Theorem nlelshi 32131

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 31074	. . 3 ⊢ 0ℎ ∈ ℋ
2		nlelsh.1	. . . 4 ⊢ 𝑇 ∈ LinFn
3	2	lnfn0i 32113	. . 3 ⊢ (𝑇‘0ℎ) = 0
4	2	lnfnfi 32112	. . . 4 ⊢ 𝑇: ℋ⟶ℂ
5		elnlfn 31999	. . . 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 712	. 2 ⊢ 0ℎ ∈ (null‘𝑇)
8		nlfnval 31952	. . . . . . . . . 10 ⊢ (𝑇: ℋ⟶ℂ → (null‘𝑇) = (◡𝑇 “ {0}))
9	4, 8	ax-mp 5	. . . . . . . . 9 ⊢ (null‘𝑇) = (◡𝑇 “ {0})
10		cnvimass 6047	. . . . . . . . 9 ⊢ (◡𝑇 “ {0}) ⊆ dom 𝑇
11	9, 10	eqsstri 3968	. . . . . . . 8 ⊢ (null‘𝑇) ⊆ dom 𝑇
12	4	fdmi 6679	. . . . . . . 8 ⊢ dom 𝑇 = ℋ
13	11, 12	sseqtri 3970	. . . . . . 7 ⊢ (null‘𝑇) ⊆ ℋ
14	13	sseli 3917	. . . . . 6 ⊢ (𝑥 ∈ (null‘𝑇) → 𝑥 ∈ ℋ)
15	13	sseli 3917	. . . . . 6 ⊢ (𝑦 ∈ (null‘𝑇) → 𝑦 ∈ ℋ)
16		hvaddcl 31083	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ)
17	14, 15, 16	syl2an 597	. . . . 5 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 +ℎ 𝑦) ∈ ℋ)
18	2	lnfnaddi 32114	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) + (𝑇‘𝑦)))
19	14, 15, 18	syl2an 597	. . . . . . 7 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) + (𝑇‘𝑦)))
20		elnlfn 31999	. . . . . . . . . 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 497	. . . . . . . 8 ⊢ (𝑥 ∈ (null‘𝑇) → (𝑇‘𝑥) = 0)
23		elnlfn 31999	. . . . . . . . . 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 497	. . . . . . . 8 ⊢ (𝑦 ∈ (null‘𝑇) → (𝑇‘𝑦) = 0)
26	22, 25	oveqan12d 7386	. . . . . . 7 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → ((𝑇‘𝑥) + (𝑇‘𝑦)) = (0 + 0))
27	19, 26	eqtrd 2771	. . . . . 6 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 +ℎ 𝑦)) = (0 + 0))
28		00id 11321	. . . . . 6 ⊢ (0 + 0) = 0
29	27, 28	eqtrdi 2787	. . . . 5 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 +ℎ 𝑦)) = 0)
30		elnlfn 31999	. . . . . 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 584	. . . 4 ⊢ ((𝑥 ∈ (null‘𝑇) ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 +ℎ 𝑦) ∈ (null‘𝑇))
33	32	rgen2 3177	. . 3 ⊢ ∀𝑥 ∈ (null‘𝑇)∀𝑦 ∈ (null‘𝑇)(𝑥 +ℎ 𝑦) ∈ (null‘𝑇)
34		hvmulcl 31084	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
35	15, 34	sylan2 594	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
36	2	lnfnmuli 32115	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·ℎ 𝑦)) = (𝑥 · (𝑇‘𝑦)))
37	15, 36	sylan2 594	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 ·ℎ 𝑦)) = (𝑥 · (𝑇‘𝑦)))
38	25	oveq2d 7383	. . . . . . 7 ⊢ (𝑦 ∈ (null‘𝑇) → (𝑥 · (𝑇‘𝑦)) = (𝑥 · 0))
39		mul01 11325	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
40	38, 39	sylan9eqr 2793	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 · (𝑇‘𝑦)) = 0)
41	37, 40	eqtrd 2771	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑇‘(𝑥 ·ℎ 𝑦)) = 0)
42		elnlfn 31999	. . . . . 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 584	. . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (null‘𝑇)) → (𝑥 ·ℎ 𝑦) ∈ (null‘𝑇))
45	44	rgen2 3177	. . 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 31290	. . 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 712	1 ⊢ (null‘𝑇) ∈ Sℋ

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889  {csn 4567  ^◡ccnv 5630  dom cdm 5631   “ cima 5634  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  ℂcc 11036  0cc0 11038   + caddc 11041   · cmul 11043   ℋchba 30990   +_ℎ cva 30991   ·_ℎ csm 30992  0_ℎc0v 30995   S_ℋ csh 30999  nullcnl 31023  LinFnclf 31025

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-hilex 31070  ax-hfvadd 31071  ax-hv0cl 31074  ax-hvaddid 31075  ax-hfvmul 31076  ax-hvmulid 31077

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-ltxr 11184  df-sub 11379  df-sh 31278  df-nlfn 31917  df-lnfn 31919

This theorem is referenced by:  nlelchi  32132