Theorem lnophmlem1 31945

Description: Lemma for lnophmi 31947. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnophmlem.1	⊢ 𝐴 ∈ ℋ
lnophmlem.2	⊢ 𝐵 ∈ ℋ
lnophmlem.3	⊢ 𝑇 ∈ LinOp
lnophmlem.4	⊢ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ

Assertion

Ref	Expression
lnophmlem1	⊢ (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑇

Proof of Theorem lnophmlem1

Step	Hyp	Ref	Expression
1		lnophmlem.1	. 2 ⊢ 𝐴 ∈ ℋ
2		lnophmlem.4	. 2 ⊢ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ
3		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4		fveq2 6858	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
5	3, 4	oveq12d 7405	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑥)) = (𝐴 ·ih (𝑇‘𝐴)))
6	5	eleq1d 2813	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ↔ (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ))
7	6	rspcv 3584	. 2 ⊢ (𝐴 ∈ ℋ → (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ → (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ))
8	1, 2, 7	mp2 9	1 ⊢ (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ

Syntax hints:   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ‘cfv 6511  (class class class)co 7387  ℝcr 11067   ℋchba 30848   ·_ih csp 30851  LinOpclo 30876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390

This theorem is referenced by:  lnophmlem2  31946