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Theorem hosmval 31492
Description: Value of the sum 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
hosmval ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
Distinct variable groups:   𝑥,𝑆   𝑥,𝑇

Proof of Theorem hosmval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hilex 30756 . . 3 ℋ ∈ V
21, 1elmap 8864 . 2 (𝑆 ∈ ( ℋ ↑m ℋ) ↔ 𝑆: ℋ⟶ ℋ)
31, 1elmap 8864 . 2 (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
4 fveq1 6883 . . . . 5 (𝑓 = 𝑆 → (𝑓𝑥) = (𝑆𝑥))
54oveq1d 7419 . . . 4 (𝑓 = 𝑆 → ((𝑓𝑥) + (𝑔𝑥)) = ((𝑆𝑥) + (𝑔𝑥)))
65mpteq2dv 5243 . . 3 (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑔𝑥))))
7 fveq1 6883 . . . . 5 (𝑔 = 𝑇 → (𝑔𝑥) = (𝑇𝑥))
87oveq2d 7420 . . . 4 (𝑔 = 𝑇 → ((𝑆𝑥) + (𝑔𝑥)) = ((𝑆𝑥) + (𝑇𝑥)))
98mpteq2dv 5243 . . 3 (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
10 df-hosum 31487 . . 3 +op = (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))
111mptex 7219 . . 3 (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))) ∈ V
126, 9, 10, 11ovmpo 7563 . 2 ((𝑆 ∈ ( ℋ ↑m ℋ) ∧ 𝑇 ∈ ( ℋ ↑m ℋ)) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
132, 3, 12syl2anbr 598 1 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cmpt 5224  wf 6532  cfv 6536  (class class class)co 7404  m cmap 8819  chba 30676   + cva 30677   +op chos 30695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-hilex 30756
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-hosum 31487
This theorem is referenced by:  hosval  31497  hoaddcl  31515
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