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Theorem lnof 29126

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 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 29124	. . 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 499	. 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 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ⟶wf 6428 ‘cfv 6432 (class class class)co 7272 ℂcc 10880 NrmCVeccnv 28955 +_𝑣 cpv 28956 BaseSetcba 28957 ·_𝑠OLD cns 28958 LnOp clno 29111

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-fv 6440 df-ov 7275 df-oprab 7276 df-mpo 7277 df-map 8609 df-lno 29115

This theorem is referenced by: lno0 29127 lnocoi 29128 lnoadd 29129 lnosub 29130 lnomul 29131 isblo2 29154 blof 29156 nmlno0lem 29164 nmlnoubi 29167 nmlnogt0 29168 lnon0 29169 isblo3i 29172 blocnilem 29175 blocni 29176 htthlem 29288

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