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Theorem hoeqi 28960
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 28959 . 2 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑆𝑥) = (𝑇𝑥) ↔ 𝑆 = 𝑇))
41, 2, 3mp2an 672 1 (∀𝑥 ∈ ℋ (𝑆𝑥) = (𝑇𝑥) ↔ 𝑆 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  wral 3061  wf 6027  cfv 6031  chil 28116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039
This theorem is referenced by:  hoaddcomi  28971  hodsi  28974  hoaddassi  28975  hocadddiri  28978  hocsubdiri  28979  hoaddid1i  28985  ho0coi  28987  hoid1i  28988  hoid1ri  28989  honegsubi  28995  hoddii  29188  pjsdii  29354  pjddii  29355  pjss1coi  29362  pjss2coi  29363  pjorthcoi  29368  pjscji  29369  pjtoi  29378  pjclem4  29398  pj3si  29406  pj3cor1i  29408
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