Theorem mapdh6iN

Description: Lemmma for mapdh6N . Eliminate auxiliary vector w . (Contributed by NM, 1-May-2015) (New usage is discouraged.)

	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 } ) )
	mapdh.p	⊢ + = ( +g ‘ 𝑈 )
	mapdh.a	⊢ ✚ = ( +g ‘ 𝐶 )
	mapdh6i.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
	mapdh6i.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh6i.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh6i.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
Assertion	mapdh6iN	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )

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		mapdh.p	⊢ + = ( +g ‘ 𝑈 )
19		mapdh.a	⊢ ✚ = ( +g ‘ 𝐶 )
20		mapdh6i.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
21		mapdh6i.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
22		mapdh6i.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
23		mapdh6i.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
24	17	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
25	21	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
26	3 5 6 9 14 24 25	dvh3dim	⊢ ( 𝜑 → ∃ 𝑤 ∈ 𝑉 ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
27	14	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
28	15	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝐹 ∈ 𝐷 )
29	16	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
30	17	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
31	20	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
32	23	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
33	21	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
34	22	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
35		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
36	3 5 14	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
37	36	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑈 ∈ LMod )
38	6 35 9 36 24 25	lspprcl	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
39	38	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
40		simp2	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑤 ∈ 𝑉 )
41		simp3	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
42	8 35 37 39 40 41	lssneln0	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
43	1 2 3 4 5 6 7 8 9 10 11 12 13 27 28 29 30 18 19 31 32 33 34 42 41	mapdh6hN	⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
44	43	rexlimdv3a	⊢ ( 𝜑 → ( ∃ 𝑤 ∈ 𝑉 ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) )
45	26 44	mpd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )