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Theorem unopf1o 29106
 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.
StepHypRef Expression
1 elunop 29062 . . . . 5 (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
21simplbi 478 . . . 4 (𝑇 ∈ UniOp → 𝑇: ℋ–onto→ ℋ)
3 fof 6278 . . . 4 (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ)
42, 3syl 17 . . 3 (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
5 unop 29105 . . . . . . . . . . . . 13 ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇𝑥) ·ih (𝑇𝑥)) = (𝑥 ·ih 𝑥))
653anidm23 1532 . . . . . . . . . . . 12 ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → ((𝑇𝑥) ·ih (𝑇𝑥)) = (𝑥 ·ih 𝑥))
763adant3 1127 . . . . . . . . . . 11 ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑥) ·ih (𝑇𝑥)) = (𝑥 ·ih 𝑥))
8 unop 29105 . . . . . . . . . . . . 13 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑦) ·ih (𝑇𝑦)) = (𝑦 ·ih 𝑦))
983anidm23 1532 . . . . . . . . . . . 12 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((𝑇𝑦) ·ih (𝑇𝑦)) = (𝑦 ·ih 𝑦))
1093adant2 1126 . . . . . . . . . . 11 ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑦) ·ih (𝑇𝑦)) = (𝑦 ·ih 𝑦))
117, 10oveq12d 6833 . . . . . . . . . 10 ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑇𝑥) ·ih (𝑇𝑥)) + ((𝑇𝑦) ·ih (𝑇𝑦))) = ((𝑥 ·ih 𝑥) + (𝑦 ·ih 𝑦)))
12 unop 29105 . . . . . . . . . . 11 ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦))
13 unop 29105 . . . . . . . . . . . 12 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇𝑦) ·ih (𝑇𝑥)) = (𝑦 ·ih 𝑥))
14133com23 1121 . . . . . . . . . . 11 ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑦) ·ih (𝑇𝑥)) = (𝑦 ·ih 𝑥))
1512, 14oveq12d 6833 . . . . . . . . . 10 ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑇𝑥) ·ih (𝑇𝑦)) + ((𝑇𝑦) ·ih (𝑇𝑥))) = ((𝑥 ·ih 𝑦) + (𝑦 ·ih 𝑥)))
1611, 15oveq12d 6833 . . . . . . . . 9 ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((((𝑇𝑥) ·ih (𝑇𝑥)) + ((𝑇𝑦) ·ih (𝑇𝑦))) − (((𝑇𝑥) ·ih (𝑇𝑦)) + ((𝑇𝑦) ·ih (𝑇𝑥)))) = (((𝑥 ·ih 𝑥) + (𝑦 ·ih 𝑦)) − ((𝑥 ·ih 𝑦) + (𝑦 ·ih 𝑥))))
17163expb 1114 . . . . . . . 8 ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((((𝑇𝑥) ·ih (𝑇𝑥)) + ((𝑇𝑦) ·ih (𝑇𝑦))) − (((𝑇𝑥) ·ih (𝑇𝑦)) + ((𝑇𝑦) ·ih (𝑇𝑥)))) = (((𝑥 ·ih 𝑥) + (𝑦 ·ih 𝑦)) − ((𝑥 ·ih 𝑦) + (𝑦 ·ih 𝑥))))
18 ffvelrn 6522 . . . . . . . . . . 11 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
19 ffvelrn 6522 . . . . . . . . . . 11 ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇𝑦) ∈ ℋ)
2018, 19anim12dan 918 . . . . . . . . . 10 ((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ∈ ℋ ∧ (𝑇𝑦) ∈ ℋ))
214, 20sylan 489 . . . . . . . . 9 ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ∈ ℋ ∧ (𝑇𝑦) ∈ ℋ))
22 normlem9at 28309 . . . . . . . . 9 (((𝑇𝑥) ∈ ℋ ∧ (𝑇𝑦) ∈ ℋ) → (((𝑇𝑥) − (𝑇𝑦)) ·ih ((𝑇𝑥) − (𝑇𝑦))) = ((((𝑇𝑥) ·ih (𝑇𝑥)) + ((𝑇𝑦) ·ih (𝑇𝑦))) − (((𝑇𝑥) ·ih (𝑇𝑦)) + ((𝑇𝑦) ·ih (𝑇𝑥)))))
2321, 22syl 17 . . . . . . . 8 ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑇𝑥) − (𝑇𝑦)) ·ih ((𝑇𝑥) − (𝑇𝑦))) = ((((𝑇𝑥) ·ih (𝑇𝑥)) + ((𝑇𝑦) ·ih (𝑇𝑦))) − (((𝑇𝑥) ·ih (𝑇𝑦)) + ((𝑇𝑦) ·ih (𝑇𝑥)))))
24 normlem9at 28309 . . . . . . . . 9 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 𝑦) ·ih (𝑥 𝑦)) = (((𝑥 ·ih 𝑥) + (𝑦 ·ih 𝑦)) − ((𝑥 ·ih 𝑦) + (𝑦 ·ih 𝑥))))
2524adantl 473 . . . . . . . 8 ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 𝑦) ·ih (𝑥 𝑦)) = (((𝑥 ·ih 𝑥) + (𝑦 ·ih 𝑦)) − ((𝑥 ·ih 𝑦) + (𝑦 ·ih 𝑥))))
2617, 23, 253eqtr4rd 2806 . . . . . . 7 ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 𝑦) ·ih (𝑥 𝑦)) = (((𝑇𝑥) − (𝑇𝑦)) ·ih ((𝑇𝑥) − (𝑇𝑦))))
2726eqeq1d 2763 . . . . . 6 ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 𝑦) ·ih (𝑥 𝑦)) = 0 ↔ (((𝑇𝑥) − (𝑇𝑦)) ·ih ((𝑇𝑥) − (𝑇𝑦))) = 0))
28 hvsubcl 28205 . . . . . . . . 9 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 𝑦) ∈ ℋ)
29 his6 28287 . . . . . . . . 9 ((𝑥 𝑦) ∈ ℋ → (((𝑥 𝑦) ·ih (𝑥 𝑦)) = 0 ↔ (𝑥 𝑦) = 0))
3028, 29syl 17 . . . . . . . 8 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑥 𝑦) ·ih (𝑥 𝑦)) = 0 ↔ (𝑥 𝑦) = 0))
31 hvsubeq0 28256 . . . . . . . 8 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 𝑦) = 0𝑥 = 𝑦))
3230, 31bitrd 268 . . . . . . 7 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑥 𝑦) ·ih (𝑥 𝑦)) = 0 ↔ 𝑥 = 𝑦))
3332adantl 473 . . . . . 6 ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 𝑦) ·ih (𝑥 𝑦)) = 0 ↔ 𝑥 = 𝑦))
34 hvsubcl 28205 . . . . . . . . 9 (((𝑇𝑥) ∈ ℋ ∧ (𝑇𝑦) ∈ ℋ) → ((𝑇𝑥) − (𝑇𝑦)) ∈ ℋ)
35 his6 28287 . . . . . . . . 9 (((𝑇𝑥) − (𝑇𝑦)) ∈ ℋ → ((((𝑇𝑥) − (𝑇𝑦)) ·ih ((𝑇𝑥) − (𝑇𝑦))) = 0 ↔ ((𝑇𝑥) − (𝑇𝑦)) = 0))
3634, 35syl 17 . . . . . . . 8 (((𝑇𝑥) ∈ ℋ ∧ (𝑇𝑦) ∈ ℋ) → ((((𝑇𝑥) − (𝑇𝑦)) ·ih ((𝑇𝑥) − (𝑇𝑦))) = 0 ↔ ((𝑇𝑥) − (𝑇𝑦)) = 0))
37 hvsubeq0 28256 . . . . . . . 8 (((𝑇𝑥) ∈ ℋ ∧ (𝑇𝑦) ∈ ℋ) → (((𝑇𝑥) − (𝑇𝑦)) = 0 ↔ (𝑇𝑥) = (𝑇𝑦)))
3836, 37bitrd 268 . . . . . . 7 (((𝑇𝑥) ∈ ℋ ∧ (𝑇𝑦) ∈ ℋ) → ((((𝑇𝑥) − (𝑇𝑦)) ·ih ((𝑇𝑥) − (𝑇𝑦))) = 0 ↔ (𝑇𝑥) = (𝑇𝑦)))
3921, 38syl 17 . . . . . 6 ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((((𝑇𝑥) − (𝑇𝑦)) ·ih ((𝑇𝑥) − (𝑇𝑦))) = 0 ↔ (𝑇𝑥) = (𝑇𝑦)))
4027, 33, 393bitr3rd 299 . . . . 5 ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) = (𝑇𝑦) ↔ 𝑥 = 𝑦))
4140biimpd 219 . . . 4 ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) = (𝑇𝑦) → 𝑥 = 𝑦))
4241ralrimivva 3110 . . 3 (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) = (𝑇𝑦) → 𝑥 = 𝑦))
43 dff13 6677 . . 3 (𝑇: ℋ–1-1→ ℋ ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) = (𝑇𝑦) → 𝑥 = 𝑦)))
444, 42, 43sylanbrc 701 . 2 (𝑇 ∈ UniOp → 𝑇: ℋ–1-1→ ℋ)
45 df-f1o 6057 . 2 (𝑇: ℋ–1-1-onto→ ℋ ↔ (𝑇: ℋ–1-1→ ℋ ∧ 𝑇: ℋ–onto→ ℋ))
4644, 2, 45sylanbrc 701 1 (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2140  ∀wral 3051  ⟶wf 6046  –1-1→wf1 6047  –onto→wfo 6048  –1-1-onto→wf1o 6049  ‘cfv 6050  (class class class)co 6815  0cc0 10149   + caddc 10152   − cmin 10479   ℋchil 28107   ·ih csp 28110  0ℎc0v 28112   −ℎ cmv 28113  UniOpcuo 28137 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226  ax-hilex 28187  ax-hfvadd 28188  ax-hvcom 28189  ax-hvass 28190  ax-hv0cl 28191  ax-hvaddid 28192  ax-hfvmul 28193  ax-hvmulid 28194  ax-hvdistr2 28197  ax-hvmul0 28198  ax-hfi 28267  ax-his1 28270  ax-his2 28271  ax-his3 28272  ax-his4 28273 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-po 5188  df-so 5189  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-er 7914  df-en 8125  df-dom 8126  df-sdom 8127  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-div 10898  df-2 11292  df-cj 14059  df-re 14060  df-im 14061  df-hvsub 28159  df-unop 29033 This theorem is referenced by:  unopnorm  29107  cnvunop  29108  unopadj  29109  unoplin  29110  counop  29111  unopbd  29205
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