Theorem tendoplcbv 36063

Description: Define sum operation for trace-perserving 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 6190	. . . . 5 ⊢ (𝑠 = 𝑢 → (𝑠‘𝑓) = (𝑢‘𝑓))
3	2	coeq1d 5283	. . . 4 ⊢ (𝑠 = 𝑢 → ((𝑠‘𝑓) ∘ (𝑡‘𝑓)) = ((𝑢‘𝑓) ∘ (𝑡‘𝑓)))
4	3	mpteq2dv 4745	. . 3 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))))
5		fveq1 6190	. . . . . 6 ⊢ (𝑡 = 𝑣 → (𝑡‘𝑓) = (𝑣‘𝑓))
6	5	coeq2d 5284	. . . . 5 ⊢ (𝑡 = 𝑣 → ((𝑢‘𝑓) ∘ (𝑡‘𝑓)) = ((𝑢‘𝑓) ∘ (𝑣‘𝑓)))
7	6	mpteq2dv 4745	. . . 4 ⊢ (𝑡 = 𝑣 → (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑣‘𝑓))))
8		fveq2 6191	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑢‘𝑓) = (𝑢‘𝑔))
9		fveq2 6191	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑣‘𝑓) = (𝑣‘𝑔))
10	8, 9	coeq12d 5286	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑢‘𝑓) ∘ (𝑣‘𝑓)) = ((𝑢‘𝑔) ∘ (𝑣‘𝑔)))
11	10	cbvmptv 4750	. . . 4 ⊢ (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑣‘𝑓))) = (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔)))
12	7, 11	syl6eq 2672	. . 3 ⊢ (𝑡 = 𝑣 → (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))) = (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))
13	4, 12	cbvmpt2v 6735	. 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 2644	1 ⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))

Syntax hints:   = wceq 1483   ↦ cmpt 4729   ∘ ccom 5118  ‘cfv 5888   ↦ cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-co 5123  df-iota 5851  df-fv 5896  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  tendopl  36064