Theorem tendoplcbv 41527

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 6882	. . . . 5 ⊢ (𝑠 = 𝑢 → (𝑠‘𝑓) = (𝑢‘𝑓))
3	2	coeq1d 5849	. . . 4 ⊢ (𝑠 = 𝑢 → ((𝑠‘𝑓) ∘ (𝑡‘𝑓)) = ((𝑢‘𝑓) ∘ (𝑡‘𝑓)))
4	3	mpteq2dv 5206	. . 3 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))))
5		fveq1 6882	. . . . . 6 ⊢ (𝑡 = 𝑣 → (𝑡‘𝑓) = (𝑣‘𝑓))
6	5	coeq2d 5850	. . . . 5 ⊢ (𝑡 = 𝑣 → ((𝑢‘𝑓) ∘ (𝑡‘𝑓)) = ((𝑢‘𝑓) ∘ (𝑣‘𝑓)))
7	6	mpteq2dv 5206	. . . 4 ⊢ (𝑡 = 𝑣 → (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑣‘𝑓))))
8		fveq2 6883	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑢‘𝑓) = (𝑢‘𝑔))
9		fveq2 6883	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑣‘𝑓) = (𝑣‘𝑔))
10	8, 9	coeq12d 5852	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑢‘𝑓) ∘ (𝑣‘𝑓)) = ((𝑢‘𝑔) ∘ (𝑣‘𝑔)))
11	10	cbvmptv 5216	. . . 4 ⊢ (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑣‘𝑓))) = (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔)))
12	7, 11	eqtrdi 2814	. . 3 ⊢ (𝑡 = 𝑣 → (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))) = (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))
13	4, 12	cbvmpov 7507	. 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 2786	1 ⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))

Syntax hints:   = wceq 1570   ↦ cmpt 5193   ∘ ccom 5667  ‘cfv 6538   ∈ cmpo 7414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-co 5672  df-iota 6494  df-fv 6546  df-oprab 7416  df-mpo 7417

This theorem is referenced by:  tendopl  41528