Theorem lnophmlem1 32112

Description: Lemma for lnophmi 32114. (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 6834	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
5	3, 4	oveq12d 7381	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑥)) = (𝐴 ·ih (𝑇‘𝐴)))
6	5	eleq1d 2825	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ↔ (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ))
7	6	rspcv 3563	. 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 1547   ∈ wcel 2119  ∀wral 3054  ‘cfv 6492  (class class class)co 7363  ℝcr 11035   ℋchba 31015   ·_ih csp 31018  LinOpclo 31043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366

This theorem is referenced by:  lnophmlem2  32113