Metamath Proof Explorer


Theorem mapdheq

Description: Lemmma for ~? mapdh . The defining equation for h(x,x',y)=y' in part (2) in Baer p. 45 line 24. (Contributed by NM, 4-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 } ) )
mapdhe.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
mapdhe.g ( 𝜑𝐺𝐷 )
mapdh.ne2 ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
Assertion mapdheq ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) )

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 mapdhe.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
19 mapdhe.g ( 𝜑𝐺𝐷 )
20 mapdh.ne2 ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
21 1 2 17 15 18 mapdhval2 ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ) ) )
22 21 eqeq1d ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ↔ ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ) ) = 𝐺 ) )
23 3 4 5 6 7 8 9 10 11 12 13 14 17 18 15 20 16 mapdpg ( 𝜑 → ∃! 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ) )
24 nfv 𝜑
25 nfcvd ( 𝜑 𝐺 )
26 nfvd ( 𝜑 → Ⅎ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) )
27 sneq ( = 𝐺 → { } = { 𝐺 } )
28 27 fveq2d ( = 𝐺 → ( 𝐽 ‘ { } ) = ( 𝐽 ‘ { 𝐺 } ) )
29 28 eqeq2d ( = 𝐺 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ) )
30 oveq2 ( = 𝐺 → ( 𝐹 𝑅 ) = ( 𝐹 𝑅 𝐺 ) )
31 30 sneqd ( = 𝐺 → { ( 𝐹 𝑅 ) } = { ( 𝐹 𝑅 𝐺 ) } )
32 31 fveq2d ( = 𝐺 → ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) )
33 32 eqeq2d ( = 𝐺 → ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) )
34 29 33 anbi12d ( = 𝐺 → ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ) ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) )
35 34 adantl ( ( 𝜑 = 𝐺 ) → ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ) ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) )
36 24 25 26 19 35 riota2df ( ( 𝜑 ∧ ∃! 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ) ) → ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ↔ ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ) ) = 𝐺 ) )
37 23 36 mpdan ( 𝜑 → ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ↔ ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ) } ) ) ) = 𝐺 ) )
38 22 37 bitr4d ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) )