unopf1o - Hilbert Space Explorer

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Theorem unopf1o 31987

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 31943	. . . . 5 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦)))
2	1	simplbi 496	. . . 4 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–onto→ ℋ)
3		fof 6752	. . . 4 ⊢ (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ)
4	2, 3	syl 17	. . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
5		unop 31986	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) = (𝑥 ·_ih 𝑥))
6	5	3anidm23 1424	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) = (𝑥 ·_ih 𝑥))
7	6	3adant3 1133	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑥)) = (𝑥 ·_ih 𝑥))
8		unop 31986	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑦)) = (𝑦 ·_ih 𝑦))
9	8	3anidm23 1424	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑦)) = (𝑦 ·_ih 𝑦))
10	9	3adant2 1132	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑦)) = (𝑦 ·_ih 𝑦))
11	7, 10	oveq12d 7385	. . . . . . . . . 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 31986	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
13		unop 31986	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑥)) = (𝑦 ·_ih 𝑥))
14	13	3com23 1127	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑦) ·_ih (𝑇‘𝑥)) = (𝑦 ·_ih 𝑥))
15	12, 14	oveq12d 7385	. . . . . . . . . 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 7385	. . . . . . . . 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 1121	. . . . . . . 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 7033	. . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
19		ffvelcdm 7033	. . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
20	18, 19	anim12dan 620	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
21	4, 20	sylan 581	. . . . . . . . 9 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
22		normlem9at 31192	. . . . . . . . 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 31192	. . . . . . . . 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 31088	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 −_ℎ 𝑦) ∈ ℋ)
29		his6 31170	. . . . . . . . 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 31139	. . . . . . . 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 31088	. . . . . . . . 9 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((𝑇‘𝑥) −_ℎ (𝑇‘𝑦)) ∈ ℋ)
35		his6 31170	. . . . . . . . 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 31139	. . . . . . . 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 3180	. . 3 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
43		dff13 7209	. . 3 ⊢ (𝑇: ℋ–1-1→ ℋ ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦)))
44	4, 42, 43	sylanbrc 584	. 2 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1→ ℋ)
45		df-f1o 6505	. 2 ⊢ (𝑇: ℋ–1-1-onto→ ℋ ↔ (𝑇: ℋ–1-1→ ℋ ∧ 𝑇: ℋ–onto→ ℋ))
46	44, 2, 45	sylanbrc 584	1 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3051 ⟶wf 6494 –1-1→wf1 6495 –onto→wfo 6496 –1-1-onto→wf1o 6497 ‘cfv 6498 (class class class)co 7367 0cc0 11038 + caddc 11041 − cmin 11377 ℋchba 30990 ·_ih csp 30993 0_ℎc0v 30995 −_ℎ cmv 30996 UniOpcuo 31020

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-hilex 31070 ax-hfvadd 31071 ax-hvcom 31072 ax-hvass 31073 ax-hv0cl 31074 ax-hvaddid 31075 ax-hfvmul 31076 ax-hvmulid 31077 ax-hvdistr2 31080 ax-hvmul0 31081 ax-hfi 31150 ax-his1 31153 ax-his2 31154 ax-his3 31155 ax-his4 31156

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 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 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-cj 15061 df-re 15062 df-im 15063 df-hvsub 31042 df-unop 31914

This theorem is referenced by: unopnorm 31988 cnvunop 31989 unopadj 31990 unoplin 31991 counop 31992 unopbd 32086

W3C validator