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Theorem elunop2 29397
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 29304 . . 3 (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2 elunop 29256 . . . 4 (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
32simplbi 492 . . 3 (𝑇 ∈ UniOp → 𝑇: ℋ–onto→ ℋ)
4 unopnorm 29301 . . . 4 ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → (norm‘(𝑇𝑥)) = (norm𝑥))
54ralrimiva 3147 . . 3 (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥))
61, 3, 53jca 1159 . 2 (𝑇 ∈ UniOp → (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)))
7 eleq1 2866 . . 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 2866 . . . . . . 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 6327 . . . . . . 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 6416 . . . . . . . . . 10 (𝑥 = 𝑦 → (norm‘(𝑇𝑥)) = (norm‘(𝑇𝑦)))
11 fveq2 6411 . . . . . . . . . 10 (𝑥 = 𝑦 → (norm𝑥) = (norm𝑦))
1210, 11eqeq12d 2814 . . . . . . . . 9 (𝑥 = 𝑦 → ((norm‘(𝑇𝑥)) = (norm𝑥) ↔ (norm‘(𝑇𝑦)) = (norm𝑦)))
1312cbvralv 3354 . . . . . . . 8 (∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥) ↔ ∀𝑦 ∈ ℋ (norm‘(𝑇𝑦)) = (norm𝑦))
14 fveq1 6410 . . . . . . . . . 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 6419 . . . . . . . . 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 3167 . . . . . . . 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 275 . . . . . . 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 1561 . . . . . 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 2866 . . . . . . 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 6327 . . . . . . 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 6410 . . . . . . . . 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 6419 . . . . . . . 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 3167 . . . . . . 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 1561 . . . . . 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 29376 . . . . . . 7 ( I ↾ ℋ) ∈ LinOp
26 f1oi 6393 . . . . . . . 8 ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
27 f1ofo 6363 . . . . . . . 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 6668 . . . . . . . . 9 (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
3029fveq2d 6415 . . . . . . . 8 (𝑦 ∈ ℋ → (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦))
3130rgen 3103 . . . . . . 7 𝑦 ∈ ℋ (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦)
3225, 28, 313pm3.2i 1439 . . . . . 6 (( I ↾ ℋ) ∈ LinOp ∧ ( I ↾ ℋ): ℋ–onto→ ℋ ∧ ∀𝑦 ∈ ℋ (norm‘(( I ↾ ℋ)‘𝑦)) = (norm𝑦))
3318, 24, 32elimhyp 4340 . . . . 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 1170 . . . 4 if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ LinOp
3533simp2i 1171 . . . 4 if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)): ℋ–onto→ ℋ
3633simp3i 1172 . . . 4 𝑦 ∈ ℋ (norm‘(if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ))‘𝑦)) = (norm𝑦)
3734, 35, 36lnopunii 29396 . . 3 if((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)), 𝑇, ( I ↾ ℋ)) ∈ UniOp
387, 37dedth 4333 . 2 ((𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)) → 𝑇 ∈ UniOp)
396, 38impbii 201 1 (𝑇 ∈ UniOp ↔ (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) = (norm𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  w3a 1108   = wceq 1653  wcel 2157  wral 3089  ifcif 4277   I cid 5219  cres 5314  ontowfo 6099  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6878  chba 28301   ·ih csp 28304  normcno 28305  LinOpclo 28329  UniOpcuo 28331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302  ax-hilex 28381  ax-hfvadd 28382  ax-hvcom 28383  ax-hvass 28384  ax-hv0cl 28385  ax-hvaddid 28386  ax-hfvmul 28387  ax-hvmulid 28388  ax-hvdistr2 28391  ax-hvmul0 28392  ax-hfi 28461  ax-his1 28464  ax-his2 28465  ax-his3 28466  ax-his4 28467
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-sup 8590  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-div 10977  df-nn 11313  df-2 11376  df-3 11377  df-n0 11581  df-z 11667  df-uz 11931  df-rp 12075  df-seq 13056  df-exp 13115  df-cj 14180  df-re 14181  df-im 14182  df-sqrt 14316  df-hnorm 28350  df-hvsub 28353  df-lnop 29225  df-unop 29227
This theorem is referenced by: (None)
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