Theorem imaelshi 31987

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 6042	. . . 4 ⊢ (𝑇 “ 𝐴) ⊆ ran 𝑇
2		rnelsh.1	. . . . . 6 ⊢ 𝑇 ∈ LinOp
3	2	lnopfi 31898	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
4		frn 6695	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → ran 𝑇 ⊆ ℋ)
5	3, 4	ax-mp 5	. . . 4 ⊢ ran 𝑇 ⊆ ℋ
6	1, 5	sstri 3956	. . 3 ⊢ (𝑇 “ 𝐴) ⊆ ℋ
7	2	lnop0i 31899	. . . 4 ⊢ (𝑇‘0ℎ) = 0ℎ
8		imaelsh.2	. . . . . 6 ⊢ 𝐴 ∈ Sℋ
9		sh0 31145	. . . . . 6 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴)
10	8, 9	ax-mp 5	. . . . 5 ⊢ 0ℎ ∈ 𝐴
11		ffun 6691	. . . . . . 7 ⊢ (𝑇: ℋ⟶ ℋ → Fun 𝑇)
12	3, 11	ax-mp 5	. . . . . 6 ⊢ Fun 𝑇
13	8	shssii 31142	. . . . . . 7 ⊢ 𝐴 ⊆ ℋ
14	3	fdmi 6699	. . . . . . 7 ⊢ dom 𝑇 = ℋ
15	13, 14	sseqtrri 3996	. . . . . 6 ⊢ 𝐴 ⊆ dom 𝑇
16		funfvima2 7205	. . . . . 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 2825	. . 3 ⊢ 0ℎ ∈ (𝑇 “ 𝐴)
20	6, 19	pm3.2i 470	. 2 ⊢ ((𝑇 “ 𝐴) ⊆ ℋ ∧ 0ℎ ∈ (𝑇 “ 𝐴))
21		ffn 6688	. . . . . 6 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
22	3, 21	ax-mp 5	. . . . 5 ⊢ 𝑇 Fn ℋ
23		oveq1 7394	. . . . . . . 8 ⊢ (𝑢 = (𝑇‘𝑥) → (𝑢 +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ 𝑣))
24	23	eleq1d 2813	. . . . . . 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 7211	. . . . 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 31143	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ)
29	8	sheli 31143	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
30	2	lnopaddi 31900	. . . . . . . 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 31146	. . . . . . . . 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 7205	. . . . . . . . 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 2829	. . . . . 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 7395	. . . . . . . 8 ⊢ (𝑣 = (𝑇‘𝑦) → ((𝑇‘𝑥) +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)))
40	39	eleq1d 2813	. . . . . . 7 ⊢ (𝑣 = (𝑇‘𝑦) → (((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)))
41	40	ralima 7211	. . . . . 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 3051	. . 3 ⊢ ∀𝑢 ∈ (𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴)
45	2	lnopmuli 31901	. . . . . . . 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 31147	. . . . . . . . 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 7205	. . . . . . . . 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 2829	. . . . . 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 7395	. . . . . . . 8 ⊢ (𝑣 = (𝑇‘𝑦) → (𝑢 ·ℎ 𝑣) = (𝑢 ·ℎ (𝑇‘𝑦)))
55	54	eleq1d 2813	. . . . . . 7 ⊢ (𝑣 = (𝑇‘𝑦) → ((𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)))
56	55	ralima 7211	. . . . . 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 3046	. . 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 31138	. 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 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3914  dom cdm 5638  ran crn 5639   “ cima 5641  Fun wfun 6505   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  ℂcc 11066   ℋchba 30848   +_ℎ cva 30849   ·_ℎ csm 30850  0_ℎc0v 30853   S_ℋ csh 30857  LinOpclo 30876

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-hilex 30928  ax-hfvadd 30929  ax-hvass 30931  ax-hv0cl 30932  ax-hvaddid 30933  ax-hfvmul 30934  ax-hvmulid 30935  ax-hvdistr2 30938  ax-hvmul0 30939

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-po 5546  df-so 5547  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-ltxr 11213  df-sub 11407  df-neg 11408  df-hvsub 30900  df-sh 31136  df-lnop 31770

This theorem is referenced by:  rnelshi  31988