unopf1o - Hilbert Space Explorer

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Theorem unopf1o 31864

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 31820	. . . . 5 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦)))
2	1	simplbi 497	. . . 4 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–onto→ ℋ)
3		fof 6736	. . . 4 ⊢ (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ)
4	2, 3	syl 17	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
5		unop 31863	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) = (𝑥 ·_ih 𝑥))
6	5	3anidm23 1423	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) = (𝑥 ·_ih 𝑥))
7	6	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) = (𝑥 ·_ih 𝑥))
8		unop 31863	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑦)) = (𝑦 ·_ih 𝑦))
9	8	3anidm23 1423	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑦)) = (𝑦 ·_ih 𝑦))
10	9	3adant2 1131	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑦)) = (𝑦 ·_ih 𝑦))
11	7, 10	oveq12d 7367	. . . . . . . . . 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 31863	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
13		unop 31863	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑥)) = (𝑦 ·_ih 𝑥))
14	13	3com23 1126	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑥)) = (𝑦 ·_ih 𝑥))
15	12, 14	oveq12d 7367	. . . . . . . . . 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 7367	. . . . . . . . 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 1120	. . . . . . . 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		ffvelcdm 7015	. . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
19		ffvelcdm 7015	. . . . . . . . . . 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 31069	. . . . . . . . 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 31069	. . . . . . . . 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 481	. . . . . . . 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 2775	. . . . . . 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 2731	. . . . . 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 30965	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 −_ℎ 𝑦) ∈ ℋ)
29		his6 31047	. . . . . . . . 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 31016	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 −_ℎ 𝑦) = 0_ℎ ↔ 𝑥 = 𝑦))
32	30, 31	bitrd 279	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑥 −_ℎ 𝑦) ·_ih (𝑥 −_ℎ 𝑦)) = 0 ↔ 𝑥 = 𝑦))
33	32	adantl 481	. . . . . 6 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 −_ℎ 𝑦) ·_ih (𝑥 −_ℎ 𝑦)) = 0 ↔ 𝑥 = 𝑦))
34		hvsubcl 30965	. . . . . . . . 9 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ∈ ℋ)
35		his6 31047	. . . . . . . . 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 31016	. . . . . . . 8 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) = 0_ℎ ↔ (𝑇‘𝑥) = (𝑇‘𝑦)))
38	36, 37	bitrd 279	. . . . . . 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 229	. . . 4 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
42	41	ralrimivva 3172	. . 3 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
43		dff13 7191	. . 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 6489	. 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⟶wf 6478 –1-1→wf1 6479 –onto→wfo 6480 –1-1-onto→wf1o 6481 ‘cfv 6482 (class class class)co 7349 0cc0 11009 + caddc 11012 − cmin 11347 ℋchba 30867 ·_ih csp 30870 0_ℎc0v 30872 −_ℎ cmv 30873 UniOpcuo 30897

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-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-hilex 30947 ax-hfvadd 30948 ax-hvcom 30949 ax-hvass 30950 ax-hv0cl 30951 ax-hvaddid 30952 ax-hfvmul 30953 ax-hvmulid 30954 ax-hvdistr2 30957 ax-hvmul0 30958 ax-hfi 31027 ax-his1 31030 ax-his2 31031 ax-his3 31032 ax-his4 31033

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-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-cj 15006 df-re 15007 df-im 15008 df-hvsub 30919 df-unop 31791

This theorem is referenced by: unopnorm 31865 cnvunop 31866 unopadj 31867 unoplin 31868 counop 31869 unopbd 31963

W3C validator