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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 2740 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
4 | eqid 2740 | . . . 4 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
5 | eqid 2740 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
6 | eqid 2740 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
7 | lnof.7 | . . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | islno 30785 | . . 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 1537 ∈ wcel 2108 ∀wral 3067 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 NrmCVeccnv 30616 +𝑣 cpv 30617 BaseSetcba 30618 ·𝑠OLD cns 30619 LnOp clno 30772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-map 8886 df-lno 30776 |
This theorem is referenced by: lno0 30788 lnocoi 30789 lnoadd 30790 lnosub 30791 lnomul 30792 isblo2 30815 blof 30817 nmlno0lem 30825 nmlnoubi 30828 nmlnogt0 30829 lnon0 30830 isblo3i 30833 blocnilem 30836 blocni 30837 htthlem 30949 |
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