Theorem imaelshi 29835

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 5940	. . . 4 ⊢ (𝑇 “ 𝐴) ⊆ ran 𝑇
2		rnelsh.1	. . . . . 6 ⊢ 𝑇 ∈ LinOp
3	2	lnopfi 29746	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
4		frn 6520	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → ran 𝑇 ⊆ ℋ)
5	3, 4	ax-mp 5	. . . 4 ⊢ ran 𝑇 ⊆ ℋ
6	1, 5	sstri 3976	. . 3 ⊢ (𝑇 “ 𝐴) ⊆ ℋ
7	2	lnop0i 29747	. . . 4 ⊢ (𝑇‘0ℎ) = 0ℎ
8		imaelsh.2	. . . . . 6 ⊢ 𝐴 ∈ Sℋ
9		sh0 28993	. . . . . 6 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴)
10	8, 9	ax-mp 5	. . . . 5 ⊢ 0ℎ ∈ 𝐴
11		ffun 6517	. . . . . . 7 ⊢ (𝑇: ℋ⟶ ℋ → Fun 𝑇)
12	3, 11	ax-mp 5	. . . . . 6 ⊢ Fun 𝑇
13	8	shssii 28990	. . . . . . 7 ⊢ 𝐴 ⊆ ℋ
14	3	fdmi 6524	. . . . . . 7 ⊢ dom 𝑇 = ℋ
15	13, 14	sseqtrri 4004	. . . . . 6 ⊢ 𝐴 ⊆ dom 𝑇
16		funfvima2 6993	. . . . . 6 ⊢ ((Fun 𝑇 ∧ 𝐴 ⊆ dom 𝑇) → (0ℎ ∈ 𝐴 → (𝑇‘0ℎ) ∈ (𝑇 “ 𝐴)))
17	12, 15, 16	mp2an 690	. . . . 5 ⊢ (0ℎ ∈ 𝐴 → (𝑇‘0ℎ) ∈ (𝑇 “ 𝐴))
18	10, 17	ax-mp 5	. . . 4 ⊢ (𝑇‘0ℎ) ∈ (𝑇 “ 𝐴)
19	7, 18	eqeltrri 2910	. . 3 ⊢ 0ℎ ∈ (𝑇 “ 𝐴)
20	6, 19	pm3.2i 473	. 2 ⊢ ((𝑇 “ 𝐴) ⊆ ℋ ∧ 0ℎ ∈ (𝑇 “ 𝐴))
21		ffn 6514	. . . . . 6 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
22	3, 21	ax-mp 5	. . . . 5 ⊢ 𝑇 Fn ℋ
23		oveq1 7163	. . . . . . . 8 ⊢ (𝑢 = (𝑇‘𝑥) → (𝑢 +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ 𝑣))
24	23	eleq1d 2897	. . . . . . 7 ⊢ (𝑢 = (𝑇‘𝑥) → ((𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴)))
25	24	ralbidv 3197	. . . . . 6 ⊢ (𝑢 = (𝑇‘𝑥) → (∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴)))
26	25	ralima 7000	. . . . 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 690	. . . 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 28991	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ)
29	8	sheli 28991	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
30	2	lnopaddi 29748	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)))
31	28, 29, 30	syl2an 597	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑇‘(𝑥 +ℎ 𝑦)) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)))
32		shaddcl 28994	. . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) ∈ 𝐴)
33	8, 32	mp3an1 1444	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) ∈ 𝐴)
34		funfvima2 6993	. . . . . . . . 9 ⊢ ((Fun 𝑇 ∧ 𝐴 ⊆ dom 𝑇) → ((𝑥 +ℎ 𝑦) ∈ 𝐴 → (𝑇‘(𝑥 +ℎ 𝑦)) ∈ (𝑇 “ 𝐴)))
35	12, 15, 34	mp2an 690	. . . . . . . 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 2914	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
38	37	ralrimiva 3182	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
39		oveq2 7164	. . . . . . . 8 ⊢ (𝑣 = (𝑇‘𝑦) → ((𝑇‘𝑥) +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)))
40	39	eleq1d 2897	. . . . . . 7 ⊢ (𝑣 = (𝑇‘𝑦) → (((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)))
41	40	ralima 7000	. . . . . 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 690	. . . . 5 ⊢ (∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑦 ∈ 𝐴 ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
43	38, 42	sylibr 236	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴))
44	27, 43	mprgbir 3153	. . 3 ⊢ ∀𝑢 ∈ (𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴)
45	2	lnopmuli 29749	. . . . . . . 8 ⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑢 ·ℎ 𝑦)) = (𝑢 ·ℎ (𝑇‘𝑦)))
46	29, 45	sylan2 594	. . . . . . 7 ⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑇‘(𝑢 ·ℎ 𝑦)) = (𝑢 ·ℎ (𝑇‘𝑦)))
47		shmulcl 28995	. . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ 𝑦) ∈ 𝐴)
48	8, 47	mp3an1 1444	. . . . . . . 8 ⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ 𝑦) ∈ 𝐴)
49		funfvima2 6993	. . . . . . . . 9 ⊢ ((Fun 𝑇 ∧ 𝐴 ⊆ dom 𝑇) → ((𝑢 ·ℎ 𝑦) ∈ 𝐴 → (𝑇‘(𝑢 ·ℎ 𝑦)) ∈ (𝑇 “ 𝐴)))
50	12, 15, 49	mp2an 690	. . . . . . . 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 2914	. . . . . 6 ⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
53	52	ralrimiva 3182	. . . . 5 ⊢ (𝑢 ∈ ℂ → ∀𝑦 ∈ 𝐴 (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
54		oveq2 7164	. . . . . . . 8 ⊢ (𝑣 = (𝑇‘𝑦) → (𝑢 ·ℎ 𝑣) = (𝑢 ·ℎ (𝑇‘𝑦)))
55	54	eleq1d 2897	. . . . . . 7 ⊢ (𝑣 = (𝑇‘𝑦) → ((𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)))
56	55	ralima 7000	. . . . . 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 690	. . . . 5 ⊢ (∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑦 ∈ 𝐴 (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
58	53, 57	sylibr 236	. . . 4 ⊢ (𝑢 ∈ ℂ → ∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴))
59	58	rgen 3148	. . 3 ⊢ ∀𝑢 ∈ ℂ ∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴)
60	44, 59	pm3.2i 473	. 2 ⊢ (∀𝑢 ∈ (𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ∧ ∀𝑢 ∈ ℂ ∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴))
61		issh2 28986	. 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 709	1 ⊢ (𝑇 “ 𝐴) ∈ Sℋ

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138   ⊆ wss 3936  dom cdm 5555  ran crn 5556   “ cima 5558  Fun wfun 6349   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  ℂcc 10535   ℋchba 28696   +_ℎ cva 28697   ·_ℎ csm 28698  0_ℎc0v 28701   S_ℋ csh 28705  LinOpclo 28724

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-hilex 28776  ax-hfvadd 28777  ax-hvass 28779  ax-hv0cl 28780  ax-hvaddid 28781  ax-hfvmul 28782  ax-hvmulid 28783  ax-hvdistr2 28786  ax-hvmul0 28787

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-po 5474  df-so 5475  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-ltxr 10680  df-sub 10872  df-neg 10873  df-hvsub 28748  df-sh 28984  df-lnop 29618

This theorem is referenced by:  rnelshi  29836