Theorem imaelshi 31063

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 6029	. . . 4 ⊢ (𝑇 “ 𝐴) ⊆ ran 𝑇
2		rnelsh.1	. . . . . 6 ⊢ 𝑇 ∈ LinOp
3	2	lnopfi 30974	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
4		frn 6680	. . . . 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 30975	. . . 4 ⊢ (𝑇‘0ℎ) = 0ℎ
8		imaelsh.2	. . . . . 6 ⊢ 𝐴 ∈ Sℋ
9		sh0 30221	. . . . . 6 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴)
10	8, 9	ax-mp 5	. . . . 5 ⊢ 0ℎ ∈ 𝐴
11		ffun 6676	. . . . . . 7 ⊢ (𝑇: ℋ⟶ ℋ → Fun 𝑇)
12	3, 11	ax-mp 5	. . . . . 6 ⊢ Fun 𝑇
13	8	shssii 30218	. . . . . . 7 ⊢ 𝐴 ⊆ ℋ
14	3	fdmi 6685	. . . . . . 7 ⊢ dom 𝑇 = ℋ
15	13, 14	sseqtrri 3984	. . . . . 6 ⊢ 𝐴 ⊆ dom 𝑇
16		funfvima2 7186	. . . . . 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 2829	. . 3 ⊢ 0ℎ ∈ (𝑇 “ 𝐴)
20	6, 19	pm3.2i 471	. 2 ⊢ ((𝑇 “ 𝐴) ⊆ ℋ ∧ 0ℎ ∈ (𝑇 “ 𝐴))
21		ffn 6673	. . . . . 6 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
22	3, 21	ax-mp 5	. . . . 5 ⊢ 𝑇 Fn ℋ
23		oveq1 7369	. . . . . . . 8 ⊢ (𝑢 = (𝑇‘𝑥) → (𝑢 +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ 𝑣))
24	23	eleq1d 2817	. . . . . . 7 ⊢ (𝑢 = (𝑇‘𝑥) → ((𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴)))
25	24	ralbidv 3170	. . . . . 6 ⊢ (𝑢 = (𝑇‘𝑥) → (∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴)))
26	25	ralima 7193	. . . . 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 30219	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ)
29	8	sheli 30219	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
30	2	lnopaddi 30976	. . . . . . . 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 30222	. . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) ∈ 𝐴)
33	8, 32	mp3an1 1448	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) ∈ 𝐴)
34		funfvima2 7186	. . . . . . . . 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 2833	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
38	37	ralrimiva 3139	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
39		oveq2 7370	. . . . . . . 8 ⊢ (𝑣 = (𝑇‘𝑦) → ((𝑇‘𝑥) +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)))
40	39	eleq1d 2817	. . . . . . 7 ⊢ (𝑣 = (𝑇‘𝑦) → (((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)))
41	40	ralima 7193	. . . . . 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 233	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴))
44	27, 43	mprgbir 3067	. . 3 ⊢ ∀𝑢 ∈ (𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴)
45	2	lnopmuli 30977	. . . . . . . 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 30223	. . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ 𝑦) ∈ 𝐴)
48	8, 47	mp3an1 1448	. . . . . . . 8 ⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ 𝑦) ∈ 𝐴)
49		funfvima2 7186	. . . . . . . . 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 2833	. . . . . 6 ⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
53	52	ralrimiva 3139	. . . . 5 ⊢ (𝑢 ∈ ℂ → ∀𝑦 ∈ 𝐴 (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
54		oveq2 7370	. . . . . . . 8 ⊢ (𝑣 = (𝑇‘𝑦) → (𝑢 ·ℎ 𝑣) = (𝑢 ·ℎ (𝑇‘𝑦)))
55	54	eleq1d 2817	. . . . . . 7 ⊢ (𝑣 = (𝑇‘𝑦) → ((𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)))
56	55	ralima 7193	. . . . . 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 233	. . . 4 ⊢ (𝑢 ∈ ℂ → ∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴))
59	58	rgen 3062	. . 3 ⊢ ∀𝑢 ∈ ℂ ∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴)
60	44, 59	pm3.2i 471	. 2 ⊢ (∀𝑢 ∈ (𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ∧ ∀𝑢 ∈ ℂ ∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴))
61		issh2 30214	. 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 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060   ⊆ wss 3913  dom cdm 5638  ran crn 5639   “ cima 5641  Fun wfun 6495   Fn wfn 6496  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362  ℂcc 11058   ℋchba 29924   +_ℎ cva 29925   ·_ℎ csm 29926  0_ℎc0v 29929   S_ℋ csh 29933  LinOpclo 29952

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-hilex 30004  ax-hfvadd 30005  ax-hvass 30007  ax-hv0cl 30008  ax-hvaddid 30009  ax-hfvmul 30010  ax-hvmulid 30011  ax-hvdistr2 30014  ax-hvmul0 30015

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-po 5550  df-so 5551  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 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11200  df-mnf 11201  df-ltxr 11203  df-sub 11396  df-neg 11397  df-hvsub 29976  df-sh 30212  df-lnop 30846

This theorem is referenced by:  rnelshi  31064