Theorem 0lno 28573

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 2798	. . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		eqid 2798	. . 3 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
3		0lno.0	. . 3 ⊢ 𝑍 = (𝑈 0op 𝑊)
4	1, 2, 3	0oo 28572	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
5		simplll 774	. . . . . 6 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑈 ∈ NrmCVec)
6		simpllr 775	. . . . . 6 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑊 ∈ NrmCVec)
7		simplr 768	. . . . . . . 8 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑥 ∈ ℂ)
8		simprl 770	. . . . . . . 8 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑦 ∈ (BaseSet‘𝑈))
9		eqid 2798	. . . . . . . . 9 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
10	1, 9	nvscl 28409	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD ‘𝑈)𝑦) ∈ (BaseSet‘𝑈))
11	5, 7, 8, 10	syl3anc 1368	. . . . . . 7 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑥( ·𝑠OLD ‘𝑈)𝑦) ∈ (BaseSet‘𝑈))
12		simprr 772	. . . . . . 7 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → 𝑧 ∈ (BaseSet‘𝑈))
13		eqid 2798	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
14	1, 13	nvgcl 28403	. . . . . . 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 1368	. . . . . 6 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧) ∈ (BaseSet‘𝑈))
16		eqid 2798	. . . . . . 7 ⊢ (0vec‘𝑊) = (0vec‘𝑊)
17	1, 16, 3	0oval 28571	. . . . . 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 1368	. . . . 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 28571	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑍‘𝑦) = (0vec‘𝑊))
20	5, 6, 8, 19	syl3anc 1368	. . . . . . . 8 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍‘𝑦) = (0vec‘𝑊))
21	20	oveq2d 7151	. . . . . . 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 28571	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑍‘𝑧) = (0vec‘𝑊))
23	5, 6, 12, 22	syl3anc 1368	. . . . . . 7 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍‘𝑧) = (0vec‘𝑊))
24	21, 23	oveq12d 7153	. . . . . 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 2798	. . . . . . . . 9 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
26	25, 16	nvsz 28421	. . . . . . . 8 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ ℂ) → (𝑥( ·𝑠OLD ‘𝑊)(0vec‘𝑊)) = (0vec‘𝑊))
27	6, 7, 26	syl2anc 587	. . . . . . 7 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑥( ·𝑠OLD ‘𝑊)(0vec‘𝑊)) = (0vec‘𝑊))
28	27	oveq1d 7150	. . . . . 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 28417	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec → (0vec‘𝑊) ∈ (BaseSet‘𝑊))
30		eqid 2798	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
31	2, 30, 16	nv0rid 28418	. . . . . . 7 ⊢ ((𝑊 ∈ NrmCVec ∧ (0vec‘𝑊) ∈ (BaseSet‘𝑊)) → ((0vec‘𝑊)( +𝑣 ‘𝑊)(0vec‘𝑊)) = (0vec‘𝑊))
32	6, 29, 31	syl2anc2 588	. . . . . 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 2837	. . . . 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 2836	. . . 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 3156	. . 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 3149	. 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 28536	. 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 712	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐿)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  ℂcc 10524  NrmCVeccnv 28367   +_𝑣 cpv 28368  BaseSetcba 28369   ·_𝑠OLD cns 28370  0_veccn0v 28371   LnOp clno 28523   0_op c0o 28526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-ltxr 10669  df-grpo 28276  df-gid 28277  df-ginv 28278  df-ablo 28328  df-vc 28342  df-nv 28375  df-va 28378  df-ba 28379  df-sm 28380  df-0v 28381  df-nmcv 28383  df-lno 28527  df-0o 28530

This theorem is referenced by:  0blo  28575  nmlno0i  28577  blocn  28590