Theorem tendoplcbv 41070

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

Hypothesis

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

Assertion

Ref	Expression
tendoplcbv	⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))

Distinct variable groups:   𝑡,𝑠,𝑢,𝑣,𝐸   𝑓,𝑔,𝑠,𝑡,𝑢,𝑣,𝑇

Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑓,𝑔,𝑠)   𝐸(𝑓,𝑔)

Proof of Theorem tendoplcbv

Step	Hyp	Ref	Expression
1		tendoplcbv.p	. 2 ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
2		fveq1 6832	. . . . 5 ⊢ (𝑠 = 𝑢 → (𝑠‘𝑓) = (𝑢‘𝑓))
3	2	coeq1d 5809	. . . 4 ⊢ (𝑠 = 𝑢 → ((𝑠‘𝑓) ∘ (𝑡‘𝑓)) = ((𝑢‘𝑓) ∘ (𝑡‘𝑓)))
4	3	mpteq2dv 5191	. . 3 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))))
5		fveq1 6832	. . . . . 6 ⊢ (𝑡 = 𝑣 → (𝑡‘𝑓) = (𝑣‘𝑓))
6	5	coeq2d 5810	. . . . 5 ⊢ (𝑡 = 𝑣 → ((𝑢‘𝑓) ∘ (𝑡‘𝑓)) = ((𝑢‘𝑓) ∘ (𝑣‘𝑓)))
7	6	mpteq2dv 5191	. . . 4 ⊢ (𝑡 = 𝑣 → (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑣‘𝑓))))
8		fveq2 6833	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑢‘𝑓) = (𝑢‘𝑔))
9		fveq2 6833	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑣‘𝑓) = (𝑣‘𝑔))
10	8, 9	coeq12d 5812	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑢‘𝑓) ∘ (𝑣‘𝑓)) = ((𝑢‘𝑔) ∘ (𝑣‘𝑔)))
11	10	cbvmptv 5201	. . . 4 ⊢ (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑣‘𝑓))) = (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔)))
12	7, 11	eqtrdi 2786	. . 3 ⊢ (𝑡 = 𝑣 → (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))) = (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))
13	4, 12	cbvmpov 7453	. 2 ⊢ (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))
14	1, 13	eqtri 2758	1 ⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))

Syntax hints:   = wceq 1542   ↦ cmpt 5178   ∘ ccom 5627  ‘cfv 6491   ∈ cmpo 7360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-co 5632  df-iota 6447  df-fv 6499  df-oprab 7362  df-mpo 7363

This theorem is referenced by:  tendopl  41071