Theorem fin1a2lem3 9107

Description: Lemma for fin1a2 9120. (Contributed by Stefan O'Rear, 7-Nov-2014.)

Hypothesis

Ref	Expression
fin1a2lem.b	⊢ 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))

Assertion

Ref	Expression
fin1a2lem3	⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2𝑜 ·𝑜 𝐴))

Proof of Theorem fin1a2lem3

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6557	. 2 ⊢ (𝑎 = 𝐴 → (2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝐴))
2		fin1a2lem.b	. . 3 ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))
3		oveq2 6557	. . . 4 ⊢ (𝑥 = 𝑎 → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 𝑎))
4	3	cbvmptv 4678	. . 3 ⊢ (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥)) = (𝑎 ∈ ω ↦ (2𝑜 ·𝑜 𝑎))
5	2, 4	eqtri 2632	. 2 ⊢ 𝐸 = (𝑎 ∈ ω ↦ (2𝑜 ·𝑜 𝑎))
6		ovex 6577	. 2 ⊢ (2𝑜 ·𝑜 𝐴) ∈ V
7	1, 5, 6	fvmpt 6191	1 ⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2𝑜 ·𝑜 𝐴))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  ωcom 6957  2_𝑜c2o 7441   ·_𝑜 comu 7445

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-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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  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-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-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552

This theorem is referenced by:  fin1a2lem4  9108  fin1a2lem5  9109  fin1a2lem6  9110