Theorem tendoeq1

Description: Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013)

	Ref	Expression
Hypotheses	tendof.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	tendof.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	tendof.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion	tendoeq1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) ) → 𝑈 = 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		tendof.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		tendof.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		tendof.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4		simp3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) ) → ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) )
5		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6		simp2l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) ) → 𝑈 ∈ 𝐸 )
7	1 2 3	tendof	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ) → 𝑈 : 𝑇 ⟶ 𝑇 )
8	5 6 7	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) ) → 𝑈 : 𝑇 ⟶ 𝑇 )
9	8	ffnd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) ) → 𝑈 Fn 𝑇 )
10		simp2r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) ) → 𝑉 ∈ 𝐸 )
11	1 2 3	tendof	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑉 ∈ 𝐸 ) → 𝑉 : 𝑇 ⟶ 𝑇 )
12	5 10 11	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) ) → 𝑉 : 𝑇 ⟶ 𝑇 )
13	12	ffnd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) ) → 𝑉 Fn 𝑇 )
14		eqfnfv	⊢ ( ( 𝑈 Fn 𝑇 ∧ 𝑉 Fn 𝑇 ) → ( 𝑈 = 𝑉 ↔ ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) ) )
15	9 13 14	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) ) → ( 𝑈 = 𝑉 ↔ ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) ) )
16	4 15	mpbird	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑈 ‘ 𝑓 ) = ( 𝑉 ‘ 𝑓 ) ) → 𝑈 = 𝑉 )