Theorem mapdpglem31

Description: Lemma for mapdpg . Baer p. 45 line 19: "...and we have consequently that y' = y'', as we claimed." (Contributed by NM, 23-Mar-2015)

	Ref	Expression
Hypotheses	mapdpg.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdpg.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdpg.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdpg.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	mapdpg.s	⊢ − = ( -g ‘ 𝑈 )
	mapdpg.z	⊢ 0 = ( 0g ‘ 𝑈 )
	mapdpg.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	mapdpg.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	mapdpg.f	⊢ 𝐹 = ( Base ‘ 𝐶 )
	mapdpg.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
	mapdpg.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	mapdpg.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdpg.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdpg.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	mapdpg.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
	mapdpg.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
	mapdpg.e	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
	mapdpgem25.h1	⊢ ( 𝜑 → ( ℎ ∈ 𝐹 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ) )
	mapdpgem25.i1	⊢ ( 𝜑 → ( 𝑖 ∈ 𝐹 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) )
	mapdpglem26.a	⊢ 𝐴 = ( Scalar ‘ 𝑈 )
	mapdpglem26.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
	mapdpglem26.t	⊢ · = ( ·𝑠 ‘ 𝐶 )
	mapdpglem26.o	⊢ 𝑂 = ( 0g ‘ 𝐴 )
	mapdpglem28.ve	⊢ ( 𝜑 → 𝑣 ∈ 𝐵 )
	mapdpglem28.u1	⊢ ( 𝜑 → ℎ = ( 𝑢 · 𝑖 ) )
	mapdpglem28.u2	⊢ ( 𝜑 → ( 𝐺 𝑅 ℎ ) = ( 𝑣 · ( 𝐺 𝑅 𝑖 ) ) )
	mapdpglem28.ue	⊢ ( 𝜑 → 𝑢 ∈ 𝐵 )
Assertion	mapdpglem31	⊢ ( 𝜑 → ℎ = 𝑖 )

Proof

Step	Hyp	Ref	Expression
1		mapdpg.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdpg.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
3		mapdpg.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		mapdpg.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		mapdpg.s	⊢ − = ( -g ‘ 𝑈 )
6		mapdpg.z	⊢ 0 = ( 0g ‘ 𝑈 )
7		mapdpg.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
8		mapdpg.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
9		mapdpg.f	⊢ 𝐹 = ( Base ‘ 𝐶 )
10		mapdpg.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
11		mapdpg.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
12		mapdpg.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
13		mapdpg.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
14		mapdpg.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
15		mapdpg.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
16		mapdpg.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
17		mapdpg.e	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
18		mapdpgem25.h1	⊢ ( 𝜑 → ( ℎ ∈ 𝐹 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ) )
19		mapdpgem25.i1	⊢ ( 𝜑 → ( 𝑖 ∈ 𝐹 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) )
20		mapdpglem26.a	⊢ 𝐴 = ( Scalar ‘ 𝑈 )
21		mapdpglem26.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
22		mapdpglem26.t	⊢ · = ( ·𝑠 ‘ 𝐶 )
23		mapdpglem26.o	⊢ 𝑂 = ( 0g ‘ 𝐴 )
24		mapdpglem28.ve	⊢ ( 𝜑 → 𝑣 ∈ 𝐵 )
25		mapdpglem28.u1	⊢ ( 𝜑 → ℎ = ( 𝑢 · 𝑖 ) )
26		mapdpglem28.u2	⊢ ( 𝜑 → ( 𝐺 𝑅 ℎ ) = ( 𝑣 · ( 𝐺 𝑅 𝑖 ) ) )
27		mapdpglem28.ue	⊢ ( 𝜑 → 𝑢 ∈ 𝐵 )
28		eqid	⊢ ( 1r ‘ 𝐴 ) = ( 1r ‘ 𝐴 )
29		eqid	⊢ ( Scalar ‘ 𝐶 ) = ( Scalar ‘ 𝐶 )
30		eqid	⊢ ( 1r ‘ ( Scalar ‘ 𝐶 ) ) = ( 1r ‘ ( Scalar ‘ 𝐶 ) )
31	1 3 20 28 8 29 30 12	lcd1	⊢ ( 𝜑 → ( 1r ‘ ( Scalar ‘ 𝐶 ) ) = ( 1r ‘ 𝐴 ) )
32	31	oveq1d	⊢ ( 𝜑 → ( ( 1r ‘ ( Scalar ‘ 𝐶 ) ) · 𝑖 ) = ( ( 1r ‘ 𝐴 ) · 𝑖 ) )
33	1 8 12	lcdlmod	⊢ ( 𝜑 → 𝐶 ∈ LMod )
34	19	simpld	⊢ ( 𝜑 → 𝑖 ∈ 𝐹 )
35	9 29 22 30	lmodvs1	⊢ ( ( 𝐶 ∈ LMod ∧ 𝑖 ∈ 𝐹 ) → ( ( 1r ‘ ( Scalar ‘ 𝐶 ) ) · 𝑖 ) = 𝑖 )
36	33 34 35	syl2anc	⊢ ( 𝜑 → ( ( 1r ‘ ( Scalar ‘ 𝐶 ) ) · 𝑖 ) = 𝑖 )
37	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27	mapdpglem30	⊢ ( 𝜑 → ( 𝑣 = ( 1r ‘ 𝐴 ) ∧ 𝑣 = 𝑢 ) )
38		eqtr2	⊢ ( ( 𝑣 = ( 1r ‘ 𝐴 ) ∧ 𝑣 = 𝑢 ) → ( 1r ‘ 𝐴 ) = 𝑢 )
39	37 38	syl	⊢ ( 𝜑 → ( 1r ‘ 𝐴 ) = 𝑢 )
40	39	oveq1d	⊢ ( 𝜑 → ( ( 1r ‘ 𝐴 ) · 𝑖 ) = ( 𝑢 · 𝑖 ) )
41	32 36 40	3eqtr3rd	⊢ ( 𝜑 → ( 𝑢 · 𝑖 ) = 𝑖 )
42	25 41	eqtrd	⊢ ( 𝜑 → ℎ = 𝑖 )