unopf1o - Hilbert Space Explorer

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Theorem unopf1o 31945

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 31901	. . . . 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 6821	. . . 4 ⊢ (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ)
4	2, 3	syl 17	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
5		unop 31944	. . . . . . . . . . . . 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 31944	. . . . . . . . . . . . 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 7449	. . . . . . . . . 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 31944	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
13		unop 31944	. . . . . . . . . . . 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 7449	. . . . . . . . . 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 7449	. . . . . . . . 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		ffvelcdm 7101	. . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
19		ffvelcdm 7101	. . . . . . . . . . 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 31150	. . . . . . . . 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 31150	. . . . . . . . 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 2786	. . . . . . 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 2737	. . . . . 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 31046	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 −_ℎ 𝑦) ∈ ℋ)
29		his6 31128	. . . . . . . . 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 31097	. . . . . . . 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 31046	. . . . . . . . 9 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ∈ ℋ)
35		his6 31128	. . . . . . . . 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 31097	. . . . . . . 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 3200	. . 3 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
43		dff13 7275	. . 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 6570	. 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 1537 ∈ wcel 2106 ∀wral 3059 ⟶wf 6559 –1-1→wf1 6560 –onto→wfo 6561 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 0cc0 11153 + caddc 11156 − cmin 11490 ℋchba 30948 ·_ih csp 30951 0_ℎc0v 30953 −_ℎ cmv 30954 UniOpcuo 30978

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-hilex 31028 ax-hfvadd 31029 ax-hvcom 31030 ax-hvass 31031 ax-hv0cl 31032 ax-hvaddid 31033 ax-hfvmul 31034 ax-hvmulid 31035 ax-hvdistr2 31038 ax-hvmul0 31039 ax-hfi 31108 ax-his1 31111 ax-his2 31112 ax-his3 31113 ax-his4 31114

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 df-cj 15135 df-re 15136 df-im 15137 df-hvsub 31000 df-unop 31872

This theorem is referenced by: unopnorm 31946 cnvunop 31947 unopadj 31948 unoplin 31949 counop 31950 unopbd 32044

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