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Theorem lnophmlem1 32036
Description: Lemma for lnophmi 32038. (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 6905 . . . . 5 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
53, 4oveq12d 7450 . . . 4 (𝑥 = 𝐴 → (𝑥 ·ih (𝑇𝑥)) = (𝐴 ·ih (𝑇𝐴)))
65eleq1d 2825 . . 3 (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇𝑥)) ∈ ℝ ↔ (𝐴 ·ih (𝑇𝐴)) ∈ ℝ))
76rspcv 3617 . 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 1539  wcel 2107  wral 3060  cfv 6560  (class class class)co 7432  cr 11155  chba 30939   ·ih csp 30942  LinOpclo 30967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  lnophmlem2  32037
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