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Theorem elunop2 29704
 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 29611 . . 3 (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2 elunop 29563 . . . 4 (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
32simplbi 498 . . 3 (𝑇 ∈ UniOp → 𝑇: ℋ–onto→ ℋ)
4 unopnorm 29608 . . . 4 ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → (norm‘(𝑇𝑥)) = (norm𝑥))
54ralrimiva 3186 . . 3 (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥))
61, 3, 53jca 1122 . 2 (𝑇 ∈ UniOp → (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)))
7 eleq1 2904 . . 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 2904 . . . . . . 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 6582 . . . . . . 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 6671 . . . . . . . . . 10 (𝑥 = 𝑦 → (norm‘(𝑇𝑥)) = (norm‘(𝑇𝑦)))
11 fveq2 6666 . . . . . . . . . 10 (𝑥 = 𝑦 → (norm𝑥) = (norm𝑦))
1210, 11eqeq12d 2841 . . . . . . . . 9 (𝑥 = 𝑦 → ((norm‘(𝑇𝑥)) = (norm𝑥) ↔ (norm‘(𝑇𝑦)) = (norm𝑦)))
1312cbvralv 3457 . . . . . . . 8 (∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥) ↔ ∀𝑦 ∈ ℋ (norm‘(𝑇𝑦)) = (norm𝑦))
14 fveq1 6665 . . . . . . . . . 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 6674 . . . . . . . . 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 3201 . . . . . . . 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 284 . . . . . . 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 1429 . . . . . 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 2904 . . . . . . 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 6582 . . . . . . 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 6665 . . . . . . . . 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 6674 . . . . . . . 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 3201 . . . . . . 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 1429 . . . . . 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 29683 . . . . . . 7 ( I ↾ ℋ) ∈ LinOp
26 f1oi 6648 . . . . . . . 8 ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
27 f1ofo 6618 . . . . . . . 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 6930 . . . . . . . . 9 (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
3029fveq2d 6670 . . . . . . . 8 (𝑦 ∈ ℋ → (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦))
3130rgen 3152 . . . . . . 7 𝑦 ∈ ℋ (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦)
3225, 28, 313pm3.2i 1333 . . . . . 6 (( I ↾ ℋ) ∈ LinOp ∧ ( I ↾ ℋ): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦))
3318, 24, 32elimhyp 4532 . . . . 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 1133 . . . 4 if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp
3533simp2i 1134 . . . 4 if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ
3633simp3i 1135 . . . 4 𝑦 ∈ ℋ (norm‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (norm𝑦)
3734, 35, 36lnopunii 29703 . . 3 if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ UniOp
387, 37dedth 4525 . 2 ((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)) → 𝑇 ∈ UniOp)
396, 38impbii 210 1 (𝑇 ∈ UniOp ↔ (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107  ∀wral 3142  ifcif 4469   I cid 5457   ↾ cres 5555  –onto→wfo 6349  –1-1-onto→wf1o 6350  ‘cfv 6351  (class class class)co 7151   ℋchba 28610   ·ih csp 28613  normℎcno 28614  LinOpclo 28638  UniOpcuo 28640 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-hilex 28690  ax-hfvadd 28691  ax-hvcom 28692  ax-hvass 28693  ax-hv0cl 28694  ax-hvaddid 28695  ax-hfvmul 28696  ax-hvmulid 28697  ax-hvdistr2 28700  ax-hvmul0 28701  ax-hfi 28770  ax-his1 28773  ax-his2 28774  ax-his3 28775  ax-his4 28776 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-sup 8898  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12383  df-seq 13363  df-exp 13423  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-hnorm 28659  df-hvsub 28662  df-lnop 29532  df-unop 29534 This theorem is referenced by: (None)
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