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Metamath Proof Explorer

Theorem tendoid

Description: The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013)

Ref Expression
Hypotheses tendoid.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
tendoid.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
tendoid.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion tendoid ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑆 ‘ ( I ↾ 𝐵 ) ) = ( I ↾ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 tendoid.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 tendoid.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 tendoid.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 1 2 4 idltrn ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ 𝐵 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
6 5 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( I ↾ 𝐵 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
7 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
8 eqid ⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
9 7 2 4 8 3 tendotp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( I ↾ 𝐵 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) ( le ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( I ↾ 𝐵 ) ) )
10 6 9 mpd3an3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) ( le ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( I ↾ 𝐵 ) ) )
11 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
12 1 11 2 8 trlid0 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( I ↾ 𝐵 ) ) = ( 0. ‘ 𝐾 ) )
13 12 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( I ↾ 𝐵 ) ) = ( 0. ‘ 𝐾 ) )
14 10 13 breqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) ( le ‘ 𝐾 ) ( 0. ‘ 𝐾 ) )
15 hlop ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP )
16 15 ad2antrr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → 𝐾 ∈ OP )
17 2 4 3 tendocl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( I ↾ 𝐵 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) → ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
18 6 17 mpd3an3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
19 1 2 4 8 trlcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) ∈ 𝐵 )
20 18 19 syldan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) ∈ 𝐵 )
21 1 7 11 ople0 ⊢ ( ( 𝐾 ∈ OP ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) ∈ 𝐵 ) → ( ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) ( le ‘ 𝐾 ) ( 0. ‘ 𝐾 ) ↔ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) = ( 0. ‘ 𝐾 ) ) )
22 16 20 21 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) ( le ‘ 𝐾 ) ( 0. ‘ 𝐾 ) ↔ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) = ( 0. ‘ 𝐾 ) ) )
23 14 22 mpbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) = ( 0. ‘ 𝐾 ) )
24 1 11 2 4 8 trlid0b ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ( 𝑆 ‘ ( I ↾ 𝐵 ) ) = ( I ↾ 𝐵 ) ↔ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) = ( 0. ‘ 𝐾 ) ) )
25 18 24 syldan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( ( 𝑆 ‘ ( I ↾ 𝐵 ) ) = ( I ↾ 𝐵 ) ↔ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ ( I ↾ 𝐵 ) ) ) = ( 0. ‘ 𝐾 ) ) )
26 23 25 mpbird ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → ( 𝑆 ‘ ( I ↾ 𝐵 ) ) = ( I ↾ 𝐵 ) )