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Metamath Proof Explorer

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 )