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Theorem hosmval 30142
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 29406 . . 3 ℋ ∈ V
21, 1elmap 8690 . 2 (𝑆 ∈ ( ℋ ↑m ℋ) ↔ 𝑆: ℋ⟶ ℋ)
31, 1elmap 8690 . 2 (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
4 fveq1 6803 . . . . 5 (𝑓 = 𝑆 → (𝑓𝑥) = (𝑆𝑥))
54oveq1d 7322 . . . 4 (𝑓 = 𝑆 → ((𝑓𝑥) + (𝑔𝑥)) = ((𝑆𝑥) + (𝑔𝑥)))
65mpteq2dv 5183 . . 3 (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑔𝑥))))
7 fveq1 6803 . . . . 5 (𝑔 = 𝑇 → (𝑔𝑥) = (𝑇𝑥))
87oveq2d 7323 . . . 4 (𝑔 = 𝑇 → ((𝑆𝑥) + (𝑔𝑥)) = ((𝑆𝑥) + (𝑇𝑥)))
98mpteq2dv 5183 . . 3 (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
10 df-hosum 30137 . . 3 +op = (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))
111mptex 7131 . . 3 (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))) ∈ V
126, 9, 10, 11ovmpo 7465 . 2 ((𝑆 ∈ ( ℋ ↑m ℋ) ∧ 𝑇 ∈ ( ℋ ↑m ℋ)) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
132, 3, 12syl2anbr 600 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 397   = wceq 1539  wcel 2104  cmpt 5164  wf 6454  cfv 6458  (class class class)co 7307  m cmap 8646  chba 29326   + cva 29327   +op chos 29345
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-hilex 29406
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-map 8648  df-hosum 30137
This theorem is referenced by:  hosval  30147  hoaddcl  30165
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