Metamath Proof Explorer


Theorem mapdh6cN

Description: Lemmma for mapdh6N . (Contributed by NM, 24-Apr-2015) (New usage is discouraged.)

Ref Expression
Hypotheses mapdh.q 𝑄 = ( 0g𝐶 )
mapdh.i 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd𝑥 ) = 0 , 𝑄 , ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd𝑥 ) } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st𝑥 ) ) ( 2nd𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st𝑥 ) ) 𝑅 ) } ) ) ) ) )
mapdh.h 𝐻 = ( LHyp ‘ 𝐾 )
mapdh.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
mapdh.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
mapdh.v 𝑉 = ( Base ‘ 𝑈 )
mapdh.s = ( -g𝑈 )
mapdhc.o 0 = ( 0g𝑈 )
mapdh.n 𝑁 = ( LSpan ‘ 𝑈 )
mapdh.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
mapdh.d 𝐷 = ( Base ‘ 𝐶 )
mapdh.r 𝑅 = ( -g𝐶 )
mapdh.j 𝐽 = ( LSpan ‘ 𝐶 )
mapdh.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
mapdhc.f ( 𝜑𝐹𝐷 )
mapdh.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
mapdhcl.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh.p + = ( +g𝑈 )
mapdh.a = ( +g𝐶 )
mapdh6c.y ( 𝜑𝑌𝑉 )
mapdh6c.z ( 𝜑𝑍 = 0 )
mapdh6c.ne ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
Assertion mapdh6cN ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )

Proof

Step Hyp Ref Expression
1 mapdh.q 𝑄 = ( 0g𝐶 )
2 mapdh.i 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd𝑥 ) = 0 , 𝑄 , ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd𝑥 ) } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st𝑥 ) ) ( 2nd𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st𝑥 ) ) 𝑅 ) } ) ) ) ) )
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 = ( -g𝑈 )
8 mapdhc.o 0 = ( 0g𝑈 )
9 mapdh.n 𝑁 = ( LSpan ‘ 𝑈 )
10 mapdh.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
11 mapdh.d 𝐷 = ( Base ‘ 𝐶 )
12 mapdh.r 𝑅 = ( -g𝐶 )
13 mapdh.j 𝐽 = ( LSpan ‘ 𝐶 )
14 mapdh.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
15 mapdhc.f ( 𝜑𝐹𝐷 )
16 mapdh.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17 mapdhcl.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
18 mapdh.p + = ( +g𝑈 )
19 mapdh.a = ( +g𝐶 )
20 mapdh6c.y ( 𝜑𝑌𝑉 )
21 mapdh6c.z ( 𝜑𝑍 = 0 )
22 mapdh6c.ne ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
23 3 10 14 lcdlmod ( 𝜑𝐶 ∈ LMod )
24 lmodgrp ( 𝐶 ∈ LMod → 𝐶 ∈ Grp )
25 23 24 syl ( 𝜑𝐶 ∈ Grp )
26 3 5 14 dvhlvec ( 𝜑𝑈 ∈ LVec )
27 17 eldifad ( 𝜑𝑋𝑉 )
28 3 5 14 dvhlmod ( 𝜑𝑈 ∈ LMod )
29 6 8 lmod0vcl ( 𝑈 ∈ LMod → 0𝑉 )
30 28 29 syl ( 𝜑0𝑉 )
31 21 30 eqeltrd ( 𝜑𝑍𝑉 )
32 6 9 26 27 20 31 22 lspindpi ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
33 32 simpld ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 20 33 mapdhcl ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )
35 11 19 1 grprid ( ( 𝐶 ∈ Grp ∧ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 ) → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) 𝑄 ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) )
36 25 34 35 syl2anc ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) 𝑄 ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) )
37 21 oteq3d ( 𝜑 → ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ = ⟨ 𝑋 , 𝐹 , 0 ⟩ )
38 37 fveq2d ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = ( 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 , 𝑍 ⟩ ) = 𝑄 )
41 40 oveq2d ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) 𝑄 ) )
42 21 oveq2d ( 𝜑 → ( 𝑌 + 𝑍 ) = ( 𝑌 + 0 ) )
43 lmodgrp ( 𝑈 ∈ LMod → 𝑈 ∈ Grp )
44 28 43 syl ( 𝜑𝑈 ∈ Grp )
45 6 18 8 grprid ( ( 𝑈 ∈ Grp ∧ 𝑌𝑉 ) → ( 𝑌 + 0 ) = 𝑌 )
46 44 20 45 syl2anc ( 𝜑 → ( 𝑌 + 0 ) = 𝑌 )
47 42 46 eqtrd ( 𝜑 → ( 𝑌 + 𝑍 ) = 𝑌 )
48 47 oteq3d ( 𝜑 → ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ = ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ )
49 48 fveq2d ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) )
50 36 41 49 3eqtr4rd ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )