Theorem imaelshi 32086

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 6090	. . . 4 ⊢ (𝑇 “ 𝐴) ⊆ ran 𝑇
2		rnelsh.1	. . . . . 6 ⊢ 𝑇 ∈ LinOp
3	2	lnopfi 31997	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
4		frn 6743	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → ran 𝑇 ⊆ ℋ)
5	3, 4	ax-mp 5	. . . 4 ⊢ ran 𝑇 ⊆ ℋ
6	1, 5	sstri 4004	. . 3 ⊢ (𝑇 “ 𝐴) ⊆ ℋ
7	2	lnop0i 31998	. . . 4 ⊢ (𝑇‘0ℎ) = 0ℎ
8		imaelsh.2	. . . . . 6 ⊢ 𝐴 ∈ Sℋ
9		sh0 31244	. . . . . 6 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴)
10	8, 9	ax-mp 5	. . . . 5 ⊢ 0ℎ ∈ 𝐴
11		ffun 6739	. . . . . . 7 ⊢ (𝑇: ℋ⟶ ℋ → Fun 𝑇)
12	3, 11	ax-mp 5	. . . . . 6 ⊢ Fun 𝑇
13	8	shssii 31241	. . . . . . 7 ⊢ 𝐴 ⊆ ℋ
14	3	fdmi 6747	. . . . . . 7 ⊢ dom 𝑇 = ℋ
15	13, 14	sseqtrri 4032	. . . . . 6 ⊢ 𝐴 ⊆ dom 𝑇
16		funfvima2 7250	. . . . . 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 2835	. . 3 ⊢ 0ℎ ∈ (𝑇 “ 𝐴)
20	6, 19	pm3.2i 470	. 2 ⊢ ((𝑇 “ 𝐴) ⊆ ℋ ∧ 0ℎ ∈ (𝑇 “ 𝐴))
21		ffn 6736	. . . . . 6 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
22	3, 21	ax-mp 5	. . . . 5 ⊢ 𝑇 Fn ℋ
23		oveq1 7437	. . . . . . . 8 ⊢ (𝑢 = (𝑇‘𝑥) → (𝑢 +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ 𝑣))
24	23	eleq1d 2823	. . . . . . 7 ⊢ (𝑢 = (𝑇‘𝑥) → ((𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴)))
25	24	ralbidv 3175	. . . . . 6 ⊢ (𝑢 = (𝑇‘𝑥) → (∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ∀𝑣 ∈ (𝑇 “ 𝐴)((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴)))
26	25	ralima 7256	. . . . 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 31242	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ)
29	8	sheli 31242	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
30	2	lnopaddi 31999	. . . . . . . 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 31245	. . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) ∈ 𝐴)
33	8, 32	mp3an1 1447	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 +ℎ 𝑦) ∈ 𝐴)
34		funfvima2 7250	. . . . . . . . 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 2839	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
38	37	ralrimiva 3143	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
39		oveq2 7438	. . . . . . . 8 ⊢ (𝑣 = (𝑇‘𝑦) → ((𝑇‘𝑥) +ℎ 𝑣) = ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)))
40	39	eleq1d 2823	. . . . . . 7 ⊢ (𝑣 = (𝑇‘𝑦) → (((𝑇‘𝑥) +ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ ((𝑇‘𝑥) +ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)))
41	40	ralima 7256	. . . . . 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 3065	. . 3 ⊢ ∀𝑢 ∈ (𝑇 “ 𝐴)∀𝑣 ∈ (𝑇 “ 𝐴)(𝑢 +ℎ 𝑣) ∈ (𝑇 “ 𝐴)
45	2	lnopmuli 32000	. . . . . . . 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 31246	. . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ 𝑦) ∈ 𝐴)
48	8, 47	mp3an1 1447	. . . . . . . 8 ⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ 𝑦) ∈ 𝐴)
49		funfvima2 7250	. . . . . . . . 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 2839	. . . . . 6 ⊢ ((𝑢 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
53	52	ralrimiva 3143	. . . . 5 ⊢ (𝑢 ∈ ℂ → ∀𝑦 ∈ 𝐴 (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴))
54		oveq2 7438	. . . . . . . 8 ⊢ (𝑣 = (𝑇‘𝑦) → (𝑢 ·ℎ 𝑣) = (𝑢 ·ℎ (𝑇‘𝑦)))
55	54	eleq1d 2823	. . . . . . 7 ⊢ (𝑣 = (𝑇‘𝑦) → ((𝑢 ·ℎ 𝑣) ∈ (𝑇 “ 𝐴) ↔ (𝑢 ·ℎ (𝑇‘𝑦)) ∈ (𝑇 “ 𝐴)))
56	55	ralima 7256	. . . . . 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 3060	. . 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 31237	. 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 1536   ∈ wcel 2105  ∀wral 3058   ⊆ wss 3962  dom cdm 5688  ran crn 5689   “ cima 5691  Fun wfun 6556   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  ℂcc 11150   ℋchba 30947   +_ℎ cva 30948   ·_ℎ csm 30949  0_ℎc0v 30952   S_ℋ csh 30956  LinOpclo 30975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-hilex 31027  ax-hfvadd 31028  ax-hvass 31030  ax-hv0cl 31031  ax-hvaddid 31032  ax-hfvmul 31033  ax-hvmulid 31034  ax-hvdistr2 31037  ax-hvmul0 31038

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-ltxr 11297  df-sub 11491  df-neg 11492  df-hvsub 30999  df-sh 31235  df-lnop 31869

This theorem is referenced by:  rnelshi  32087