Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2610 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
4 | eqid 2610 | . . . 4 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
5 | eqid 2610 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
6 | eqid 2610 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
7 | lnof.7 | . . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | islno 26992 | . . 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 651 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌) |
10 | 9 | 3impa 1251 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 NrmCVeccnv 26823 +𝑣 cpv 26824 BaseSetcba 26825 ·𝑠OLD cns 26826 LnOp clno 26979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-lno 26983 |
This theorem is referenced by: lno0 26995 lnocoi 26996 lnoadd 26997 lnosub 26998 lnomul 26999 isblo2 27022 blof 27024 nmlno0lem 27032 nmlnoubi 27035 nmlnogt0 27036 lnon0 27037 isblo3i 27040 blocnilem 27043 blocni 27044 htthlem 27158 |
Copyright terms: Public domain | W3C validator |