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Mirrors > Home > HSE Home > Th. List > lnophmlem1 | Structured version Visualization version GIF version |
Description: Lemma for lnophmi 28261. (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 6103 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴)) | |
5 | 3, 4 | oveq12d 6567 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑥)) = (𝐴 ·ih (𝑇‘𝐴))) |
6 | 5 | eleq1d 2672 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ↔ (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ)) |
7 | 6 | rspcv 3278 | . 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 1475 ∈ wcel 1977 ∀wral 2896 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 ℋchil 27160 ·ih csp 27163 LinOpclo 27188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: lnophmlem2 28260 |
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