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