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Theorem lnof 30756

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 2733	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4		eqid 2733	. . . 4 ⊢ ( +_𝑣 ‘𝑊) = ( +_𝑣 ‘𝑊)
5		eqid 2733	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
6		eqid 2733	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
7		lnof.7	. . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
8	1, 2, 3, 4, 5, 6, 7	islno 30754	. . 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 1541 ∈ wcel 2113 ∀wral 3048 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 ℂcc 11015 NrmCVeccnv 30585 +_𝑣 cpv 30586 BaseSetcba 30587 ·_𝑠OLD cns 30588 LnOp clno 30741

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-sbc 3738 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-br 5096 df-opab 5158 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-fv 6497 df-ov 7358 df-oprab 7359 df-mpo 7360 df-map 8761 df-lno 30745

This theorem is referenced by: lno0 30757 lnocoi 30758 lnoadd 30759 lnosub 30760 lnomul 30761 isblo2 30784 blof 30786 nmlno0lem 30794 nmlnoubi 30797 nmlnogt0 30798 lnon0 30799 isblo3i 30802 blocnilem 30805 blocni 30806 htthlem 30918

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