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Theorem elunop2 31735
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 31642 . . 3 (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2 elunop 31594 . . . 4 (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
32simplbi 497 . . 3 (𝑇 ∈ UniOp → 𝑇: ℋ–onto→ ℋ)
4 unopnorm 31639 . . . 4 ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → (norm‘(𝑇𝑥)) = (norm𝑥))
54ralrimiva 3138 . . 3 (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥))
61, 3, 53jca 1125 . 2 (𝑇 ∈ UniOp → (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)))
7 eleq1 2813 . . 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 2813 . . . . . . 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 6791 . . . . . . 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 6886 . . . . . . . . . 10 (𝑥 = 𝑦 → (norm‘(𝑇𝑥)) = (norm‘(𝑇𝑦)))
11 fveq2 6881 . . . . . . . . . 10 (𝑥 = 𝑦 → (norm𝑥) = (norm𝑦))
1210, 11eqeq12d 2740 . . . . . . . . 9 (𝑥 = 𝑦 → ((norm‘(𝑇𝑥)) = (norm𝑥) ↔ (norm‘(𝑇𝑦)) = (norm𝑦)))
1312cbvralvw 3226 . . . . . . . 8 (∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥) ↔ ∀𝑦 ∈ ℋ (norm‘(𝑇𝑦)) = (norm𝑦))
14 fveq1 6880 . . . . . . . . . 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 6889 . . . . . . . . 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 3169 . . . . . . . 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, 16bitrid 283 . . . . . . 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 1432 . . . . . 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 2813 . . . . . . 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 6791 . . . . . . 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 6880 . . . . . . . . 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 6889 . . . . . . . 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 3169 . . . . . . 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 1432 . . . . . 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 31714 . . . . . . 7 ( I ↾ ℋ) ∈ LinOp
26 f1oi 6861 . . . . . . . 8 ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
27 f1ofo 6830 . . . . . . . 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 7163 . . . . . . . . 9 (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
3029fveq2d 6885 . . . . . . . 8 (𝑦 ∈ ℋ → (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦))
3130rgen 3055 . . . . . . 7 𝑦 ∈ ℋ (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦)
3225, 28, 313pm3.2i 1336 . . . . . 6 (( I ↾ ℋ) ∈ LinOp ∧ ( I ↾ ℋ): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦))
3318, 24, 32elimhyp 4585 . . . . 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 1136 . . . 4 if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp
3533simp2i 1137 . . . 4 if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ
3633simp3i 1138 . . . 4 𝑦 ∈ ℋ (norm‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (norm𝑦)
3734, 35, 36lnopunii 31734 . . 3 if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ UniOp
387, 37dedth 4578 . 2 ((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)) → 𝑇 ∈ UniOp)
396, 38impbii 208 1 (𝑇 ∈ UniOp ↔ (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  w3a 1084   = wceq 1533  wcel 2098  wral 3053  ifcif 4520   I cid 5563  cres 5668  ontowfo 6531  1-1-ontowf1o 6532  cfv 6533  (class class class)co 7401  chba 30641   ·ih csp 30644  normcno 30645  LinOpclo 30669  UniOpcuo 30671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-hilex 30721  ax-hfvadd 30722  ax-hvcom 30723  ax-hvass 30724  ax-hv0cl 30725  ax-hvaddid 30726  ax-hfvmul 30727  ax-hvmulid 30728  ax-hvdistr2 30731  ax-hvmul0 30732  ax-hfi 30801  ax-his1 30804  ax-his2 30805  ax-his3 30806  ax-his4 30807
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-rp 12972  df-seq 13964  df-exp 14025  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-hnorm 30690  df-hvsub 30693  df-lnop 31563  df-unop 31565
This theorem is referenced by: (None)
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