Theorem ishmo 30693

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 30692	. . 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 6896	. . . 4 ⊢ (𝑡 = 𝑇 → (𝐴‘𝑡) = (𝐴‘𝑇))
6		id 22	. . . 4 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇)
7	5, 6	eqeq12d 2741	. . 3 ⊢ (𝑡 = 𝑇 → ((𝐴‘𝑡) = 𝑡 ↔ (𝐴‘𝑇) = 𝑇))
8	7	elrab 3679	. 2 ⊢ (𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡} ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇))
9	4, 8	bitrdi 286	1 ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {crab 3418  dom cdm 5678  ‘cfv 6549  (class class class)co 7419  NrmCVeccnv 30466  adjcaj 30630  HmOpchmo 30631

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 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-hmo 30633

This theorem is referenced by: (None)