Theorem tendopl 38476

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 6694	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢‘𝑔) = (𝑈‘𝑔))
2	1	coeq1d 5715	. . 3 ⊢ (𝑢 = 𝑈 → ((𝑢‘𝑔) ∘ (𝑣‘𝑔)) = ((𝑈‘𝑔) ∘ (𝑣‘𝑔)))
3	2	mpteq2dv 5136	. 2 ⊢ (𝑢 = 𝑈 → (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑣‘𝑔))))
4		fveq1 6694	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣‘𝑔) = (𝑉‘𝑔))
5	4	coeq2d 5716	. . 3 ⊢ (𝑣 = 𝑉 → ((𝑈‘𝑔) ∘ (𝑣‘𝑔)) = ((𝑈‘𝑔) ∘ (𝑉‘𝑔)))
6	5	mpteq2dv 5136	. 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 38475	. 2 ⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))
9		tendopl2.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
10	9	fvexi 6709	. . 3 ⊢ 𝑇 ∈ V
11	10	mptex 7017	. 2 ⊢ (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔))) ∈ V
12	3, 6, 8, 11	ovmpo 7347	1 ⊢ ((𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑈𝑃𝑉) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112   ↦ cmpt 5120   ∘ ccom 5540  ‘cfv 6358  (class class class)co 7191   ∈ cmpo 7193  LTrncltrn 37801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196

This theorem is referenced by:  tendopl2  38477  tendoplcl  38481  erngplus  38503  erngplus-rN  38511  dvaplusg  38709