Theorem ishmo 31300

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 31299	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})
4	3	eleq2d 2848	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ 𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}))
5		fveq2 6882	. . . 4 ⊢ (𝑡 = 𝑇 → (𝐴‘𝑡) = (𝐴‘𝑇))
6		id 23	. . . 4 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇)
7	5, 6	eqeq12d 2778	. . 3 ⊢ (𝑡 = 𝑇 → ((𝐴‘𝑡) = 𝑡 ↔ (𝐴‘𝑇) = 𝑇))
8	7	elrab 3648	. 2 ⊢ (𝑇 ∈ {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡} ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇))
9	4, 8	bitrdi 290	1 ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3414  dom cdm 5659  ‘cfv 6537  (class class class)co 7417  NrmCVeccnv 31073  adjcaj 31237  HmOpchmo 31238

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-hmo 31240

This theorem is used by: (None)