Theorem lnophmlem1 32045

Description: Lemma for lnophmi 32047. (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 6907	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
5	3, 4	oveq12d 7449	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑥)) = (𝐴 ·ih (𝑇‘𝐴)))
6	5	eleq1d 2824	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ↔ (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ))
7	6	rspcv 3618	. 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 1537   ∈ wcel 2106  ∀wral 3059  ‘cfv 6563  (class class class)co 7431  ℝcr 11152   ℋchba 30948   ·_ih csp 30951  LinOpclo 30976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434

This theorem is referenced by:  lnophmlem2  32046