Theorem lnophmlem1 31264

Description: Lemma for lnophmi 31266. (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 6891	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
5	3, 4	oveq12d 7426	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑥)) = (𝐴 ·ih (𝑇‘𝐴)))
6	5	eleq1d 2818	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ↔ (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ))
7	6	rspcv 3608	. 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 1541   ∈ wcel 2106  ∀wral 3061  ‘cfv 6543  (class class class)co 7408  ℝcr 11108   ℋchba 30167   ·_ih csp 30170  LinOpclo 30195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411

This theorem is referenced by:  lnophmlem2  31265