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| Mirrors > Home > MPE Home > Th. List > lnof | Structured version Visualization version GIF version | ||
| Description: A linear operator is a mapping. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| lnof.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
| lnof.2 | ⊢ 𝑌 = (BaseSet‘𝑊) |
| lnof.7 | ⊢ 𝐿 = (𝑈 LnOp 𝑊) |
| Ref | Expression |
|---|---|
| lnof | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lnof.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 2 | lnof.2 | . . . 4 ⊢ 𝑌 = (BaseSet‘𝑊) | |
| 3 | eqid 2737 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
| 4 | eqid 2737 | . . . 4 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
| 5 | eqid 2737 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
| 6 | eqid 2737 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
| 7 | lnof.7 | . . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | islno 30772 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐿 ↔ (𝑇:𝑋⟶𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑇‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑊)(𝑇‘𝑦))( +𝑣 ‘𝑊)(𝑇‘𝑧))))) |
| 9 | 8 | simprbda 498 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌) |
| 10 | 9 | 3impa 1110 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 NrmCVeccnv 30603 +𝑣 cpv 30604 BaseSetcba 30605 ·𝑠OLD cns 30606 LnOp clno 30759 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-lno 30763 |
| This theorem is referenced by: lno0 30775 lnocoi 30776 lnoadd 30777 lnosub 30778 lnomul 30779 isblo2 30802 blof 30804 nmlno0lem 30812 nmlnoubi 30815 nmlnogt0 30816 lnon0 30817 isblo3i 30820 blocnilem 30823 blocni 30824 htthlem 30936 |
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