unopf1o - Hilbert Space Explorer

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Theorem unopf1o 31991

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 31947	. . . . 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 6746	. . . 4 ⊢ (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ)
4	2, 3	syl 17	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
5		unop 31990	. . . . . . . . . . . . 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 31990	. . . . . . . . . . . . 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 7376	. . . . . . . . . 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 31990	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
13		unop 31990	. . . . . . . . . . . 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 7376	. . . . . . . . . 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 7376	. . . . . . . . 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 7026	. . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
19		ffvelcdm 7026	. . . . . . . . . . 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 31196	. . . . . . . . 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 31196	. . . . . . . . 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 2782	. . . . . . 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 2738	. . . . . 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 31092	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 −_ℎ 𝑦) ∈ ℋ)
29		his6 31174	. . . . . . . . 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 31143	. . . . . . . 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 31092	. . . . . . . . 9 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ∈ ℋ)
35		his6 31174	. . . . . . . . 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 31143	. . . . . . . 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 3179	. . 3 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
43		dff13 7200	. . 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 6499	. 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 1541 ∈ wcel 2113 ∀wral 3051 ⟶wf 6488 –1-1→wf1 6489 –onto→wfo 6490 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7358 0cc0 11026 + caddc 11029 − cmin 11364 ℋchba 30994 ·_ih csp 30997 0_ℎc0v 30999 −_ℎ cmv 31000 UniOpcuo 31024

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-hilex 31074 ax-hfvadd 31075 ax-hvcom 31076 ax-hvass 31077 ax-hv0cl 31078 ax-hvaddid 31079 ax-hfvmul 31080 ax-hvmulid 31081 ax-hvdistr2 31084 ax-hvmul0 31085 ax-hfi 31154 ax-his1 31157 ax-his2 31158 ax-his3 31159 ax-his4 31160

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-cj 15022 df-re 15023 df-im 15024 df-hvsub 31046 df-unop 31918

This theorem is referenced by: unopnorm 31992 cnvunop 31993 unopadj 31994 unoplin 31995 counop 31996 unopbd 32090

W3C validator