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| Description: Lemma for lnophmi 32038. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| lnophmlem.1 | ⊢ 𝐴 ∈ ℋ | 
| lnophmlem.2 | ⊢ 𝐵 ∈ ℋ | 
| lnophmlem.3 | ⊢ 𝑇 ∈ LinOp | 
| lnophmlem.4 | ⊢ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ | 
| Ref | Expression | 
|---|---|
| lnophmlem1 | ⊢ (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ | 
| 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 6905 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴)) | |
| 5 | 3, 4 | oveq12d 7450 | . . . 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 3617 | . 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 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 ∀wral 3060 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 ℋchba 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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