Theorem tendopl 41036

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 6833	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢‘𝑔) = (𝑈‘𝑔))
2	1	coeq1d 5810	. . 3 ⊢ (𝑢 = 𝑈 → ((𝑢‘𝑔) ∘ (𝑣‘𝑔)) = ((𝑈‘𝑔) ∘ (𝑣‘𝑔)))
3	2	mpteq2dv 5192	. 2 ⊢ (𝑢 = 𝑈 → (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑣‘𝑔))))
4		fveq1 6833	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣‘𝑔) = (𝑉‘𝑔))
5	4	coeq2d 5811	. . 3 ⊢ (𝑣 = 𝑉 → ((𝑈‘𝑔) ∘ (𝑣‘𝑔)) = ((𝑈‘𝑔) ∘ (𝑉‘𝑔)))
6	5	mpteq2dv 5192	. 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 41035	. 2 ⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))
9		tendopl2.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
10	9	fvexi 6848	. . 3 ⊢ 𝑇 ∈ V
11	10	mptex 7169	. 2 ⊢ (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔))) ∈ V
12	3, 6, 8, 11	ovmpo 7518	1 ⊢ ((𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑈𝑃𝑉) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ↦ cmpt 5179   ∘ ccom 5628  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  LTrncltrn 40361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363

This theorem is referenced by:  tendopl2  41037  tendoplcl  41041  erngplus  41063  erngplus-rN  41071  dvaplusg  41269