Theorem ishmo 30713

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 30712	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})
4	3	eleq2d 2814	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ 𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}))
5		fveq2 6840	. . . 4 ⊢ (𝑡 = 𝑇 → (𝐴‘𝑡) = (𝐴‘𝑇))
6		id 22	. . . 4 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇)
7	5, 6	eqeq12d 2745	. . 3 ⊢ (𝑡 = 𝑇 → ((𝐴‘𝑡) = 𝑡 ↔ (𝐴‘𝑇) = 𝑇))
8	7	elrab 3656	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3402  dom cdm 5631  ‘cfv 6499  (class class class)co 7369  NrmCVeccnv 30486  adjcaj 30650  HmOpchmo 30651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-hmo 30653

This theorem is referenced by: (None)