Metamath Proof Explorer

Theorem tendocan

Description: Cancellation law: if the values of two trace-preserving endormorphisms are equal, so are the endormorphisms. Lemma J of Crawley p. 118. (Contributed by NM, 21-Jun-2013)

Ref Expression
Hypotheses tendocan.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
tendocan.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
tendocan.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
tendocan.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion tendocan ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → 𝑈 = 𝑉 )

Proof

Step Hyp Ref Expression
1 tendocan.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 tendocan.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 tendocan.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 tendocan.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
5 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → 𝐾 ∈ HL )
6 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → 𝑊 ∈ 𝐻 )
7 simp21 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → 𝑈 ∈ 𝐸 )
8 simp22 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → 𝑉 ∈ 𝐸 )
9 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) )
11 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → 𝐹 ∈ 𝑇 )
12 simp13r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → 𝐹 ≠ ( I ↾ 𝐵 ) )
13 simp2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ℎ ∈ 𝑇 )
14 11 12 13 3jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ∧ ℎ ∈ 𝑇 ) )
15 simp3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ℎ ≠ ( I ↾ 𝐵 ) )
16 eqid ⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
17 1 2 3 16 4 cdlemj3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ∧ ℎ ∈ 𝑇 ) ) ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ( 𝑈 ‘ ℎ ) = ( 𝑉 ‘ ℎ ) )
18 9 10 14 15 17 syl31anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) ∧ ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵 ) ) → ( 𝑈 ‘ ℎ ) = ( 𝑉 ‘ ℎ ) )
19 18 3exp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → ( ℎ ∈ 𝑇 → ( ℎ ≠ ( I ↾ 𝐵 ) → ( 𝑈 ‘ ℎ ) = ( 𝑉 ‘ ℎ ) ) ) )
20 19 ralrimiv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → ∀ ℎ ∈ 𝑇 ( ℎ ≠ ( I ↾ 𝐵 ) → ( 𝑈 ‘ ℎ ) = ( 𝑉 ‘ ℎ ) ) )
21 1 2 3 4 tendoeq2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ∀ ℎ ∈ 𝑇 ( ℎ ≠ ( I ↾ 𝐵 ) → ( 𝑈 ‘ ℎ ) = ( 𝑉 ‘ ℎ ) ) ) → 𝑈 = 𝑉 )
22 5 6 7 8 20 21 syl221anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐹 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → 𝑈 = 𝑉 )