Theorem 0lno 29053

Description: The zero operator is linear. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
0lno.0	⊢ 𝑍 = (𝑈 0op 𝑊)
0lno.7	⊢ 𝐿 = (𝑈 LnOp 𝑊)

Assertion

Ref	Expression
0lno	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐿)

Proof of Theorem 0lno

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		eqid 2738	. . 3 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
3		0lno.0	. . 3 ⊢ 𝑍 = (𝑈 0op 𝑊)
4	1, 2, 3	0oo 29052	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
5		simplll 771	. . . . . 6 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑈 ∈ NrmCVec)
6		simpllr 772	. . . . . 6 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑊 ∈ NrmCVec)
7		simplr 765	. . . . . . . 8 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑥 ∈ ℂ)
8		simprl 767	. . . . . . . 8 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑦 ∈ (BaseSet‘𝑈))
9		eqid 2738	. . . . . . . . 9 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
10	1, 9	nvscl 28889	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD ‘𝑈)𝑦) ∈ (BaseSet‘𝑈))
11	5, 7, 8, 10	syl3anc 1369	. . . . . . 7 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑥( ·𝑠OLD ‘𝑈)𝑦) ∈ (BaseSet‘𝑈))
12		simprr 769	. . . . . . 7 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑧 ∈ (BaseSet‘𝑈))
13		eqid 2738	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
14	1, 13	nvgcl 28883	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝑥( ·𝑠OLD ‘𝑈)𝑦) ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧) ∈ (BaseSet‘𝑈))
15	5, 11, 12, 14	syl3anc 1369	. . . . . 6 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧) ∈ (BaseSet‘𝑈))
16		eqid 2738	. . . . . . 7 ⊢ (0vec‘𝑊) = (0vec‘𝑊)
17	1, 16, 3	0oval 29051	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ ((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧) ∈ (BaseSet‘𝑈)) → (𝑍‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = (0vec‘𝑊))
18	5, 6, 15, 17	syl3anc 1369	. . . . 5 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = (0vec‘𝑊))
19	1, 16, 3	0oval 29051	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑍‘𝑦) = (0vec‘𝑊))
20	5, 6, 8, 19	syl3anc 1369	. . . . . . . 8 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍‘𝑦) = (0vec‘𝑊))
21	20	oveq2d 7271	. . . . . . 7 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑥( ·𝑠OLD ‘𝑊)(𝑍‘𝑦)) = (𝑥( ·𝑠OLD ‘𝑊)(0vec‘𝑊)))
22	1, 16, 3	0oval 29051	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑍‘𝑧) = (0vec‘𝑊))
23	5, 6, 12, 22	syl3anc 1369	. . . . . . 7 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍‘𝑧) = (0vec‘𝑊))
24	21, 23	oveq12d 7273	. . . . . 6 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((𝑥( ·𝑠OLD ‘𝑊)(𝑍‘𝑦))( +𝑣 ‘𝑊)(𝑍‘𝑧)) = ((𝑥( ·𝑠OLD ‘𝑊)(0vec‘𝑊))( +𝑣 ‘𝑊)(0vec‘𝑊)))
25		eqid 2738	. . . . . . . . 9 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
26	25, 16	nvsz 28901	. . . . . . . 8 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ ℂ) → (𝑥( ·𝑠OLD ‘𝑊)(0vec‘𝑊)) = (0vec‘𝑊))
27	6, 7, 26	syl2anc 583	. . . . . . 7 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑥( ·𝑠OLD ‘𝑊)(0vec‘𝑊)) = (0vec‘𝑊))
28	27	oveq1d 7270	. . . . . 6 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((𝑥( ·𝑠OLD ‘𝑊)(0vec‘𝑊))( +𝑣 ‘𝑊)(0vec‘𝑊)) = ((0vec‘𝑊)( +𝑣 ‘𝑊)(0vec‘𝑊)))
29	2, 16	nvzcl 28897	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec → (0vec‘𝑊) ∈ (BaseSet‘𝑊))
30		eqid 2738	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
31	2, 30, 16	nv0rid 28898	. . . . . . 7 ⊢ ((𝑊 ∈ NrmCVec ∧ (0vec‘𝑊) ∈ (BaseSet‘𝑊)) → ((0vec‘𝑊)( +𝑣 ‘𝑊)(0vec‘𝑊)) = (0vec‘𝑊))
32	6, 29, 31	syl2anc2 584	. . . . . 6 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((0vec‘𝑊)( +𝑣 ‘𝑊)(0vec‘𝑊)) = (0vec‘𝑊))
33	24, 28, 32	3eqtrd 2782	. . . . 5 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((𝑥( ·𝑠OLD ‘𝑊)(𝑍‘𝑦))( +𝑣 ‘𝑊)(𝑍‘𝑧)) = (0vec‘𝑊))
34	18, 33	eqtr4d 2781	. . . 4 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑊)(𝑍‘𝑦))( +𝑣 ‘𝑊)(𝑍‘𝑧)))
35	34	ralrimivva 3114	. . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) → ∀𝑦 ∈ (BaseSet‘𝑈)∀𝑧 ∈ (BaseSet‘𝑈)(𝑍‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑊)(𝑍‘𝑦))( +𝑣 ‘𝑊)(𝑍‘𝑧)))
36	35	ralrimiva 3107	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑈)∀𝑧 ∈ (BaseSet‘𝑈)(𝑍‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑊)(𝑍‘𝑦))( +𝑣 ‘𝑊)(𝑍‘𝑧)))
37		0lno.7	. . 3 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
38	1, 2, 13, 30, 9, 25, 37	islno 29016	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑍 ∈ 𝐿 ↔ (𝑍:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑈)∀𝑧 ∈ (BaseSet‘𝑈)(𝑍‘((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑊)(𝑍‘𝑦))( +𝑣 ‘𝑊)(𝑍‘𝑧)))))
39	4, 36, 38	mpbir2and 709	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐿)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  NrmCVeccnv 28847   +_𝑣 cpv 28848  BaseSetcba 28849   ·_𝑠OLD cns 28850  0_veccn0v 28851   LnOp clno 29003   0_op c0o 29006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-ltxr 10945  df-grpo 28756  df-gid 28757  df-ginv 28758  df-ablo 28808  df-vc 28822  df-nv 28855  df-va 28858  df-ba 28859  df-sm 28860  df-0v 28861  df-nmcv 28863  df-lno 29007  df-0o 29010

This theorem is referenced by:  0blo  29055  nmlno0i  29057  blocn  29070