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Mirrors > Home > HSE Home > Th. List > lnophmlem1 | Structured version Visualization version GIF version |
Description: Lemma for lnophmi 29795. (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 6670 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴)) | |
5 | 3, 4 | oveq12d 7174 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑥)) = (𝐴 ·ih (𝑇‘𝐴))) |
6 | 5 | eleq1d 2897 | . . 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 2114 ∀wral 3138 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 ℋchba 28696 ·ih csp 28699 LinOpclo 28724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 |
This theorem is referenced by: lnophmlem2 29794 |
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