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Theorem lnof 27890

Description: A linear operator is a mapping. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
lnof.1	⊢ 𝑋 = (BaseSet‘𝑈)
lnof.2	⊢ 𝑌 = (BaseSet‘𝑊)
lnof.7	⊢ 𝐿 = (𝑈 LnOp 𝑊)

**Assertion**
Ref	Expression
lnof	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌)

Proof of Theorem lnof Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnof.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
2		lnof.2	. . . 4 ⊢ 𝑌 = (BaseSet‘𝑊)
3		eqid 2748	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4		eqid 2748	. . . 4 ⊢ ( +_𝑣 ‘𝑊) = ( +_𝑣 ‘𝑊)
5		eqid 2748	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
6		eqid 2748	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
7		lnof.7	. . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
8	1, 2, 3, 4, 5, 6, 7	islno 27888	. . 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 654	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌)
10	9	3impa 1100	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1620 ∈ wcel 2127 ∀wral 3038 ⟶wf 6033 ‘cfv 6037 (class class class)co 6801 ℂcc 10097 NrmCVeccnv 27719 +_𝑣 cpv 27720 BaseSetcba 27721 ·_𝑠OLD cns 27722 LnOp clno 27875

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ral 3043 df-rex 3044 df-rab 3047 df-v 3330 df-sbc 3565 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-op 4316 df-uni 4577 df-br 4793 df-opab 4853 df-id 5162 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-fv 6045 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-map 8013 df-lno 27879

This theorem is referenced by: lno0 27891 lnocoi 27892 lnoadd 27893 lnosub 27894 lnomul 27895 isblo2 27918 blof 27920 nmlno0lem 27928 nmlnoubi 27931 nmlnogt0 27932 lnon0 27933 isblo3i 27936 blocnilem 27939 blocni 27940 htthlem 28054

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