Theorem fin1a2lem3 9224

Description: Lemma for fin1a2 9237. (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 6658	. 2 ⊢ (𝑎 = 𝐴 → (2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝐴))
2		fin1a2lem.b	. . 3 ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))
3		oveq2 6658	. . . 4 ⊢ (𝑥 = 𝑎 → (2𝑜 ·𝑜 𝑥) = (2𝑜 ·𝑜 𝑎))
4	3	cbvmptv 4750	. . 3 ⊢ (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥)) = (𝑎 ∈ ω ↦ (2𝑜 ·𝑜 𝑎))
5	2, 4	eqtri 2644	. 2 ⊢ 𝐸 = (𝑎 ∈ ω ↦ (2𝑜 ·𝑜 𝑎))
6		ovex 6678	. 2 ⊢ (2𝑜 ·𝑜 𝐴) ∈ V
7	1, 5, 6	fvmpt 6282	1 ⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2𝑜 ·𝑜 𝐴))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  ωcom 7065  2_𝑜c2o 7554   ·_𝑜 comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  fin1a2lem4  9225  fin1a2lem5  9226  fin1a2lem6  9227