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Theorem hoeq 31779
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 6736 . 2 (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
2 ffn 6736 . 2 (𝑈: ℋ⟶ ℋ → 𝑈 Fn ℋ)
3 eqfnfv 7051 . . 3 ((𝑇 Fn ℋ ∧ 𝑈 Fn ℋ) → (𝑇 = 𝑈 ↔ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑈𝑥)))
43bicomd 223 . 2 ((𝑇 Fn ℋ ∧ 𝑈 Fn ℋ) → (∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑈𝑥) ↔ 𝑇 = 𝑈))
51, 2, 4syl2an 596 1 ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑈𝑥) ↔ 𝑇 = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wral 3061   Fn wfn 6556  wf 6557  cfv 6561  chba 30938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  hoeqi  31780  homullid  31819  homco1  31820  homulass  31821  hoadddi  31822  hoadddir  31823  homco2  31996
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