	⊢ ✚ = ( +g ‘ 𝐶 )
	hdmap14lem8.e	⊢ ∙ = ( ·𝑠 ‘ 𝐶 )
	hdmap14lem8.p	⊢ 𝑃 = ( Scalar ‘ 𝐶 )
	hdmap14lem8.a	⊢ 𝐴 = ( Base ‘ 𝑃 )
	hdmap14lem8.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
	hdmap14lem8.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmap14lem8.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap14lem8.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap14lem8.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
	hdmap14lem8.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐴 )
	hdmap14lem8.i	⊢ ( 𝜑 → 𝐼 ∈ 𝐴 )
	hdmap14lem8.xx	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) )
	hdmap14lem8.yy	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑌 ) ) = ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) )
	hdmap14lem8.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
	hdmap14lem8.j	⊢ ( 𝜑 → 𝐽 ∈ 𝐴 )
	hdmap14lem8.xy	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝐽 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) )
Assertion	hdmap14lem8	⊢ ( 𝜑 → ( ( 𝐽 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐽 ∙ ( 𝑆 ‘ 𝑌 ) ) ) = ( ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		hdmap14lem8.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmap14lem8.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmap14lem8.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmap14lem8.q	⊢ + = ( +g ‘ 𝑈 )
5		hdmap14lem8.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
6		hdmap14lem8.o	⊢ 0 = ( 0g ‘ 𝑈 )
7		hdmap14lem8.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
8		hdmap14lem8.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
9		hdmap14lem8.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
10		hdmap14lem8.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
11		hdmap14lem8.d	⊢ ✚ = ( +g ‘ 𝐶 )
12		hdmap14lem8.e	⊢ ∙ = ( ·𝑠 ‘ 𝐶 )
13		hdmap14lem8.p	⊢ 𝑃 = ( Scalar ‘ 𝐶 )
14		hdmap14lem8.a	⊢ 𝐴 = ( Base ‘ 𝑃 )
15		hdmap14lem8.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
16		hdmap14lem8.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
17		hdmap14lem8.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
18		hdmap14lem8.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
19		hdmap14lem8.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
20		hdmap14lem8.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐴 )
21		hdmap14lem8.i	⊢ ( 𝜑 → 𝐼 ∈ 𝐴 )
22		hdmap14lem8.xx	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) )
23		hdmap14lem8.yy	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 𝑌 ) ) = ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) )
24		hdmap14lem8.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
25		hdmap14lem8.j	⊢ ( 𝜑 → 𝐽 ∈ 𝐴 )
26		hdmap14lem8.xy	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝐽 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) )
27	1 10 16	lcdlmod	⊢ ( 𝜑 → 𝐶 ∈ LMod )
28		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
29	17	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
30	1 2 3 10 28 15 16 29	hdmapcl	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑋 ) ∈ ( Base ‘ 𝐶 ) )
31	18	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
32	1 2 3 10 28 15 16 31	hdmapcl	⊢ ( 𝜑 → ( 𝑆 ‘ 𝑌 ) ∈ ( Base ‘ 𝐶 ) )
33	28 11 13 12 14	lmodvsdi	⊢ ( ( 𝐶 ∈ LMod ∧ ( 𝐽 ∈ 𝐴 ∧ ( 𝑆 ‘ 𝑋 ) ∈ ( Base ‘ 𝐶 ) ∧ ( 𝑆 ‘ 𝑌 ) ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝐽 ∙ ( ( 𝑆 ‘ 𝑋 ) ✚ ( 𝑆 ‘ 𝑌 ) ) ) = ( ( 𝐽 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐽 ∙ ( 𝑆 ‘ 𝑌 ) ) ) )
34	27 25 30 32 33	syl13anc	⊢ ( 𝜑 → ( 𝐽 ∙ ( ( 𝑆 ‘ 𝑋 ) ✚ ( 𝑆 ‘ 𝑌 ) ) ) = ( ( 𝐽 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐽 ∙ ( 𝑆 ‘ 𝑌 ) ) ) )
35	1 2 3 4 10 11 15 16 29 31	hdmapadd	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝑆 ‘ 𝑋 ) ✚ ( 𝑆 ‘ 𝑌 ) ) )
36	35	oveq2d	⊢ ( 𝜑 → ( 𝐽 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) = ( 𝐽 ∙ ( ( 𝑆 ‘ 𝑋 ) ✚ ( 𝑆 ‘ 𝑌 ) ) ) )
37	1 2 16	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
38	3 4 8 5 9	lmodvsdi	⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐹 · ( 𝑋 + 𝑌 ) ) = ( ( 𝐹 · 𝑋 ) + ( 𝐹 · 𝑌 ) ) )
39	37 19 29 31 38	syl13anc	⊢ ( 𝜑 → ( 𝐹 · ( 𝑋 + 𝑌 ) ) = ( ( 𝐹 · 𝑋 ) + ( 𝐹 · 𝑌 ) ) )
40	39	fveq2d	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( 𝑆 ‘ ( ( 𝐹 · 𝑋 ) + ( 𝐹 · 𝑌 ) ) ) )
41	3 8 5 9	lmodvscl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ) → ( 𝐹 · 𝑋 ) ∈ 𝑉 )
42	37 19 29 41	syl3anc	⊢ ( 𝜑 → ( 𝐹 · 𝑋 ) ∈ 𝑉 )
43	3 8 5 9	lmodvscl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉 ) → ( 𝐹 · 𝑌 ) ∈ 𝑉 )
44	37 19 31 43	syl3anc	⊢ ( 𝜑 → ( 𝐹 · 𝑌 ) ∈ 𝑉 )
45	1 2 3 4 10 11 15 16 42 44	hdmapadd	⊢ ( 𝜑 → ( 𝑆 ‘ ( ( 𝐹 · 𝑋 ) + ( 𝐹 · 𝑌 ) ) ) = ( ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) ✚ ( 𝑆 ‘ ( 𝐹 · 𝑌 ) ) ) )
46	22 23	oveq12d	⊢ ( 𝜑 → ( ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) ✚ ( 𝑆 ‘ ( 𝐹 · 𝑌 ) ) ) = ( ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) ) )
47	40 45 46	3eqtrd	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · ( 𝑋 + 𝑌 ) ) ) = ( ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) ) )
48	26 47	eqtr3d	⊢ ( 𝜑 → ( 𝐽 ∙ ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) ) = ( ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) ) )
49	36 48	eqtr3d	⊢ ( 𝜑 → ( 𝐽 ∙ ( ( 𝑆 ‘ 𝑋 ) ✚ ( 𝑆 ‘ 𝑌 ) ) ) = ( ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) ) )
50	34 49	eqtr3d	⊢ ( 𝜑 → ( ( 𝐽 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐽 ∙ ( 𝑆 ‘ 𝑌 ) ) ) = ( ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ✚ ( 𝐼 ∙ ( 𝑆 ‘ 𝑌 ) ) ) )