Theorem 0lno 30726

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 2730	. . 3 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		eqid 2730	. . 3 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
3		0lno.0	. . 3 ⊢ 𝑍 = (𝑈 0op 𝑊)
4	1, 2, 3	0oo 30725	. 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 2730	. . . . . . . . 9 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
10	1, 9	nvscl 30562	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD ‘𝑈)𝑦) ∈ (BaseSet‘𝑈))
11	5, 7, 8, 10	syl3anc 1373	. . . . . . 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 2730	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
14	1, 13	nvgcl 30556	. . . . . . 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 1373	. . . . . 6 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → ((𝑥( ·𝑠OLD ‘𝑈)𝑦)( +𝑣 ‘𝑈)𝑧) ∈ (BaseSet‘𝑈))
16		eqid 2730	. . . . . . 7 ⊢ (0vec‘𝑊) = (0vec‘𝑊)
17	1, 16, 3	0oval 30724	. . . . . 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 1373	. . . . 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 30724	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑍‘𝑦) = (0vec‘𝑊))
20	5, 6, 8, 19	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍‘𝑦) = (0vec‘𝑊))
21	20	oveq2d 7406	. . . . . . 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 30724	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑍‘𝑧) = (0vec‘𝑊))
23	5, 6, 12, 22	syl3anc 1373	. . . . . . 7 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑍‘𝑧) = (0vec‘𝑊))
24	21, 23	oveq12d 7408	. . . . . 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 2730	. . . . . . . . 9 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
26	25, 16	nvsz 30574	. . . . . . . 8 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ ℂ) → (𝑥( ·𝑠OLD ‘𝑊)(0vec‘𝑊)) = (0vec‘𝑊))
27	6, 7, 26	syl2anc 584	. . . . . . 7 ⊢ ((((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑥 ∈ ℂ) ∧ (𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑥( ·𝑠OLD ‘𝑊)(0vec‘𝑊)) = (0vec‘𝑊))
28	27	oveq1d 7405	. . . . . 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 30570	. . . . . . 7 ⊢ (𝑊 ∈ NrmCVec → (0vec‘𝑊) ∈ (BaseSet‘𝑊))
30		eqid 2730	. . . . . . . 8 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊)
31	2, 30, 16	nv0rid 30571	. . . . . . 7 ⊢ ((𝑊 ∈ NrmCVec ∧ (0vec‘𝑊) ∈ (BaseSet‘𝑊)) → ((0vec‘𝑊)( +𝑣 ‘𝑊)(0vec‘𝑊)) = (0vec‘𝑊))
32	6, 29, 31	syl2anc2 585	. . . . . 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 2769	. . . . 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 2768	. . . 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 3181	. . 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 3126	. 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 30689	. 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 713	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐿)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  ℂcc 11073  NrmCVeccnv 30520   +_𝑣 cpv 30521  BaseSetcba 30522   ·_𝑠OLD cns 30523  0_veccn0v 30524   LnOp clno 30676   0_op c0o 30679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-po 5549  df-so 5550  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-ltxr 11220  df-grpo 30429  df-gid 30430  df-ginv 30431  df-ablo 30481  df-vc 30495  df-nv 30528  df-va 30531  df-ba 30532  df-sm 30533  df-0v 30534  df-nmcv 30536  df-lno 30680  df-0o 30683

This theorem is referenced by:  0blo  30728  nmlno0i  30730  blocn  30743