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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 (𝑇𝑥)) ∈ ℝ
Assertion
Ref Expression
lnophmlem1 (𝐴 ·ih (𝑇𝐴)) ∈ ℝ
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑇

Proof of Theorem lnophmlem1
StepHypRef Expression
1 lnophmlem.1 . 2 𝐴 ∈ ℋ
2 lnophmlem.4 . 2 𝑥 ∈ ℋ (𝑥 ·ih (𝑇𝑥)) ∈ ℝ
3 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
4 fveq2 6907 . . . . 5 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
53, 4oveq12d 7449 . . . 4 (𝑥 = 𝐴 → (𝑥 ·ih (𝑇𝑥)) = (𝐴 ·ih (𝑇𝐴)))
65eleq1d 2824 . . 3 (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇𝑥)) ∈ ℝ ↔ (𝐴 ·ih (𝑇𝐴)) ∈ ℝ))
76rspcv 3618 . 2 (𝐴 ∈ ℋ → (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇𝑥)) ∈ ℝ → (𝐴 ·ih (𝑇𝐴)) ∈ ℝ))
81, 2, 7mp2 9 1 (𝐴 ·ih (𝑇𝐴)) ∈ ℝ
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wral 3059  cfv 6563  (class class class)co 7431  cr 11152  chba 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
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