Metamath Proof Explorer

Theorem tendocnv

Description: Converse of a trace-preserving endomorphism value. (Contributed by NM, 7-Apr-2014)

Ref Expression
Hypotheses tendosp.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
tendosp.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
tendosp.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion tendocnv ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ◡ ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ ◡ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 tendosp.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 tendosp.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 tendosp.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
5 1 2 3 tendocl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑆 ‘ 𝐹 ) ∈ 𝑇 )
6 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
7 6 1 2 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ‘ 𝐹 ) ∈ 𝑇 ) → ( 𝑆 ‘ 𝐹 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
8 4 5 7 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑆 ‘ 𝐹 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
9 f1ococnv1 ⊢ ( ( 𝑆 ‘ 𝐹 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) → ( ◡ ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ 𝐹 ) ) = ( I ↾ ( Base ‘ 𝐾 ) ) )
10 8 9 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( ◡ ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ 𝐹 ) ) = ( I ↾ ( Base ‘ 𝐾 ) ) )
11 10 coeq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( ( ◡ ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ 𝐹 ) ) ∘ ◡ ( 𝑆 ‘ 𝐹 ) ) = ( ( I ↾ ( Base ‘ 𝐾 ) ) ∘ ◡ ( 𝑆 ‘ 𝐹 ) ) )
12 simp2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → 𝑆 ∈ 𝐸 )
13 6 1 3 tendoid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑆 ‘ ( I ↾ ( Base ‘ 𝐾 ) ) ) = ( I ↾ ( Base ‘ 𝐾 ) ) )
14 4 12 13 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑆 ‘ ( I ↾ ( Base ‘ 𝐾 ) ) ) = ( I ↾ ( Base ‘ 𝐾 ) ) )
15 6 1 2 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
16 15 3adant2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
17 f1ococnv2 ⊢ ( 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) → ( 𝐹 ∘ ◡ 𝐹 ) = ( I ↾ ( Base ‘ 𝐾 ) ) )
18 16 17 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ∘ ◡ 𝐹 ) = ( I ↾ ( Base ‘ 𝐾 ) ) )
19 18 fveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑆 ‘ ( 𝐹 ∘ ◡ 𝐹 ) ) = ( 𝑆 ‘ ( I ↾ ( Base ‘ 𝐾 ) ) ) )
20 f1ococnv2 ⊢ ( ( 𝑆 ‘ 𝐹 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) → ( ( 𝑆 ‘ 𝐹 ) ∘ ◡ ( 𝑆 ‘ 𝐹 ) ) = ( I ↾ ( Base ‘ 𝐾 ) ) )
21 8 20 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑆 ‘ 𝐹 ) ∘ ◡ ( 𝑆 ‘ 𝐹 ) ) = ( I ↾ ( Base ‘ 𝐾 ) ) )
22 14 19 21 3eqtr4rd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑆 ‘ 𝐹 ) ∘ ◡ ( 𝑆 ‘ 𝐹 ) ) = ( 𝑆 ‘ ( 𝐹 ∘ ◡ 𝐹 ) ) )
23 simp3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ 𝑇 )
24 1 2 ltrncnv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ◡ 𝐹 ∈ 𝑇 )
25 24 3adant2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ◡ 𝐹 ∈ 𝑇 )
26 1 2 3 tendospdi1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ ◡ 𝐹 ∈ 𝑇 ) ) → ( 𝑆 ‘ ( 𝐹 ∘ ◡ 𝐹 ) ) = ( ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ ◡ 𝐹 ) ) )
27 4 12 23 25 26 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑆 ‘ ( 𝐹 ∘ ◡ 𝐹 ) ) = ( ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ ◡ 𝐹 ) ) )
28 22 27 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑆 ‘ 𝐹 ) ∘ ◡ ( 𝑆 ‘ 𝐹 ) ) = ( ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ ◡ 𝐹 ) ) )
29 28 coeq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( ◡ ( 𝑆 ‘ 𝐹 ) ∘ ( ( 𝑆 ‘ 𝐹 ) ∘ ◡ ( 𝑆 ‘ 𝐹 ) ) ) = ( ◡ ( 𝑆 ‘ 𝐹 ) ∘ ( ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ ◡ 𝐹 ) ) ) )
30 coass ⊢ ( ( ◡ ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ 𝐹 ) ) ∘ ◡ ( 𝑆 ‘ 𝐹 ) ) = ( ◡ ( 𝑆 ‘ 𝐹 ) ∘ ( ( 𝑆 ‘ 𝐹 ) ∘ ◡ ( 𝑆 ‘ 𝐹 ) ) )
31 coass ⊢ ( ( ◡ ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ 𝐹 ) ) ∘ ( 𝑆 ‘ ◡ 𝐹 ) ) = ( ◡ ( 𝑆 ‘ 𝐹 ) ∘ ( ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ ◡ 𝐹 ) ) )
32 29 30 31 3eqtr4g ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( ( ◡ ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ 𝐹 ) ) ∘ ◡ ( 𝑆 ‘ 𝐹 ) ) = ( ( ◡ ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ 𝐹 ) ) ∘ ( 𝑆 ‘ ◡ 𝐹 ) ) )
33 10 coeq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( ( ◡ ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ 𝐹 ) ) ∘ ( 𝑆 ‘ ◡ 𝐹 ) ) = ( ( I ↾ ( Base ‘ 𝐾 ) ) ∘ ( 𝑆 ‘ ◡ 𝐹 ) ) )
34 1 2 3 tendocl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ◡ 𝐹 ∈ 𝑇 ) → ( 𝑆 ‘ ◡ 𝐹 ) ∈ 𝑇 )
35 25 34 syld3an3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑆 ‘ ◡ 𝐹 ) ∈ 𝑇 )
36 6 1 2 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ‘ ◡ 𝐹 ) ∈ 𝑇 ) → ( 𝑆 ‘ ◡ 𝐹 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
37 4 35 36 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑆 ‘ ◡ 𝐹 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
38 f1of ⊢ ( ( 𝑆 ‘ ◡ 𝐹 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) → ( 𝑆 ‘ ◡ 𝐹 ) : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) )
39 fcoi2 ⊢ ( ( 𝑆 ‘ ◡ 𝐹 ) : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) → ( ( I ↾ ( Base ‘ 𝐾 ) ) ∘ ( 𝑆 ‘ ◡ 𝐹 ) ) = ( 𝑆 ‘ ◡ 𝐹 ) )
40 37 38 39 3syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( ( I ↾ ( Base ‘ 𝐾 ) ) ∘ ( 𝑆 ‘ ◡ 𝐹 ) ) = ( 𝑆 ‘ ◡ 𝐹 ) )
41 32 33 40 3eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( ( ◡ ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ 𝐹 ) ) ∘ ◡ ( 𝑆 ‘ 𝐹 ) ) = ( 𝑆 ‘ ◡ 𝐹 ) )
42 1 2 ltrncnv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ‘ 𝐹 ) ∈ 𝑇 ) → ◡ ( 𝑆 ‘ 𝐹 ) ∈ 𝑇 )
43 4 5 42 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ◡ ( 𝑆 ‘ 𝐹 ) ∈ 𝑇 )
44 6 1 2 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ◡ ( 𝑆 ‘ 𝐹 ) ∈ 𝑇 ) → ◡ ( 𝑆 ‘ 𝐹 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
45 4 43 44 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ◡ ( 𝑆 ‘ 𝐹 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
46 f1of ⊢ ( ◡ ( 𝑆 ‘ 𝐹 ) : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) → ◡ ( 𝑆 ‘ 𝐹 ) : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) )
47 fcoi2 ⊢ ( ◡ ( 𝑆 ‘ 𝐹 ) : ( Base ‘ 𝐾 ) ⟶ ( Base ‘ 𝐾 ) → ( ( I ↾ ( Base ‘ 𝐾 ) ) ∘ ◡ ( 𝑆 ‘ 𝐹 ) ) = ◡ ( 𝑆 ‘ 𝐹 ) )
48 45 46 47 3syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( ( I ↾ ( Base ‘ 𝐾 ) ) ∘ ◡ ( 𝑆 ‘ 𝐹 ) ) = ◡ ( 𝑆 ‘ 𝐹 ) )
49 11 41 48 3eqtr3rd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ◡ ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ ◡ 𝐹 ) )