mapdh8

Description: Part (8) in Baer p. 48. Given a reference vector X , the value of function I at a vector T is independent of the choice of auxiliary vectors Y and Z . Unlike Baer's, our version does not require X , Y , and Z to be independent, and also is defined for all Y and Z that are not colinear with X or T . We do this to make the definition of Baer's sigma function more straightforward. (This part eliminates T =/= .0. .) (Contributed by NM, 13-May-2015)

	Ref	Expression
Hypotheses	mapdh8a.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdh8a.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdh8a.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	mapdh8a.s	⊢ − = ( -_g ‘ 𝑈 )
	mapdh8a.o	⊢ 0 = ( 0_g ‘ 𝑈 )
	mapdh8a.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	mapdh8a.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	mapdh8a.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
	mapdh8a.r	⊢ 𝑅 = ( -_g ‘ 𝐶 )
	mapdh8a.q	⊢ 𝑄 = ( 0_g ‘ 𝐶 )
	mapdh8a.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	mapdh8a.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdh8a.i	⊢ 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) )
	mapdh8a.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdh8h.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	mapdh8h.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
	mapdh8i.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8i.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8i.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8i.xy	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
	mapdh8i.xz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
	mapdh8i.yt	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
	mapdh8i.zt	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
	mapdh8.t	⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
Assertion	mapdh8	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) , 𝑇 ⟩ ) )

Proof

Step	Hyp	Ref	Expression
1		mapdh8a.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdh8a.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		mapdh8a.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		mapdh8a.s	⊢ − = ( -_g ‘ 𝑈 )
5		mapdh8a.o	⊢ 0 = ( 0_g ‘ 𝑈 )
6		mapdh8a.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7		mapdh8a.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8		mapdh8a.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
9		mapdh8a.r	⊢ 𝑅 = ( -_g ‘ 𝐶 )
10		mapdh8a.q	⊢ 𝑄 = ( 0_g ‘ 𝐶 )
11		mapdh8a.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
12		mapdh8a.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
13		mapdh8a.i	⊢ 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) − ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) )
14		mapdh8a.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15		mapdh8h.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
16		mapdh8h.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17		mapdh8i.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
18		mapdh8i.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
19		mapdh8i.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
20		mapdh8i.xy	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
21		mapdh8i.xz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
22		mapdh8i.yt	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
23		mapdh8i.zt	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
24		mapdh8.t	⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
25		fvexd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ V )
26	10 13 5 18 25	mapdhval0	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) , 0 ⟩ ) = 𝑄 )
27		fvexd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ∈ V )
28	10 13 5 19 27	mapdhval0	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑍 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) , 0 ⟩ ) = 𝑄 )
29	26 28	eqtr4d	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) , 0 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) , 0 ⟩ ) )
30	29	adantr	⊢ ( ( 𝜑 ∧ 𝑇 = 0 ) → ( 𝐼 ‘ ⟨ 𝑌 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) , 0 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) , 0 ⟩ ) )
31		oteq3	⊢ ( 𝑇 = 0 → ⟨ 𝑌 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) , 𝑇 ⟩ = ⟨ 𝑌 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) , 0 ⟩ )
32	31	fveq2d	⊢ ( 𝑇 = 0 → ( 𝐼 ‘ ⟨ 𝑌 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑌 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) , 0 ⟩ ) )
33	32	adantl	⊢ ( ( 𝜑 ∧ 𝑇 = 0 ) → ( 𝐼 ‘ ⟨ 𝑌 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑌 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) , 0 ⟩ ) )
34		oteq3	⊢ ( 𝑇 = 0 → ⟨ 𝑍 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) , 𝑇 ⟩ = ⟨ 𝑍 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) , 0 ⟩ )
35	34	fveq2d	⊢ ( 𝑇 = 0 → ( 𝐼 ‘ ⟨ 𝑍 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) , 0 ⟩ ) )
36	35	adantl	⊢ ( ( 𝜑 ∧ 𝑇 = 0 ) → ( 𝐼 ‘ ⟨ 𝑍 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) , 0 ⟩ ) )
37	30 33 36	3eqtr4d	⊢ ( ( 𝜑 ∧ 𝑇 = 0 ) → ( 𝐼 ‘ ⟨ 𝑌 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) , 𝑇 ⟩ ) )
38	14	adantr	⊢ ( ( 𝜑 ∧ 𝑇 ≠ 0 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
39	15	adantr	⊢ ( ( 𝜑 ∧ 𝑇 ≠ 0 ) → 𝐹 ∈ 𝐷 )
40	16	adantr	⊢ ( ( 𝜑 ∧ 𝑇 ≠ 0 ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
41	17	adantr	⊢ ( ( 𝜑 ∧ 𝑇 ≠ 0 ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
42	18	adantr	⊢ ( ( 𝜑 ∧ 𝑇 ≠ 0 ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
43	19	adantr	⊢ ( ( 𝜑 ∧ 𝑇 ≠ 0 ) → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
44	20	adantr	⊢ ( ( 𝜑 ∧ 𝑇 ≠ 0 ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
45	21	adantr	⊢ ( ( 𝜑 ∧ 𝑇 ≠ 0 ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
46	22	adantr	⊢ ( ( 𝜑 ∧ 𝑇 ≠ 0 ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
47	23	adantr	⊢ ( ( 𝜑 ∧ 𝑇 ≠ 0 ) → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
48	24	anim1i	⊢ ( ( 𝜑 ∧ 𝑇 ≠ 0 ) → ( 𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 ) )
49		eldifsn	⊢ ( 𝑇 ∈ ( 𝑉 ∖ { 0 } ) ↔ ( 𝑇 ∈ 𝑉 ∧ 𝑇 ≠ 0 ) )
50	48 49	sylibr	⊢ ( ( 𝜑 ∧ 𝑇 ≠ 0 ) → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
51	1 2 3 4 5 6 7 8 9 10 11 12 13 38 39 40 41 42 43 44 45 46 47 50	mapdh8j	⊢ ( ( 𝜑 ∧ 𝑇 ≠ 0 ) → ( 𝐼 ‘ ⟨ 𝑌 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) , 𝑇 ⟩ ) )
52	37 51	pm2.61dane	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑍 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) , 𝑇 ⟩ ) )

Metamath Proof Explorer

Theorem mapdh8

Proof