Theorem nlelshi 32047

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 30990	. . 3 ⊢ 0ℎ ∈ ℋ
2		nlelsh.1	. . . 4 ⊢ 𝑇 ∈ LinFn
3	2	lnfn0i 32029	. . 3 ⊢ (𝑇‘0ℎ) = 0
4	2	lnfnfi 32028	. . . 4 ⊢ 𝑇: ℋ⟶ℂ
5		elnlfn 31915	. . . 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 31868	. . . . . . . . . 10 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0}))
9	4, 8	ax-mp 5	. . . . . . . . 9 ⊢ (null‘𝑇) = (◡𝑇 “ {0})
10		cnvimass 6036	. . . . . . . . 9 ⊢ (◡𝑇 “ {0}) ⊆ dom 𝑇
11	9, 10	eqsstri 3976	. . . . . . . 8 ⊢ (null‘𝑇) ⊆ dom 𝑇
12	4	fdmi 6668	. . . . . . . 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 30999	. . . . . 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 32030	. . . . . . . 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 31915	. . . . . . . . . 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 31915	. . . . . . . . . 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 7371	. . . . . . 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 11294	. . . . . 6 ⊢ (0 + 0) = 0
29	27, 28	eqtrdi 2782	. . . . 5 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 +ℎ 𝑦)) = 0)
30		elnlfn 31915	. . . . . 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 31000	. . . . . 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 32031	. . . . . . 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 7368	. . . . . . 7 ⊢ (𝑦 ∈ (null‘𝑇) → (𝑥 · (𝑇‘𝑦)) = (𝑥 · 0))
39		mul01 11298	. . . . . . 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 31915	. . . . . 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 31206	. . 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 4575  ^◡ccnv 5618  dom cdm 5619   “ cima 5622  ⟶wf 6483  ‘cfv 6487  (class class class)co 7352  ℂcc 11010  0cc0 11012   + caddc 11015   · cmul 11017   ℋchba 30906   +_ℎ cva 30907   ·_ℎ csm 30908  0_ℎc0v 30911   S_ℋ csh 30915  nullcnl 30939  LinFnclf 30941

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 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11068  ax-resscn 11069  ax-1cn 11070  ax-icn 11071  ax-addcl 11072  ax-addrcl 11073  ax-mulcl 11074  ax-mulrcl 11075  ax-mulcom 11076  ax-addass 11077  ax-mulass 11078  ax-distr 11079  ax-i2m1 11080  ax-1ne0 11081  ax-1rid 11082  ax-rnegex 11083  ax-rrecex 11084  ax-cnre 11085  ax-pre-lttri 11086  ax-pre-lttrn 11087  ax-pre-ltadd 11088  ax-hilex 30986  ax-hfvadd 30987  ax-hv0cl 30990  ax-hvaddid 30991  ax-hfvmul 30992  ax-hvmulid 30993

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 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-pnf 11154  df-mnf 11155  df-ltxr 11157  df-sub 11352  df-sh 31194  df-nlfn 31833  df-lnfn 31835

This theorem is referenced by:  nlelchi  32048