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Mirrors > Home > HSE Home > Th. List > lnophmlem1 | Structured version Visualization version GIF version |
Description: Lemma for lnophmi 28877. (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 6191 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴)) | |
5 | 3, 4 | oveq12d 6668 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑥)) = (𝐴 ·ih (𝑇‘𝐴))) |
6 | 5 | eleq1d 2686 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ↔ (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ)) |
7 | 6 | rspcv 3305 | . 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 1483 ∈ wcel 1990 ∀wral 2912 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 ℋchil 27776 ·ih csp 27779 LinOpclo 27804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: lnophmlem2 28876 |
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