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Theorem lnof 30915

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 2761	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4		eqid 2761	. . . 4 ⊢ ( +_𝑣 ‘𝑊) = ( +_𝑣 ‘𝑊)
5		eqid 2761	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
6		eqid 2761	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
7		lnof.7	. . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
8	1, 2, 3, 4, 5, 6, 7	islno 30913	. . 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 502	. 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌)
10	9	3impa 1121	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 ℂcc 11065 NrmCVeccnv 30744 +_𝑣 cpv 30745 BaseSetcba 30746 ·_𝑠OLD cns 30747 LnOp clno 30900

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3743 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-fv 6524 df-ov 7394 df-oprab 7395 df-mpo 7396 df-map 8804 df-lno 30904

This theorem is referenced by: lno0 30916 lnocoi 30917 lnoadd 30918 lnosub 30919 lnomul 30920 isblo2 30943 blof 30945 nmlno0lem 30953 nmlnoubi 30956 nmlnogt0 30957 lnon0 30958 isblo3i 30961 blocnilem 30964 blocni 30965 htthlem 31077

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