Theorem tendopl 35082

Description: Value of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)

Hypotheses

Ref	Expression
tendoplcbv.p	⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
tendopl2.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
tendopl	⊢ ((𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑈𝑃𝑉) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔))))

Distinct variable groups:   𝑡,𝑠,𝐸   𝑓,𝑔,𝑠,𝑡,𝑇   𝑓,𝑊,𝑔,𝑠,𝑡   𝑈,𝑔   𝑔,𝑉

Allowed substitution hints:   𝑃(𝑡,𝑓,𝑔,𝑠)   𝑈(𝑡,𝑓,𝑠)   𝐸(𝑓,𝑔)   𝐾(𝑡,𝑓,𝑔,𝑠)   𝑉(𝑡,𝑓,𝑠)

Proof of Theorem tendopl

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6102	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢‘𝑔) = (𝑈‘𝑔))
2	1	coeq1d 5205	. . 3 ⊢ (𝑢 = 𝑈 → ((𝑢‘𝑔) ∘ (𝑣‘𝑔)) = ((𝑈‘𝑔) ∘ (𝑣‘𝑔)))
3	2	mpteq2dv 4673	. 2 ⊢ (𝑢 = 𝑈 → (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑣‘𝑔))))
4		fveq1 6102	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣‘𝑔) = (𝑉‘𝑔))
5	4	coeq2d 5206	. . 3 ⊢ (𝑣 = 𝑉 → ((𝑈‘𝑔) ∘ (𝑣‘𝑔)) = ((𝑈‘𝑔) ∘ (𝑉‘𝑔)))
6	5	mpteq2dv 4673	. 2 ⊢ (𝑣 = 𝑉 → (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑣‘𝑔))) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔))))
7		tendoplcbv.p	. . 3 ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
8	7	tendoplcbv 35081	. 2 ⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))
9		tendopl2.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
10		fvex 6113	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
11	9, 10	eqeltri 2684	. . 3 ⊢ 𝑇 ∈ V
12	11	mptex 6390	. 2 ⊢ (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔))) ∈ V
13	3, 6, 8, 12	ovmpt2 6694	1 ⊢ ((𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑈𝑃𝑉) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔))))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  LTrncltrn 34405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  tendopl2  35083  tendoplcl  35087  erngplus  35109  erngplus-rN  35117  dvaplusg  35315