Theorem lnophmlem1 32103

Description: Lemma for lnophmi 32105. (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 6842	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
5	3, 4	oveq12d 7386	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑥)) = (𝐴 ·ih (𝑇‘𝐴)))
6	5	eleq1d 2822	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ↔ (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ))
7	6	rspcv 3574	. 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 1542   ∈ wcel 2114  ∀wral 3052  ‘cfv 6500  (class class class)co 7368  ℝcr 11037   ℋchba 31006   ·_ih csp 31009  LinOpclo 31034

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371

This theorem is referenced by:  lnophmlem2  32104