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Theorem elunop2 29948
Description: An operator is unitary iff it is linear, onto, and idempotent in the norm. Similar to theorem in [AkhiezerGlazman] p. 73, and its converse. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
elunop2 (𝑇 ∈ UniOp ↔ (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)))
Distinct variable group:   𝑥,𝑇

Proof of Theorem elunop2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 unoplin 29855 . . 3 (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2 elunop 29807 . . . 4 (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
32simplbi 501 . . 3 (𝑇 ∈ UniOp → 𝑇: ℋ–onto→ ℋ)
4 unopnorm 29852 . . . 4 ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → (norm‘(𝑇𝑥)) = (norm𝑥))
54ralrimiva 3096 . . 3 (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥))
61, 3, 53jca 1129 . 2 (𝑇 ∈ UniOp → (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)))
7 eleq1 2820 . . 3 (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → (𝑇 ∈ UniOp ↔ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ UniOp))
8 eleq1 2820 . . . . . . 7 (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → (𝑇 ∈ LinOp ↔ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp))
9 foeq1 6588 . . . . . . 7 (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → (𝑇: ℋ–onto→ ℋ ↔ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ))
10 2fveq3 6679 . . . . . . . . . 10 (𝑥 = 𝑦 → (norm‘(𝑇𝑥)) = (norm‘(𝑇𝑦)))
11 fveq2 6674 . . . . . . . . . 10 (𝑥 = 𝑦 → (norm𝑥) = (norm𝑦))
1210, 11eqeq12d 2754 . . . . . . . . 9 (𝑥 = 𝑦 → ((norm‘(𝑇𝑥)) = (norm𝑥) ↔ (norm‘(𝑇𝑦)) = (norm𝑦)))
1312cbvralvw 3349 . . . . . . . 8 (∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥) ↔ ∀𝑦 ∈ ℋ (norm‘(𝑇𝑦)) = (norm𝑦))
14 fveq1 6673 . . . . . . . . . 10 (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → (𝑇𝑦) = (if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦))
1514fveqeq2d 6682 . . . . . . . . 9 (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → ((norm‘(𝑇𝑦)) = (norm𝑦) ↔ (norm‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (norm𝑦)))
1615ralbidv 3109 . . . . . . . 8 (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → (∀𝑦 ∈ ℋ (norm‘(𝑇𝑦)) = (norm𝑦) ↔ ∀𝑦 ∈ ℋ (norm‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (norm𝑦)))
1713, 16syl5bb 286 . . . . . . 7 (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → (∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥) ↔ ∀𝑦 ∈ ℋ (norm‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (norm𝑦)))
188, 9, 173anbi123d 1437 . . . . . 6 (𝑇 = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → ((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)) ↔ (if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp ∧ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (norm‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (norm𝑦))))
19 eleq1 2820 . . . . . . 7 (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → (( I ↾ ℋ) ∈ LinOp ↔ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp))
20 foeq1 6588 . . . . . . 7 (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → (( I ↾ ℋ): ℋ–onto→ ℋ ↔ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ))
21 fveq1 6673 . . . . . . . . 9 (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → (( I ↾ ℋ)‘𝑦) = (if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦))
2221fveqeq2d 6682 . . . . . . . 8 (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → ((norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦) ↔ (norm‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (norm𝑦)))
2322ralbidv 3109 . . . . . . 7 (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → (∀𝑦 ∈ ℋ (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦) ↔ ∀𝑦 ∈ ℋ (norm‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (norm𝑦)))
2419, 20, 233anbi123d 1437 . . . . . 6 (( I ↾ ℋ) = if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) → ((( I ↾ ℋ) ∈ LinOp ∧ ( I ↾ ℋ): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦)) ↔ (if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp ∧ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (norm‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (norm𝑦))))
25 idlnop 29927 . . . . . . 7 ( I ↾ ℋ) ∈ LinOp
26 f1oi 6655 . . . . . . . 8 ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
27 f1ofo 6625 . . . . . . . 8 (( I ↾ ℋ): ℋ–1-1-onto→ ℋ → ( I ↾ ℋ): ℋ–onto→ ℋ)
2826, 27ax-mp 5 . . . . . . 7 ( I ↾ ℋ): ℋ–onto→ ℋ
29 fvresi 6945 . . . . . . . . 9 (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
3029fveq2d 6678 . . . . . . . 8 (𝑦 ∈ ℋ → (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦))
3130rgen 3063 . . . . . . 7 𝑦 ∈ ℋ (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦)
3225, 28, 313pm3.2i 1340 . . . . . 6 (( I ↾ ℋ) ∈ LinOp ∧ ( I ↾ ℋ): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦))
3318, 24, 32elimhyp 4479 . . . . 5 (if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp ∧ if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (norm‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (norm𝑦))
3433simp1i 1140 . . . 4 if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp
3533simp2i 1141 . . . 4 if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ
3633simp3i 1142 . . . 4 𝑦 ∈ ℋ (norm‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (norm𝑦)
3734, 35, 36lnopunii 29947 . . 3 if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ UniOp
387, 37dedth 4472 . 2 ((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)) → 𝑇 ∈ UniOp)
396, 38impbii 212 1 (𝑇 ∈ UniOp ↔ (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  w3a 1088   = wceq 1542  wcel 2114  wral 3053  ifcif 4414   I cid 5428  cres 5527  ontowfo 6337  1-1-ontowf1o 6338  cfv 6339  (class class class)co 7170  chba 28854   ·ih csp 28857  normcno 28858  LinOpclo 28882  UniOpcuo 28884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693  ax-hilex 28934  ax-hfvadd 28935  ax-hvcom 28936  ax-hvass 28937  ax-hv0cl 28938  ax-hvaddid 28939  ax-hfvmul 28940  ax-hvmulid 28941  ax-hvdistr2 28944  ax-hvmul0 28945  ax-hfi 29014  ax-his1 29017  ax-his2 29018  ax-his3 29019  ax-his4 29020
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-er 8320  df-map 8439  df-en 8556  df-dom 8557  df-sdom 8558  df-sup 8979  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-2 11779  df-3 11780  df-n0 11977  df-z 12063  df-uz 12325  df-rp 12473  df-seq 13461  df-exp 13522  df-cj 14548  df-re 14549  df-im 14550  df-sqrt 14684  df-hnorm 28903  df-hvsub 28906  df-lnop 29776  df-unop 29778
This theorem is referenced by: (None)
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