Theorem tendopl 40515

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 6902	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢‘𝑔) = (𝑈‘𝑔))
2	1	coeq1d 5870	. . 3 ⊢ (𝑢 = 𝑈 → ((𝑢‘𝑔) ∘ (𝑣‘𝑔)) = ((𝑈‘𝑔) ∘ (𝑣‘𝑔)))
3	2	mpteq2dv 5256	. 2 ⊢ (𝑢 = 𝑈 → (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑣‘𝑔))))
4		fveq1 6902	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣‘𝑔) = (𝑉‘𝑔))
5	4	coeq2d 5871	. . 3 ⊢ (𝑣 = 𝑉 → ((𝑈‘𝑔) ∘ (𝑣‘𝑔)) = ((𝑈‘𝑔) ∘ (𝑉‘𝑔)))
6	5	mpteq2dv 5256	. 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 40514	. 2 ⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))
9		tendopl2.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
10	9	fvexi 6917	. . 3 ⊢ 𝑇 ∈ V
11	10	mptex 7242	. 2 ⊢ (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔))) ∈ V
12	3, 6, 8, 11	ovmpo 7588	1 ⊢ ((𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) → (𝑈𝑃𝑉) = (𝑔 ∈ 𝑇 ↦ ((𝑈‘𝑔) ∘ (𝑉‘𝑔))))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2100   ↦ cmpt 5237   ∘ ccom 5688  ‘cfv 6556  (class class class)co 7427   ∈ cmpo 7429  LTrncltrn 39840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-rep 5291  ax-sep 5305  ax-nul 5312  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-reu 3374  df-rab 3429  df-v 3474  df-sbc 3788  df-csb 3904  df-dif 3961  df-un 3963  df-in 3965  df-ss 3975  df-nul 4334  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4917  df-iun 5006  df-br 5155  df-opab 5217  df-mpt 5238  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-ov 7430  df-oprab 7431  df-mpo 7432

This theorem is referenced by:  tendopl2  40516  tendoplcl  40520  erngplus  40542  erngplus-rN  40550  dvaplusg  40748