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Metamath Proof Explorer

Theorem tendoplcbv

Description: Define sum operation for trace-preserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013)

Ref Expression
Hypothesis tendoplcbv.p ⊢ 𝑃 = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) )
Assertion tendoplcbv 𝑃 = ( 𝑢 ∈ 𝐸 , 𝑣 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑢 ‘ 𝑔 ) ∘ ( 𝑣 ‘ 𝑔 ) ) ) )

Proof

Step Hyp Ref Expression
1 tendoplcbv.p ⊢ 𝑃 = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) )
2 fveq1 ⊢ ( 𝑠 = 𝑢 → ( 𝑠 ‘ 𝑓 ) = ( 𝑢 ‘ 𝑓 ) )
3 2 coeq1d ⊢ ( 𝑠 = 𝑢 → ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) = ( ( 𝑢 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) )
4 3 mpteq2dv ⊢ ( 𝑠 = 𝑢 → ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑢 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) )
5 fveq1 ⊢ ( 𝑡 = 𝑣 → ( 𝑡 ‘ 𝑓 ) = ( 𝑣 ‘ 𝑓 ) )
6 5 coeq2d ⊢ ( 𝑡 = 𝑣 → ( ( 𝑢 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) = ( ( 𝑢 ‘ 𝑓 ) ∘ ( 𝑣 ‘ 𝑓 ) ) )
7 6 mpteq2dv ⊢ ( 𝑡 = 𝑣 → ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑢 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) = ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑢 ‘ 𝑓 ) ∘ ( 𝑣 ‘ 𝑓 ) ) ) )
8 fveq2 ⊢ ( 𝑓 = 𝑔 → ( 𝑢 ‘ 𝑓 ) = ( 𝑢 ‘ 𝑔 ) )
9 fveq2 ⊢ ( 𝑓 = 𝑔 → ( 𝑣 ‘ 𝑓 ) = ( 𝑣 ‘ 𝑔 ) )
10 8 9 coeq12d ⊢ ( 𝑓 = 𝑔 → ( ( 𝑢 ‘ 𝑓 ) ∘ ( 𝑣 ‘ 𝑓 ) ) = ( ( 𝑢 ‘ 𝑔 ) ∘ ( 𝑣 ‘ 𝑔 ) ) )
11 10 cbvmptv ⊢ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑢 ‘ 𝑓 ) ∘ ( 𝑣 ‘ 𝑓 ) ) ) = ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑢 ‘ 𝑔 ) ∘ ( 𝑣 ‘ 𝑔 ) ) )
12 7 11 eqtrdi ⊢ ( 𝑡 = 𝑣 → ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑢 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) = ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑢 ‘ 𝑔 ) ∘ ( 𝑣 ‘ 𝑔 ) ) ) )
13 4 12 cbvmpov ⊢ ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) = ( 𝑢 ∈ 𝐸 , 𝑣 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑢 ‘ 𝑔 ) ∘ ( 𝑣 ‘ 𝑔 ) ) ) )
14 1 13 eqtri ⊢ 𝑃 = ( 𝑢 ∈ 𝐸 , 𝑣 ∈ 𝐸 ↦ ( 𝑔 ∈ 𝑇 ↦ ( ( 𝑢 ‘ 𝑔 ) ∘ ( 𝑣 ‘ 𝑔 ) ) ) )