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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 2735 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
4 | eqid 2735 | . . . 4 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
5 | eqid 2735 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
6 | eqid 2735 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
7 | lnof.7 | . . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | islno 30782 | . . 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 1109 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 NrmCVeccnv 30613 +𝑣 cpv 30614 BaseSetcba 30615 ·𝑠OLD cns 30616 LnOp clno 30769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-lno 30773 |
This theorem is referenced by: lno0 30785 lnocoi 30786 lnoadd 30787 lnosub 30788 lnomul 30789 isblo2 30812 blof 30814 nmlno0lem 30822 nmlnoubi 30825 nmlnogt0 30826 lnon0 30827 isblo3i 30830 blocnilem 30833 blocni 30834 htthlem 30946 |
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