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Theorem hoeq 28949
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 6206 . 2 (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
2 ffn 6206 . 2 (𝑈: ℋ⟶ ℋ → 𝑈 Fn ℋ)
3 eqfnfv 6475 . . 3 ((𝑇 Fn ℋ ∧ 𝑈 Fn ℋ) → (𝑇 = 𝑈 ↔ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑈𝑥)))
43bicomd 213 . 2 ((𝑇 Fn ℋ ∧ 𝑈 Fn ℋ) → (∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑈𝑥) ↔ 𝑇 = 𝑈))
51, 2, 4syl2an 495 1 ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑈𝑥) ↔ 𝑇 = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wral 3050   Fn wfn 6044  wf 6045  cfv 6049  chil 28106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057
This theorem is referenced by:  hoeqi  28950  homulid2  28989  homco1  28990  homulass  28991  hoadddi  28992  hoadddir  28993  homco2  29166
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