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Theorem hoeqi 29465
Description: Equality of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
hoeq.1 𝑆: ℋ⟶ ℋ
hoeq.2 𝑇: ℋ⟶ ℋ
Assertion
Ref Expression
hoeqi (∀𝑥 ∈ ℋ (𝑆𝑥) = (𝑇𝑥) ↔ 𝑆 = 𝑇)
Distinct variable groups:   𝑥,𝑆   𝑥,𝑇

Proof of Theorem hoeqi
StepHypRef Expression
1 hoeq.1 . 2 𝑆: ℋ⟶ ℋ
2 hoeq.2 . 2 𝑇: ℋ⟶ ℋ
3 hoeq 29464 . 2 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑆𝑥) = (𝑇𝑥) ↔ 𝑆 = 𝑇))
41, 2, 3mp2an 688 1 (∀𝑥 ∈ ℋ (𝑆𝑥) = (𝑇𝑥) ↔ 𝑆 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wral 3135  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:  hoaddcomi  29476  hodsi  29479  hoaddassi  29480  hocadddiri  29483  hocsubdiri  29484  hoaddid1i  29490  ho0coi  29492  hoid1i  29493  hoid1ri  29494  honegsubi  29500  hoddii  29693  pjsdii  29859  pjddii  29860  pjss1coi  29867  pjss2coi  29868  pjorthcoi  29873  pjscji  29874  pjtoi  29883  pjclem4  29903  pj3si  29911  pj3cor1i  29913
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