Theorem imaelshi 32059

Description: The image of a subspace under a linear operator is a subspace. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rnelsh.1	⊢ 𝑇 ∈ LinOp
imaelsh.2	⊢ 𝐴 ∈ Sℋ

Assertion

Ref	Expression
imaelshi	⊢ (𝑇 “ 𝐴) ∈ Sℋ

Proof of Theorem imaelshi

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

Step	Hyp	Ref	Expression
1		imassrn 6027	. . . 4 ⊢ (𝑇 “ 𝐴) ⊆ ran 𝑇
2		rnelsh.1	. . . . . 6 ⊢ 𝑇 ∈ LinOp
3	2	lnopfi 31970	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
4		frn 6666	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → ran 𝑇 ⊆ ℋ)
5	3, 4	ax-mp 5	. . . 4 ⊢ ran 𝑇 ⊆ ℋ
6	1, 5	sstri 3940	. . 3 ⊢ (𝑇 “ 𝐴) ⊆ ℋ
7	2	lnop0i 31971	. . . 4 ⊢ (𝑇‘0ℎ) = 0ℎ
8		imaelsh.2	. . . . . 6 ⊢ 𝐴 ∈ Sℋ
9		sh0 31217	. . . . . 6 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴)
10	8, 9	ax-mp 5	. . . . 5 ⊢ 0ℎ ∈ 𝐴
11		ffun 6662	. . . . . . 7 ⊢ (𝑇: ℋ⟶ ℋ → Fun 𝑇)
12	3, 11	ax-mp 5	. . . . . 6 ⊢ Fun 𝑇
13	8	shssii 31214	. . . . . . 7 ⊢ 𝐴 ⊆ ℋ
14	3	fdmi 6670	. . . . . . 7 ⊢ dom 𝑇 = ℋ
15	13, 14	sseqtrri 3980	. . . . . 6 ⊢ 𝐴 ⊆ dom 𝑇
16		funfvima2 7174	. . . . . 6 ⊢ ((Fun 𝑇 ∧ 𝐴 ⊆ dom 𝑇) → (0ℎ ∈ 𝐴 → (𝑇‘0ℎ) ∈ (𝑇 “ 𝐴)))
17	12, 15, 16	mp2an 692	. . . . 5 ⊢ (0ℎ ∈ 𝐴 → (𝑇‘0ℎ) ∈ (𝑇 “ 𝐴))
18	10, 17	ax-mp 5	. . . 4 ⊢ (𝑇‘0ℎ) ∈ (𝑇 “ 𝐴)
19	7, 18	eqeltrri 2830	. . 3 ⊢ 0ℎ ∈ (𝑇 “ 𝐴)
20	6, 19	pm3.2i 470	. 2 ⊢ ((𝑇 “ 𝐴) ⊆ ℋ ∧ 0ℎ ∈ (𝑇 “ 𝐴))
21		ffn 6659	. . . . . 6 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
22	3, 21	ax-mp 5	. . . . 5 ⊢ 𝑇 Fn ℋ
23		oveq1 7362	. . . . . . . 8 ⊢ (𝑢 = (𝑇‘𝑥) → (𝑢 +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ 𝑣))
24	23	eleq1d 2818	. . . . . . 7 ⊢ (𝑢 = (𝑇‘𝑥) → ((𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴)))
25	24	ralbidv 3156	. . . . . 6 ⊢ (𝑢 = (𝑇‘𝑥) → (∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴)))
26	25	ralima 7180	. . . . 5 ⊢ ((𝑇 Fn ℋ ∧ 𝐴 ⊆ ℋ) → (∀𝑢 ∈ (𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴)))
27	22, 13, 26	mp2an 692	. . . 4 ⊢ (∀𝑢 ∈ (𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴))
28	8	sheli 31215	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ)
29	8	sheli 31215	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
30	2	lnopaddi 31972	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)))
31	28, 29, 30	syl2an 596	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)))
32		shaddcl 31218	. . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) ∈ 𝐴)
33	8, 32	mp3an1 1450	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) ∈ 𝐴)
34		funfvima2 7174	. . . . . . . . 9 ⊢ ((Fun 𝑇 ∧ 𝐴 ⊆ dom 𝑇) → ((𝑥 +ℎ 𝑦) ∈ 𝐴 → (𝑇‘(𝑥 +ℎ 𝑦)) ∈ (𝑇 “ 𝐴)))
35	12, 15, 34	mp2an 692	. . . . . . . 8 ⊢ ((𝑥 +ℎ 𝑦) ∈ 𝐴 → (𝑇‘(𝑥 +ℎ 𝑦)) ∈ (𝑇 “ 𝐴))
36	33, 35	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑇‘(𝑥 +ℎ 𝑦)) ∈ (𝑇 “ 𝐴))
37	31, 36	eqeltrrd 2834	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
38	37	ralrimiva 3125	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
39		oveq2 7363	. . . . . . . 8 ⊢ (𝑣 = (𝑇‘𝑦) → ((𝑇‘𝑥) +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)))
40	39	eleq1d 2818	. . . . . . 7 ⊢ (𝑣 = (𝑇‘𝑦) → (((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)))
41	40	ralima 7180	. . . . . 6 ⊢ ((𝑇 Fn ℋ ∧ 𝐴 ⊆ ℋ) → (∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑦 ∈ 𝐴 ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)))
42	22, 13, 41	mp2an 692	. . . . 5 ⊢ (∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑦 ∈ 𝐴 ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
43	38, 42	sylibr 234	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴))
44	27, 43	mprgbir 3055	. . 3 ⊢ ∀𝑢 ∈ (𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴)
45	2	lnopmuli 31973	. . . . . . . 8 ⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑢 ·ℎ 𝑦)) = (𝑢 ·ℎ (𝑇‘𝑦)))
46	29, 45	sylan2 593	. . . . . . 7 ⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑇‘(𝑢 ·ℎ 𝑦)) = (𝑢 ·ℎ (𝑇‘𝑦)))
47		shmulcl 31219	. . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ 𝑦) ∈ 𝐴)
48	8, 47	mp3an1 1450	. . . . . . . 8 ⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ 𝑦) ∈ 𝐴)
49		funfvima2 7174	. . . . . . . . 9 ⊢ ((Fun 𝑇 ∧ 𝐴 ⊆ dom 𝑇) → ((𝑢 ·ℎ 𝑦) ∈ 𝐴 → (𝑇‘(𝑢 ·ℎ 𝑦)) ∈ (𝑇 “ 𝐴)))
50	12, 15, 49	mp2an 692	. . . . . . . 8 ⊢ ((𝑢 ·ℎ 𝑦) ∈ 𝐴 → (𝑇‘(𝑢 ·ℎ 𝑦)) ∈ (𝑇 “ 𝐴))
51	48, 50	syl 17	. . . . . . 7 ⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑇‘(𝑢 ·ℎ 𝑦)) ∈ (𝑇 “ 𝐴))
52	46, 51	eqeltrrd 2834	. . . . . 6 ⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
53	52	ralrimiva 3125	. . . . 5 ⊢ (𝑢 ∈ ℂ → ∀𝑦 ∈ 𝐴 (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
54		oveq2 7363	. . . . . . . 8 ⊢ (𝑣 = (𝑇‘𝑦) → (𝑢 ·ℎ 𝑣) = (𝑢 ·ℎ (𝑇‘𝑦)))
55	54	eleq1d 2818	. . . . . . 7 ⊢ (𝑣 = (𝑇‘𝑦) → ((𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)))
56	55	ralima 7180	. . . . . 6 ⊢ ((𝑇 Fn ℋ ∧ 𝐴 ⊆ ℋ) → (∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑦 ∈ 𝐴 (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)))
57	22, 13, 56	mp2an 692	. . . . 5 ⊢ (∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑦 ∈ 𝐴 (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
58	53, 57	sylibr 234	. . . 4 ⊢ (𝑢 ∈ ℂ → ∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴))
59	58	rgen 3050	. . 3 ⊢ ∀𝑢 ∈ ℂ ∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴)
60	44, 59	pm3.2i 470	. 2 ⊢ (∀𝑢 ∈ (𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ∧ ∀𝑢 ∈ ℂ ∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴))
61		issh2 31210	. 2 ⊢ ((𝑇 “ 𝐴) ∈ Sℋ ↔ (((𝑇 “ 𝐴) ⊆ ℋ ∧ 0ℎ ∈ (𝑇 “ 𝐴)) ∧ (∀𝑢 ∈ (𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ∧ ∀𝑢 ∈ ℂ ∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴))))
62	20, 60, 61	mpbir2an 711	1 ⊢ (𝑇 “ 𝐴) ∈ Sℋ

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ⊆ wss 3898  dom cdm 5621  ran crn 5622   “ cima 5624  Fun wfun 6483   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355  ℂcc 11015   ℋchba 30920   +_ℎ cva 30921   ·_ℎ csm 30922  0_ℎc0v 30925   S_ℋ csh 30929  LinOpclo 30948

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-hilex 31000  ax-hfvadd 31001  ax-hvass 31003  ax-hv0cl 31004  ax-hvaddid 31005  ax-hfvmul 31006  ax-hvmulid 31007  ax-hvdistr2 31010  ax-hvmul0 31011

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-po 5529  df-so 5530  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-er 8631  df-map 8761  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11159  df-mnf 11160  df-ltxr 11162  df-sub 11357  df-neg 11358  df-hvsub 30972  df-sh 31208  df-lnop 31842

This theorem is referenced by:  rnelshi  32060