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Mirrors > Home > HSE Home > Th. List > lnophmlem1 | Structured version Visualization version GIF version |
Description: Lemma for lnophmi 32047. (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 6907 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴)) | |
5 | 3, 4 | oveq12d 7449 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑥)) = (𝐴 ·ih (𝑇‘𝐴))) |
6 | 5 | eleq1d 2824 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ↔ (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ)) |
7 | 6 | rspcv 3618 | . 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 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 ℋchba 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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