Theorem nlelshi 32041

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 30984	. . 3 ⊢ 0ℎ ∈ ℋ
2		nlelsh.1	. . . 4 ⊢ 𝑇 ∈ LinFn
3	2	lnfn0i 32023	. . 3 ⊢ (𝑇‘0ℎ) = 0
4	2	lnfnfi 32022	. . . 4 ⊢ 𝑇: ℋ⟶ℂ
5		elnlfn 31909	. . . 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 31862	. . . . . . . . . 10 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0}))
9	4, 8	ax-mp 5	. . . . . . . . 9 ⊢ (null‘𝑇) = (◡𝑇 “ {0})
10		cnvimass 6069	. . . . . . . . 9 ⊢ (◡𝑇 “ {0}) ⊆ dom 𝑇
11	9, 10	eqsstri 4005	. . . . . . . 8 ⊢ (null‘𝑇) ⊆ dom 𝑇
12	4	fdmi 6717	. . . . . . . 8 ⊢ dom 𝑇 = ℋ
13	11, 12	sseqtri 4007	. . . . . . 7 ⊢ (null‘𝑇) ⊆ ℋ
14	13	sseli 3954	. . . . . 6 ⊢ (𝑥 ∈ (null‘𝑇) → 𝑥 ∈ ℋ)
15	13	sseli 3954	. . . . . 6 ⊢ (𝑦 ∈ (null‘𝑇) → 𝑦 ∈ ℋ)
16		hvaddcl 30993	. . . . . 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 32024	. . . . . . . 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 31909	. . . . . . . . . 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 31909	. . . . . . . . . 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 7424	. . . . . . 7 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → ((𝑇‘𝑥) + (𝑇‘𝑦)) = (0 + 0))
27	19, 26	eqtrd 2770	. . . . . 6 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 +ℎ 𝑦)) = (0 + 0))
28		00id 11410	. . . . . 6 ⊢ (0 + 0) = 0
29	27, 28	eqtrdi 2786	. . . . 5 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 +ℎ 𝑦)) = 0)
30		elnlfn 31909	. . . . . 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 3184	. . 3 ⊢ ∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 +ℎ 𝑦) ∈ (null‘𝑇)
34		hvmulcl 30994	. . . . . 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 32025	. . . . . . 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 7421	. . . . . . 7 ⊢ (𝑦 ∈ (null‘𝑇) → (𝑥 · (𝑇‘𝑦)) = (𝑥 · 0))
39		mul01 11414	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
40	38, 39	sylan9eqr 2792	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · (𝑇‘𝑦)) = 0)
41	37, 40	eqtrd 2770	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 ·ℎ 𝑦)) = 0)
42		elnlfn 31909	. . . . . 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 3184	. . 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 31200	. . 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 1540   ∈ wcel 2108  ∀wral 3051   ⊆ wss 3926  {csn 4601  ^◡ccnv 5653  dom cdm 5654   “ cima 5657  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  ℂcc 11127  0cc0 11129   + caddc 11132   · cmul 11134   ℋchba 30900   +_ℎ cva 30901   ·_ℎ csm 30902  0_ℎc0v 30905   S_ℋ csh 30909  nullcnl 30933  LinFnclf 30935

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-hilex 30980  ax-hfvadd 30981  ax-hv0cl 30984  ax-hvaddid 30985  ax-hfvmul 30986  ax-hvmulid 30987

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-po 5561  df-so 5562  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-ltxr 11274  df-sub 11468  df-sh 31188  df-nlfn 31827  df-lnfn 31829

This theorem is referenced by:  nlelchi  32042