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Theorem hoeqi 29192
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 29191 . 2 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑆𝑥) = (𝑇𝑥) ↔ 𝑆 = 𝑇))
41, 2, 3mp2an 682 1 (∀𝑥 ∈ ℋ (𝑆𝑥) = (𝑇𝑥) ↔ 𝑆 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1601  wral 3090  wf 6131  cfv 6135  chba 28348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143
This theorem is referenced by:  hoaddcomi  29203  hodsi  29206  hoaddassi  29207  hocadddiri  29210  hocsubdiri  29211  hoaddid1i  29217  ho0coi  29219  hoid1i  29220  hoid1ri  29221  honegsubi  29227  hoddii  29420  pjsdii  29586  pjddii  29587  pjss1coi  29594  pjss2coi  29595  pjorthcoi  29600  pjscji  29601  pjtoi  29610  pjclem4  29630  pj3si  29638  pj3cor1i  29640
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