Theorem hvmap1o2

Description: The vector to functional map provides a bijection from nonzero vectors V to nonzero functionals with closed kernels C . (Contributed by NM, 27-Mar-2015)

	Ref	Expression
Hypotheses	hvmap1o2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hvmap1o2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hvmap1o2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hvmap1o2.z	⊢ 0 = ( 0g ‘ 𝑈 )
	hvmap1o2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hvmap1o2.f	⊢ 𝐹 = ( Base ‘ 𝐶 )
	hvmap1o2.o	⊢ 𝑂 = ( 0g ‘ 𝐶 )
	hvmap1o2.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
	hvmap1o2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion	hvmap1o2	⊢ ( 𝜑 → 𝑀 : ( 𝑉 ∖ { 0 } ) –1-1-onto→ ( 𝐹 ∖ { 𝑂 } ) )

Proof

Step	Hyp	Ref	Expression
1		hvmap1o2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hvmap1o2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hvmap1o2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hvmap1o2.z	⊢ 0 = ( 0g ‘ 𝑈 )
5		hvmap1o2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
6		hvmap1o2.f	⊢ 𝐹 = ( Base ‘ 𝐶 )
7		hvmap1o2.o	⊢ 𝑂 = ( 0g ‘ 𝐶 )
8		hvmap1o2.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
9		hvmap1o2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10		eqid	⊢ ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
11		eqid	⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
12		eqid	⊢ ( LKer ‘ 𝑈 ) = ( LKer ‘ 𝑈 )
13		eqid	⊢ ( LDual ‘ 𝑈 ) = ( LDual ‘ 𝑈 )
14		eqid	⊢ ( 0g ‘ ( LDual ‘ 𝑈 ) ) = ( 0g ‘ ( LDual ‘ 𝑈 ) )
15		eqid	⊢ { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) } = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) }
16	1 10 2 3 4 11 12 13 14 15 8 9	hvmap1o	⊢ ( 𝜑 → 𝑀 : ( 𝑉 ∖ { 0 } ) –1-1-onto→ ( { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) } ∖ { ( 0g ‘ ( LDual ‘ 𝑈 ) ) } ) )
17	1 10 5 6 2 11 12 15 9	lcdvbase	⊢ ( 𝜑 → 𝐹 = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) } )
18	1 2 13 14 5 7 9	lcd0v2	⊢ ( 𝜑 → 𝑂 = ( 0g ‘ ( LDual ‘ 𝑈 ) ) )
19	18	sneqd	⊢ ( 𝜑 → { 𝑂 } = { ( 0g ‘ ( LDual ‘ 𝑈 ) ) } )
20	17 19	difeq12d	⊢ ( 𝜑 → ( 𝐹 ∖ { 𝑂 } ) = ( { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) } ∖ { ( 0g ‘ ( LDual ‘ 𝑈 ) ) } ) )
21		f1oeq3	⊢ ( ( 𝐹 ∖ { 𝑂 } ) = ( { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) } ∖ { ( 0g ‘ ( LDual ‘ 𝑈 ) ) } ) → ( 𝑀 : ( 𝑉 ∖ { 0 } ) –1-1-onto→ ( 𝐹 ∖ { 𝑂 } ) ↔ 𝑀 : ( 𝑉 ∖ { 0 } ) –1-1-onto→ ( { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) } ∖ { ( 0g ‘ ( LDual ‘ 𝑈 ) ) } ) ) )
22	20 21	syl	⊢ ( 𝜑 → ( 𝑀 : ( 𝑉 ∖ { 0 } ) –1-1-onto→ ( 𝐹 ∖ { 𝑂 } ) ↔ 𝑀 : ( 𝑉 ∖ { 0 } ) –1-1-onto→ ( { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑓 ) } ∖ { ( 0g ‘ ( LDual ‘ 𝑈 ) ) } ) ) )
23	16 22	mpbird	⊢ ( 𝜑 → 𝑀 : ( 𝑉 ∖ { 0 } ) –1-1-onto→ ( 𝐹 ∖ { 𝑂 } ) )