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Theorem hodmval 28901
Description: Value of the difference of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hodmval ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) − (𝑇𝑥))))
Distinct variable groups:   𝑥,𝑆   𝑥,𝑇

Proof of Theorem hodmval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hilex 28161 . . 3 ℋ ∈ V
21, 1elmap 8048 . 2 (𝑆 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑆: ℋ⟶ ℋ)
31, 1elmap 8048 . 2 (𝑇 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ ℋ)
4 fveq1 6347 . . . . 5 (𝑓 = 𝑆 → (𝑓𝑥) = (𝑆𝑥))
54oveq1d 6824 . . . 4 (𝑓 = 𝑆 → ((𝑓𝑥) − (𝑔𝑥)) = ((𝑆𝑥) − (𝑔𝑥)))
65mpteq2dv 4893 . . 3 (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓𝑥) − (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) − (𝑔𝑥))))
7 fveq1 6347 . . . . 5 (𝑔 = 𝑇 → (𝑔𝑥) = (𝑇𝑥))
87oveq2d 6825 . . . 4 (𝑔 = 𝑇 → ((𝑆𝑥) − (𝑔𝑥)) = ((𝑆𝑥) − (𝑇𝑥)))
98mpteq2dv 4893 . . 3 (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆𝑥) − (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) − (𝑇𝑥))))
10 df-hodif 28896 . . 3 op = (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) − (𝑔𝑥))))
111mptex 6646 . . 3 (𝑥 ∈ ℋ ↦ ((𝑆𝑥) − (𝑇𝑥))) ∈ V
126, 9, 10, 11ovmpt2 6957 . 2 ((𝑆 ∈ ( ℋ ↑𝑚 ℋ) ∧ 𝑇 ∈ ( ℋ ↑𝑚 ℋ)) → (𝑆op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) − (𝑇𝑥))))
132, 3, 12syl2anbr 498 1 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) − (𝑇𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1628  wcel 2135  cmpt 4877  wf 6041  cfv 6045  (class class class)co 6809  𝑚 cmap 8019  chil 28081   cmv 28087  op chod 28102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-hilex 28161
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-map 8021  df-hodif 28896
This theorem is referenced by:  hodval  28906  hosubcli  28933
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