Metamath Proof Explorer

Theorem mapdh6bN

Description: Lemmma for mapdh6N . (Contributed by NM, 24-Apr-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 )
mapdh6b.y ⊢ ( 𝜑 → 𝑌 = 0 )
mapdh6b.z ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
mapdh6b.ne ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
Assertion mapdh6bN ( 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 mapdh6b.y ⊢ ( 𝜑 → 𝑌 = 0 )
21 mapdh6b.z ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
22 mapdh6b.ne ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
23 3 10 14 lcdlmod ⊢ ( 𝜑 → 𝐶 ∈ LMod )
24 lmodgrp ⊢ ( 𝐶 ∈ LMod → 𝐶 ∈ Grp )
25 23 24 syl ⊢ ( 𝜑 → 𝐶 ∈ Grp )
26 3 5 14 dvhlvec ⊢ ( 𝜑 → 𝑈 ∈ LVec )
27 17 eldifad ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
28 3 5 14 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
29 6 8 lmod0vcl ⊢ ( 𝑈 ∈ LMod → 0 ∈ 𝑉 )
30 28 29 syl ⊢ ( 𝜑 → 0 ∈ 𝑉 )
31 20 30 eqeltrd ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
32 6 9 26 27 31 21 22 lspindpi ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
33 32 simprd ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 21 33 mapdhcl ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ∈ 𝐷 )
35 11 19 1 grplid ⊢ ( ( 𝐶 ∈ Grp ∧ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ∈ 𝐷 ) → ( 𝑄 ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) )
36 25 34 35 syl2anc ⊢ ( 𝜑 → ( 𝑄 ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) )
37 20 oteq3d ⊢ ( 𝜑 → ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ = ⟨ 𝑋 , 𝐹 , 0 ⟩ )
38 37 fveq2d ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 0 ⟩ ) )
39 1 2 8 17 15 mapdhval0 ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 0 ⟩ ) = 𝑄 )
40 38 39 eqtrd ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝑄 )
41 40 oveq1d ⊢ ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) = ( 𝑄 ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
42 20 oveq1d ⊢ ( 𝜑 → ( 𝑌 + 𝑍 ) = ( 0 + 𝑍 ) )
43 lmodgrp ⊢ ( 𝑈 ∈ LMod → 𝑈 ∈ Grp )
44 28 43 syl ⊢ ( 𝜑 → 𝑈 ∈ Grp )
45 6 18 8 grplid ⊢ ( ( 𝑈 ∈ Grp ∧ 𝑍 ∈ 𝑉 ) → ( 0 + 𝑍 ) = 𝑍 )
46 44 21 45 syl2anc ⊢ ( 𝜑 → ( 0 + 𝑍 ) = 𝑍 )
47 42 46 eqtrd ⊢ ( 𝜑 → ( 𝑌 + 𝑍 ) = 𝑍 )
48 47 oteq3d ⊢ ( 𝜑 → ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ = ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ )
49 48 fveq2d ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) )
50 36 41 49 3eqtr4rd ⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ✚ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )