Metamath Proof Explorer

Theorem mapdheq2

Description: Lemmma for ~? mapdh . One direction of part (2) in Baer p. 45. (Contributed by NM, 4-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 } ) )
mapdhe.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
mapdhe.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐷 )
mapdh.ne2 ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
Assertion mapdheq2 ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑋 ⟩ ) = 𝐹 ) )

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 mapdhe.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
19 mapdhe.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐷 )
20 mapdh.ne2 ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 mapdheq ⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) )
22 16 adantr ⊢ ( ( 𝜑 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
23 3 5 14 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
24 17 eldifad ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
25 18 eldifad ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
26 6 7 9 23 24 25 lspsnsub ⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) = ( 𝑁 ‘ { ( 𝑌 − 𝑋 ) } ) )
27 26 fveq2d ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 − 𝑋 ) } ) ) )
28 3 10 14 lcdlmod ⊢ ( 𝜑 → 𝐶 ∈ LMod )
29 11 12 13 28 15 19 lspsnsub ⊢ ( 𝜑 → ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝐹 ) } ) )
30 27 29 eqeq12d ⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 − 𝑋 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝐹 ) } ) ) )
31 30 biimpa ⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 − 𝑋 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝐹 ) } ) )
32 31 adantrl ⊢ ( ( 𝜑 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 − 𝑋 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝐹 ) } ) )
33 14 adantr ⊢ ( ( 𝜑 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
34 19 adantr ⊢ ( ( 𝜑 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) → 𝐺 ∈ 𝐷 )
35 simprl ⊢ ( ( 𝜑 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
36 18 adantr ⊢ ( ( 𝜑 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
37 17 adantr ⊢ ( ( 𝜑 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
38 15 adantr ⊢ ( ( 𝜑 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) → 𝐹 ∈ 𝐷 )
39 20 necomd ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
40 39 adantr ⊢ ( ( 𝜑 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
41 1 2 3 4 5 6 7 8 9 10 11 12 13 33 34 35 36 37 38 40 mapdheq ⊢ ( ( 𝜑 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) → ( ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑋 ⟩ ) = 𝐹 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 − 𝑋 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝐹 ) } ) ) ) )
42 22 32 41 mpbir2and ⊢ ( ( 𝜑 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑋 ⟩ ) = 𝐹 )
43 42 ex ⊢ ( 𝜑 → ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑋 ⟩ ) = 𝐹 ) )
44 21 43 sylbid ⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑋 ⟩ ) = 𝐹 ) )