Theorem mapdh6eN

Description: Lemmma for mapdh6N . Part (6) in Baer p. 47 line 38. (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 )
	mapdh6d.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
	mapdh6d.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
	mapdh6d.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh6d.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh6d.w	⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh6d.wn	⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
Assertion	mapdh6eN	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( ( 𝑤 + 𝑌 ) + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + 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		mapdh6d.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
21		mapdh6d.yz	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
22		mapdh6d.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
23		mapdh6d.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
24		mapdh6d.w	⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
25		mapdh6d.wn	⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
26	3 5 14	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
27	24	eldifad	⊢ ( 𝜑 → 𝑤 ∈ 𝑉 )
28	22	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
29	6 18	lmodvacl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑤 + 𝑌 ) ∈ 𝑉 )
30	26 27 28 29	syl3anc	⊢ ( 𝜑 → ( 𝑤 + 𝑌 ) ∈ 𝑉 )
31	3 5 14	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
32	17	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
33	6 9 31 27 32 28 25	lspindpi	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) )
34	33	simprd	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
35	6 18 8 9 26 27 28 34	lmodindp1	⊢ ( 𝜑 → ( 𝑤 + 𝑌 ) ≠ 0 )
36		eldifsn	⊢ ( ( 𝑤 + 𝑌 ) ∈ ( 𝑉 ∖ { 0 } ) ↔ ( ( 𝑤 + 𝑌 ) ∈ 𝑉 ∧ ( 𝑤 + 𝑌 ) ≠ 0 ) )
37	30 35 36	sylanbrc	⊢ ( 𝜑 → ( 𝑤 + 𝑌 ) ∈ ( 𝑉 ∖ { 0 } ) )
38	23	eldifad	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
39	6 9 31 32 28 38 20	lspindpi	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
40	39	simpld	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
41	6 18 8 9 31 17 22 23 24 21 40 25	mapdindp3	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) )
42	6 18 8 9 31 17 22 23 24 21 40 25	mapdindp4	⊢ ( 𝜑 → ¬ 𝑍 ∈ ( 𝑁 ‘ { 𝑋 , ( 𝑤 + 𝑌 ) } ) )
43	6 8 9 31 17 30 38 41 42	lspindp1	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑍 , ( 𝑤 + 𝑌 ) } ) ) )
44	43	simprd	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑍 , ( 𝑤 + 𝑌 ) } ) )
45		prcom	⊢ { ( 𝑤 + 𝑌 ) , 𝑍 } = { 𝑍 , ( 𝑤 + 𝑌 ) }
46	45	fveq2i	⊢ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) , 𝑍 } ) = ( 𝑁 ‘ { 𝑍 , ( 𝑤 + 𝑌 ) } )
47	46	eleq2i	⊢ ( 𝑋 ∈ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) , 𝑍 } ) ↔ 𝑋 ∈ ( 𝑁 ‘ { 𝑍 , ( 𝑤 + 𝑌 ) } ) )
48	44 47	sylnibr	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) , 𝑍 } ) )
49	6 9 31 38 32 30 42	lspindpi	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) ) )
50	49	simprd	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) )
51	50	necomd	⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
52		eqidd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + 𝑌 ) ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + 𝑌 ) ⟩ ) )
53		eqidd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) )
54	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 37 23 48 51 52 53	mapdh6aN	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( ( 𝑤 + 𝑌 ) + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + 𝑌 ) ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )