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Theorem hoeq 29464
Description: Equality of Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hoeq ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑈𝑥) ↔ 𝑇 = 𝑈))
Distinct variable groups:   𝑥,𝑇   𝑥,𝑈

Proof of Theorem hoeq
StepHypRef Expression
1 ffn 6507 . 2 (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
2 ffn 6507 . 2 (𝑈: ℋ⟶ ℋ → 𝑈 Fn ℋ)
3 eqfnfv 6794 . . 3 ((𝑇 Fn ℋ ∧ 𝑈 Fn ℋ) → (𝑇 = 𝑈 ↔ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑈𝑥)))
43bicomd 224 . 2 ((𝑇 Fn ℋ ∧ 𝑈 Fn ℋ) → (∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑈𝑥) ↔ 𝑇 = 𝑈))
51, 2, 4syl2an 595 1 ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑈𝑥) ↔ 𝑇 = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wral 3135   Fn wfn 6343  wf 6344  cfv 6348  chba 28623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356
This theorem is referenced by:  hoeqi  29465  homulid2  29504  homco1  29505  homulass  29506  hoadddi  29507  hoadddir  29508  homco2  29681
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