Theorem mapdhcl

Description: Lemmma for ~? mapdh . (Contributed by NM, 3-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 } ) )
	mapdhc.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
	mapdh.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
Assertion	mapdhcl	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )

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		mapdhc.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
19		mapdh.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
20		oteq3	⊢ ( 𝑌 = 0 → ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ = ⟨ 𝑋 , 𝐹 , 0 ⟩ )
21	20	fveq2d	⊢ ( 𝑌 = 0 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 0 ⟩ ) )
22	21	eleq1d	⊢ ( 𝑌 = 0 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 ↔ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 0 ⟩ ) ∈ 𝐷 ) )
23	17	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 0 ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
24	15	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 0 ) → 𝐹 ∈ 𝐷 )
25	18	anim1i	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 0 ) → ( 𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ) )
26		eldifsn	⊢ ( 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ↔ ( 𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ) )
27	25 26	sylibr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 0 ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
28	1 2 23 24 27	mapdhval2	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) )
29	14	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 0 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
30	19	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 0 ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
31	16	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 0 ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
32	3 4 5 6 7 8 9 10 11 12 13 29 23 27 24 30 31	mapdpg	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 0 ) → ∃! ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) )
33		riotacl	⊢ ( ∃! ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) → ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) ∈ 𝐷 )
34	32 33	syl	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 0 ) → ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) ∈ 𝐷 )
35	28 34	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )
36	1 2 8 17 15	mapdhval0	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 0 ⟩ ) = 𝑄 )
37	3 10 11 1 14	lcd0vcl	⊢ ( 𝜑 → 𝑄 ∈ 𝐷 )
38	36 37	eqeltrd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 0 ⟩ ) ∈ 𝐷 )
39	22 35 38	pm2.61ne	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )