Theorem mapdheq4

Description: Lemma for ~? mapdh . Part (4) in Baer p. 46. (Contributed by NM, 12-Apr-2015)

	Ref	Expression
Hypotheses	mapdh.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
	mapdh.i	⊢ 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) − ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) )
	mapdh.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdh.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdh.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdh.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	mapdh.s	⊢ − = ( -g ‘ 𝑈 )
	mapdhc.o	⊢ 0 = ( 0g ‘ 𝑈 )
	mapdh.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	mapdh.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	mapdh.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
	mapdh.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
	mapdh.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	mapdh.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdhc.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	mapdh.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
	mapdhcl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdhe4.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdhe.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
	mapdh.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
	mapdh.eg	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
	mapdh.ee	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
Assertion	mapdheq4	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑍 ⟩ ) = 𝐸 )

Proof

Step	Hyp	Ref	Expression
1		mapdh.q	⊢ 𝑄 = ( 0g ‘ 𝐶 )
2		mapdh.i	⊢ 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) − ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) )
3		mapdh.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		mapdh.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
5		mapdh.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
6		mapdh.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
7		mapdh.s	⊢ − = ( -g ‘ 𝑈 )
8		mapdhc.o	⊢ 0 = ( 0g ‘ 𝑈 )
9		mapdh.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
10		mapdh.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
11		mapdh.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
12		mapdh.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
13		mapdh.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
14		mapdh.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15		mapdhc.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
16		mapdh.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17		mapdhcl.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
18		mapdhe4.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
19		mapdhe.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
20		mapdh.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
21		mapdh.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
22		mapdh.eg	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
23		mapdh.ee	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
24	19	eldifad	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
25	3 5 14	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
26	17	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
27	6 8 9 25 18 24 26 21 20	lspindp1	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ∧ ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) )
28	27	simpld	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
29	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 24 28	mapdhcl	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ∈ 𝐷 )
30	23 29	eqeltrrd	⊢ ( 𝜑 → 𝐸 ∈ 𝐷 )
31	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19 30 28	mapdheq	⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) = ( 𝐽 ‘ { 𝐸 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑍 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐸 ) } ) ) ) )
32	23 31	mpbid	⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) = ( 𝐽 ‘ { 𝐸 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑍 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐸 ) } ) ) )
33	32	simpld	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) = ( 𝐽 ‘ { 𝐸 } ) )
34	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23	mapdheq4lem	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 − 𝑍 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝐸 ) } ) )
35	18	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
36	6 8 9 25 35 19 26 21 20	lspindp2	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ¬ 𝑍 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) )
37	36	simpld	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
38	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 35 37	mapdhcl	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )
39	22 38	eqeltrrd	⊢ ( 𝜑 → 𝐺 ∈ 𝐷 )
40	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 39 37	mapdheq	⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) )
41	22 40	mpbid	⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) )
42	41	simpld	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
43	1 2 3 4 5 6 7 8 9 10 11 12 13 14 39 42 18 19 30 21	mapdheq	⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑍 ⟩ ) = 𝐸 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) = ( 𝐽 ‘ { 𝐸 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 − 𝑍 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝐸 ) } ) ) ) )
44	33 34 43	mpbir2and	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑍 ⟩ ) = 𝐸 )