unopf1o - Hilbert Space Explorer

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Theorem unopf1o 30278

Description: A unitary operator in Hilbert space is one-to-one and onto. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
unopf1o	⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ)

Proof of Theorem unopf1o Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elunop 30234	. . . . 5 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦)))
2	1	simplbi 498	. . . 4 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–onto→ ℋ)
3		fof 6688	. . . 4 ⊢ (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ)
4	2, 3	syl 17	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
5		unop 30277	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) = (𝑥 ·_ih 𝑥))
6	5	3anidm23 1420	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) = (𝑥 ·_ih 𝑥))
7	6	3adant3 1131	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) = (𝑥 ·_ih 𝑥))
8		unop 30277	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑦)) = (𝑦 ·_ih 𝑦))
9	8	3anidm23 1420	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑦)) = (𝑦 ·_ih 𝑦))
10	9	3adant2 1130	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑦)) = (𝑦 ·_ih 𝑦))
11	7, 10	oveq12d 7293	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) + ((𝑇‘𝑦) ·_ih (𝑇‘𝑦))) = ((𝑥 ·_ih 𝑥) + (𝑦 ·_ih 𝑦)))
12		unop 30277	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
13		unop 30277	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑥)) = (𝑦 ·_ih 𝑥))
14	13	3com23 1125	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑥)) = (𝑦 ·_ih 𝑥))
15	12, 14	oveq12d 7293	. . . . . . . . . 10 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) + ((𝑇‘𝑦) ·_ih (𝑇‘𝑥))) = ((𝑥 ·_ih 𝑦) + (𝑦 ·_ih 𝑥)))
16	11, 15	oveq12d 7293	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) + ((𝑇‘𝑦) ·_ih (𝑇‘𝑦))) − (((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) + ((𝑇‘𝑦) ·_ih (𝑇‘𝑥)))) = (((𝑥 ·_ih 𝑥) + (𝑦 ·_ih 𝑦)) − ((𝑥 ·_ih 𝑦) + (𝑦 ·_ih 𝑥))))
17	16	3expb 1119	. . . . . . . 8 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) + ((𝑇‘𝑦) ·_ih (𝑇‘𝑦))) − (((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) + ((𝑇‘𝑦) ·_ih (𝑇‘𝑥)))) = (((𝑥 ·_ih 𝑥) + (𝑦 ·_ih 𝑦)) − ((𝑥 ·_ih 𝑦) + (𝑦 ·_ih 𝑥))))
18		ffvelrn 6959	. . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
19		ffvelrn 6959	. . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
20	18, 19	anim12dan 619	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
21	4, 20	sylan 580	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
22		normlem9at 29483	. . . . . . . . 9 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ·_ih ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦))) = ((((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) + ((𝑇‘𝑦) ·_ih (𝑇‘𝑦))) − (((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) + ((𝑇‘𝑦) ·_ih (𝑇‘𝑥)))))
23	21, 22	syl 17	. . . . . . . 8 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ·_ih ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦))) = ((((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) + ((𝑇‘𝑦) ·_ih (𝑇‘𝑦))) − (((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) + ((𝑇‘𝑦) ·_ih (𝑇‘𝑥)))))
24		normlem9at 29483	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 −_ℎ 𝑦) ·_ih (𝑥 −_ℎ 𝑦)) = (((𝑥 ·_ih 𝑥) + (𝑦 ·_ih 𝑦)) − ((𝑥 ·_ih 𝑦) + (𝑦 ·_ih 𝑥))))
25	24	adantl 482	. . . . . . . 8 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 −_ℎ 𝑦) ·_ih (𝑥 −_ℎ 𝑦)) = (((𝑥 ·_ih 𝑥) + (𝑦 ·_ih 𝑦)) − ((𝑥 ·_ih 𝑦) + (𝑦 ·_ih 𝑥))))
26	17, 23, 25	3eqtr4rd 2789	. . . . . . 7 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 −_ℎ 𝑦) ·_ih (𝑥 −_ℎ 𝑦)) = (((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ·_ih ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦))))
27	26	eqeq1d 2740	. . . . . 6 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 −_ℎ 𝑦) ·_ih (𝑥 −_ℎ 𝑦)) = 0 ↔ (((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ·_ih ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦))) = 0))
28		hvsubcl 29379	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 −_ℎ 𝑦) ∈ ℋ)
29		his6 29461	. . . . . . . . 9 ⊢ ((𝑥 −_ℎ 𝑦) ∈ ℋ → (((𝑥 −_ℎ 𝑦) ·_ih (𝑥 −_ℎ 𝑦)) = 0 ↔ (𝑥 −_ℎ 𝑦) = 0_ℎ))
30	28, 29	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑥 −_ℎ 𝑦) ·_ih (𝑥 −_ℎ 𝑦)) = 0 ↔ (𝑥 −_ℎ 𝑦) = 0_ℎ))
31		hvsubeq0 29430	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 −_ℎ 𝑦) = 0_ℎ ↔ 𝑥 = 𝑦))
32	30, 31	bitrd 278	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑥 −_ℎ 𝑦) ·_ih (𝑥 −_ℎ 𝑦)) = 0 ↔ 𝑥 = 𝑦))
33	32	adantl 482	. . . . . 6 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 −_ℎ 𝑦) ·_ih (𝑥 −_ℎ 𝑦)) = 0 ↔ 𝑥 = 𝑦))
34		hvsubcl 29379	. . . . . . . . 9 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ∈ ℋ)
35		his6 29461	. . . . . . . . 9 ⊢ (((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ∈ ℋ → ((((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ·_ih ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦))) = 0 ↔ ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) = 0_ℎ))
36	34, 35	syl 17	. . . . . . . 8 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ·_ih ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦))) = 0 ↔ ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) = 0_ℎ))
37		hvsubeq0 29430	. . . . . . . 8 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) = 0_ℎ ↔ (𝑇‘𝑥) = (𝑇‘𝑦)))
38	36, 37	bitrd 278	. . . . . . 7 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ·_ih ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦))) = 0 ↔ (𝑇‘𝑥) = (𝑇‘𝑦)))
39	21, 38	syl 17	. . . . . 6 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ·_ih ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦))) = 0 ↔ (𝑇‘𝑥) = (𝑇‘𝑦)))
40	27, 33, 39	3bitr3rd 310	. . . . 5 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) = (𝑇‘𝑦) ↔ 𝑥 = 𝑦))
41	40	biimpd 228	. . . 4 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
42	41	ralrimivva 3123	. . 3 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
43		dff13 7128	. . 3 ⊢ (𝑇: ℋ–1-1→ ℋ ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦)))
44	4, 42, 43	sylanbrc 583	. 2 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1→ ℋ)
45		df-f1o 6440	. 2 ⊢ (𝑇: ℋ–1-1-onto→ ℋ ↔ (𝑇: ℋ–1-1→ ℋ ∧ 𝑇: ℋ–onto→ ℋ))
46	44, 2, 45	sylanbrc 583	1 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⟶wf 6429 –1-1→wf1 6430 –onto→wfo 6431 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 0cc0 10871 + caddc 10874 − cmin 11205 ℋchba 29281 ·_ih csp 29284 0_ℎc0v 29286 −_ℎ cmv 29287 UniOpcuo 29311

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-hilex 29361 ax-hfvadd 29362 ax-hvcom 29363 ax-hvass 29364 ax-hv0cl 29365 ax-hvaddid 29366 ax-hfvmul 29367 ax-hvmulid 29368 ax-hvdistr2 29371 ax-hvmul0 29372 ax-hfi 29441 ax-his1 29444 ax-his2 29445 ax-his3 29446 ax-his4 29447

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-2 12036 df-cj 14810 df-re 14811 df-im 14812 df-hvsub 29333 df-unop 30205

This theorem is referenced by: unopnorm 30279 cnvunop 30280 unopadj 30281 unoplin 30282 counop 30283 unopbd 30377

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