Theorem ishmo 30558

Description: The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hmoval.8	⊢ 𝐻 = (HmOp‘𝑈)
hmoval.9	⊢ 𝐴 = (𝑈adj𝑈)

Assertion

Ref	Expression
ishmo	⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇)))

Proof of Theorem ishmo

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmoval.8	. . . 4 ⊢ 𝐻 = (HmOp‘𝑈)
2		hmoval.9	. . . 4 ⊢ 𝐴 = (𝑈adj𝑈)
3	1, 2	hmoval 30557	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})
4	3	eleq2d 2811	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ 𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}))
5		fveq2 6882	. . . 4 ⊢ (𝑡 = 𝑇 → (𝐴‘𝑡) = (𝐴‘𝑇))
6		id 22	. . . 4 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇)
7	5, 6	eqeq12d 2740	. . 3 ⊢ (𝑡 = 𝑇 → ((𝐴‘𝑡) = 𝑡 ↔ (𝐴‘𝑇) = 𝑇))
8	7	elrab 3676	. 2 ⊢ (𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡} ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇))
9	4, 8	bitrdi 287	1 ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3424  dom cdm 5667  ‘cfv 6534  (class class class)co 7402  NrmCVeccnv 30331  adjcaj 30495  HmOpchmo 30496

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-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-hmo 30498

This theorem is referenced by: (None)