# Metamath Proof Explorer

## Theorem mapdhcl

Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015)

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 } ) )
mapdhc.y ( 𝜑𝑌𝑉 )
mapdh.ne ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
Assertion mapdhcl ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )

### 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 mapdhc.y ( 𝜑𝑌𝑉 )
19 mapdh.ne ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
20 oteq3 ( 𝑌 = 0 → ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ = ⟨ 𝑋 , 𝐹 , 0 ⟩ )
21 20 fveq2d ( 𝑌 = 0 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 0 ⟩ ) )
22 21 eleq1d ( 𝑌 = 0 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 ↔ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 0 ⟩ ) ∈ 𝐷 ) )
23 17 adantr ( ( 𝜑𝑌0 ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
24 15 adantr ( ( 𝜑𝑌0 ) → 𝐹𝐷 )
25 18 anim1i ( ( 𝜑𝑌0 ) → ( 𝑌𝑉𝑌0 ) )
26 eldifsn ( 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ↔ ( 𝑌𝑉𝑌0 ) )
27 25 26 sylibr ( ( 𝜑𝑌0 ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
28 1 2 23 24 27 mapdhval2 ( ( 𝜑𝑌0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ) ) )
29 14 adantr ( ( 𝜑𝑌0 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
30 19 adantr ( ( 𝜑𝑌0 ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
31 16 adantr ( ( 𝜑𝑌0 ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
32 3 4 5 6 7 8 9 10 11 12 13 29 23 27 24 30 31 mapdpg ( ( 𝜑𝑌0 ) → ∃! 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ) )
33 riotacl ( ∃! 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ) → ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ) ) ∈ 𝐷 )
34 32 33 syl ( ( 𝜑𝑌0 ) → ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ) ) ∈ 𝐷 )
35 28 34 eqeltrd ( ( 𝜑𝑌0 ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )
36 1 2 8 17 15 mapdhval0 ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 0 ⟩ ) = 𝑄 )
37 3 10 11 1 14 lcd0vcl ( 𝜑𝑄𝐷 )
38 36 37 eqeltrd ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 0 ⟩ ) ∈ 𝐷 )
39 22 35 38 pm2.61ne ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )