Theorem nlelshi 32040

Description: The null space of a linear functional is a subspace. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nlelsh.1	⊢ 𝑇 ∈ LinFn

Assertion

Ref	Expression
nlelshi	⊢ (null‘𝑇) ∈ Sℋ

Proof of Theorem nlelshi

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

Step	Hyp	Ref	Expression
1		ax-hv0cl 30983	. . 3 ⊢ 0ℎ ∈ ℋ
2		nlelsh.1	. . . 4 ⊢ 𝑇 ∈ LinFn
3	2	lnfn0i 32022	. . 3 ⊢ (𝑇‘0ℎ) = 0
4	2	lnfnfi 32021	. . . 4 ⊢ 𝑇: ℋ⟶ℂ
5		elnlfn 31908	. . . 4 ⊢ (𝑇: ℋ⟶ℂ → (0ℎ ∈ (null‘𝑇) ↔ (0ℎ ∈ ℋ ∧ (𝑇‘0ℎ) = 0)))
6	4, 5	ax-mp 5	. . 3 ⊢ (0ℎ ∈ (null‘𝑇) ↔ (0ℎ ∈ ℋ ∧ (𝑇‘0ℎ) = 0))
7	1, 3, 6	mpbir2an 711	. 2 ⊢ 0ℎ ∈ (null‘𝑇)
8		nlfnval 31861	. . . . . . . . . 10 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0}))
9	4, 8	ax-mp 5	. . . . . . . . 9 ⊢ (null‘𝑇) = (◡𝑇 “ {0})
10		cnvimass 6030	. . . . . . . . 9 ⊢ (◡𝑇 “ {0}) ⊆ dom 𝑇
11	9, 10	eqsstri 3976	. . . . . . . 8 ⊢ (null‘𝑇) ⊆ dom 𝑇
12	4	fdmi 6662	. . . . . . . 8 ⊢ dom 𝑇 = ℋ
13	11, 12	sseqtri 3978	. . . . . . 7 ⊢ (null‘𝑇) ⊆ ℋ
14	13	sseli 3925	. . . . . 6 ⊢ (𝑥 ∈ (null‘𝑇) → 𝑥 ∈ ℋ)
15	13	sseli 3925	. . . . . 6 ⊢ (𝑦 ∈ (null‘𝑇) → 𝑦 ∈ ℋ)
16		hvaddcl 30992	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ)
17	14, 15, 16	syl2an 596	. . . . 5 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 +ℎ 𝑦) ∈ ℋ)
18	2	lnfnaddi 32023	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) + (𝑇‘𝑦)))
19	14, 15, 18	syl2an 596	. . . . . . 7 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) + (𝑇‘𝑦)))
20		elnlfn 31908	. . . . . . . . . 10 ⊢ (𝑇: ℋ⟶ℂ → (𝑥 ∈ (null‘𝑇) ↔ (𝑥 ∈ ℋ ∧ (𝑇‘𝑥) = 0)))
21	4, 20	ax-mp 5	. . . . . . . . 9 ⊢ (𝑥 ∈ (null‘𝑇) ↔ (𝑥 ∈ ℋ ∧ (𝑇‘𝑥) = 0))
22	21	simprbi 496	. . . . . . . 8 ⊢ (𝑥 ∈ (null‘𝑇) → (𝑇‘𝑥) = 0)
23		elnlfn 31908	. . . . . . . . . 10 ⊢ (𝑇: ℋ⟶ℂ → (𝑦 ∈ (null‘𝑇) ↔ (𝑦 ∈ ℋ ∧ (𝑇‘𝑦) = 0)))
24	4, 23	ax-mp 5	. . . . . . . . 9 ⊢ (𝑦 ∈ (null‘𝑇) ↔ (𝑦 ∈ ℋ ∧ (𝑇‘𝑦) = 0))
25	24	simprbi 496	. . . . . . . 8 ⊢ (𝑦 ∈ (null‘𝑇) → (𝑇‘𝑦) = 0)
26	22, 25	oveqan12d 7365	. . . . . . 7 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → ((𝑇‘𝑥) + (𝑇‘𝑦)) = (0 + 0))
27	19, 26	eqtrd 2766	. . . . . 6 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 +ℎ 𝑦)) = (0 + 0))
28		00id 11288	. . . . . 6 ⊢ (0 + 0) = 0
29	27, 28	eqtrdi 2782	. . . . 5 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 +ℎ 𝑦)) = 0)
30		elnlfn 31908	. . . . . 6 ⊢ (𝑇: ℋ⟶ℂ → ((𝑥 +ℎ 𝑦) ∈ (null‘𝑇) ↔ ((𝑥 +ℎ 𝑦) ∈ ℋ ∧ (𝑇‘(𝑥 +ℎ 𝑦)) = 0)))
31	4, 30	ax-mp 5	. . . . 5 ⊢ ((𝑥 +ℎ 𝑦) ∈ (null‘𝑇) ↔ ((𝑥 +ℎ 𝑦) ∈ ℋ ∧ (𝑇‘(𝑥 +ℎ 𝑦)) = 0))
32	17, 29, 31	sylanbrc 583	. . . 4 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 +ℎ 𝑦) ∈ (null‘𝑇))
33	32	rgen2 3172	. . 3 ⊢ ∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 +ℎ 𝑦) ∈ (null‘𝑇)
34		hvmulcl 30993	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
35	15, 34	sylan2 593	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
36	2	lnfnmuli 32024	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·ℎ 𝑦)) = (𝑥 · (𝑇‘𝑦)))
37	15, 36	sylan2 593	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 ·ℎ 𝑦)) = (𝑥 · (𝑇‘𝑦)))
38	25	oveq2d 7362	. . . . . . 7 ⊢ (𝑦 ∈ (null‘𝑇) → (𝑥 · (𝑇‘𝑦)) = (𝑥 · 0))
39		mul01 11292	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
40	38, 39	sylan9eqr 2788	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · (𝑇‘𝑦)) = 0)
41	37, 40	eqtrd 2766	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 ·ℎ 𝑦)) = 0)
42		elnlfn 31908	. . . . . 6 ⊢ (𝑇: ℋ⟶ℂ → ((𝑥 ·ℎ 𝑦) ∈ (null‘𝑇) ↔ ((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ (𝑇‘(𝑥 ·ℎ 𝑦)) = 0)))
43	4, 42	ax-mp 5	. . . . 5 ⊢ ((𝑥 ·ℎ 𝑦) ∈ (null‘𝑇) ↔ ((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ (𝑇‘(𝑥 ·ℎ 𝑦)) = 0))
44	35, 41, 43	sylanbrc 583	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 ·ℎ 𝑦) ∈ (null‘𝑇))
45	44	rgen2 3172	. . 3 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 ·ℎ 𝑦) ∈ (null‘𝑇)
46	33, 45	pm3.2i 470	. 2 ⊢ (∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 +ℎ 𝑦) ∈ (null‘𝑇) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 ·ℎ 𝑦) ∈ (null‘𝑇))
47		issh3 31199	. . 3 ⊢ ((null‘𝑇) ⊆ ℋ → ((null‘𝑇) ∈ Sℋ ↔ (0ℎ ∈ (null‘𝑇) ∧ (∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 +ℎ 𝑦) ∈ (null‘𝑇) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 ·ℎ 𝑦) ∈ (null‘𝑇)))))
48	13, 47	ax-mp 5	. 2 ⊢ ((null‘𝑇) ∈ Sℋ ↔ (0ℎ ∈ (null‘𝑇) ∧ (∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 +ℎ 𝑦) ∈ (null‘𝑇) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (null‘𝑇)(𝑥 ·ℎ 𝑦) ∈ (null‘𝑇))))
49	7, 46, 48	mpbir2an 711	1 ⊢ (null‘𝑇) ∈ Sℋ

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3897  {csn 4573  ^◡ccnv 5613  dom cdm 5614   “ cima 5617  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  ℂcc 11004  0cc0 11006   + caddc 11009   · cmul 11011   ℋchba 30899   +_ℎ cva 30900   ·_ℎ csm 30901  0_ℎc0v 30904   S_ℋ csh 30908  nullcnl 30932  LinFnclf 30934

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-hilex 30979  ax-hfvadd 30980  ax-hv0cl 30983  ax-hvaddid 30984  ax-hfvmul 30985  ax-hvmulid 30986

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-po 5522  df-so 5523  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-ltxr 11151  df-sub 11346  df-sh 31187  df-nlfn 31826  df-lnfn 31828

This theorem is referenced by:  nlelchi  32041