Metamath Proof Explorer

Theorem tendoco2

Description: Distribution of compositions in preparation for endomorphism sum definition. (Contributed by NM, 10-Jun-2013)

Ref Expression
Hypotheses tendof.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
tendof.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
tendof.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion tendoco2 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑈 ‘ ( 𝐹 ∘ 𝐺 ) ) ∘ ( 𝑉 ‘ ( 𝐹 ∘ 𝐺 ) ) ) = ( ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) ∘ ( ( 𝑈 ‘ 𝐺 ) ∘ ( 𝑉 ‘ 𝐺 ) ) ) )

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 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝐾 ∈ HL )
5 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝑊 ∈ 𝐻 )
6 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝑈 ∈ 𝐸 )
7 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝐹 ∈ 𝑇 )
8 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝐺 ∈ 𝑇 )
9 1 2 3 tendovalco ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑈 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝑈 ‘ ( 𝐹 ∘ 𝐺 ) ) = ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑈 ‘ 𝐺 ) ) )
10 4 5 6 7 8 9 syl32anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝑈 ‘ ( 𝐹 ∘ 𝐺 ) ) = ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑈 ‘ 𝐺 ) ) )
11 simp2r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝑉 ∈ 𝐸 )
12 1 2 3 tendovalco ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝑉 ‘ ( 𝐹 ∘ 𝐺 ) ) = ( ( 𝑉 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐺 ) ) )
13 4 5 11 7 8 12 syl32anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝑉 ‘ ( 𝐹 ∘ 𝐺 ) ) = ( ( 𝑉 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐺 ) ) )
14 10 13 coeq12d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑈 ‘ ( 𝐹 ∘ 𝐺 ) ) ∘ ( 𝑉 ‘ ( 𝐹 ∘ 𝐺 ) ) ) = ( ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑈 ‘ 𝐺 ) ) ∘ ( ( 𝑉 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐺 ) ) ) )
15 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
16 1 2 3 tendocl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑈 ‘ 𝐺 ) ∈ 𝑇 )
17 15 6 8 16 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝑈 ‘ 𝐺 ) ∈ 𝑇 )
18 1 2 3 tendocl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑉 ‘ 𝐹 ) ∈ 𝑇 )
19 15 11 7 18 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝑉 ‘ 𝐹 ) ∈ 𝑇 )
20 1 2 ltrnco4 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ‘ 𝐺 ) ∈ 𝑇 ∧ ( 𝑉 ‘ 𝐹 ) ∈ 𝑇 ) → ( ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑈 ‘ 𝐺 ) ) ∘ ( ( 𝑉 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐺 ) ) ) = ( ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) ∘ ( ( 𝑈 ‘ 𝐺 ) ∘ ( 𝑉 ‘ 𝐺 ) ) ) )
21 15 17 19 20 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑈 ‘ 𝐺 ) ) ∘ ( ( 𝑉 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐺 ) ) ) = ( ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) ∘ ( ( 𝑈 ‘ 𝐺 ) ∘ ( 𝑉 ‘ 𝐺 ) ) ) )
22 14 21 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑈 ‘ ( 𝐹 ∘ 𝐺 ) ) ∘ ( 𝑉 ‘ ( 𝐹 ∘ 𝐺 ) ) ) = ( ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) ∘ ( ( 𝑈 ‘ 𝐺 ) ∘ ( 𝑉 ‘ 𝐺 ) ) ) )