mapdh8g

Description: Part of Part (8) in Baer p. 48. Eliminate X e. ( N{ Y , T } ) . TODO: break out T =/= .0. in mapdh8e so we can share hypotheses. Also, look at hypothesis sharing for earlier mapdh8* and mapdh75* stuff. (Contributed by NM, 10-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 ) )
	mapdh8e.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	mapdh8e.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
	mapdh8e.eg	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
	mapdh8e.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8e.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8e.t	⊢ ( 𝜑 → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdh8e.xy	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
	mapdh8e.xt	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
	mapdh8e.yt	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
Assertion	mapdh8g	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 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		mapdh8e.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
16		mapdh8e.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17		mapdh8e.eg	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
18		mapdh8e.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
19		mapdh8e.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
20		mapdh8e.t	⊢ ( 𝜑 → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
21		mapdh8e.xy	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
22		mapdh8e.xt	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
23		mapdh8e.yt	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
24	14	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
25	15	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → 𝐹 ∈ 𝐷 )
26	16	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
27	17	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
28	18	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
29	19	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
30	20	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
31	21	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
32	22	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
33	23	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
34		simpr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) )
35	1 2 3 4 5 6 7 8 9 10 11 12 13 24 25 26 27 28 29 30 31 32 33 34	mapdh8e	⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑇 ⟩ ) )
36	14	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
37	15	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → 𝐹 ∈ 𝐷 )
38	16	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
39	17	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
40	18	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
41	19	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
42	23	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) )
43	20	adantr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → 𝑇 ∈ ( 𝑉 ∖ { 0 } ) )
44		simpr	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) )
45	1 2 3 4 5 6 7 8 9 10 11 12 13 36 37 38 39 40 41 42 43 44	mapdh8a	⊢ ( ( 𝜑 ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑇 ⟩ ) )
46	35 45	pm2.61dan	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑇 ⟩ ) )

Metamath Proof Explorer

Theorem mapdh8g

Proof