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Theorem hosmval 27780
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 27042 . . 3 ℋ ∈ V
21, 1elmap 7745 . 2 (𝑆 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑆: ℋ⟶ ℋ)
31, 1elmap 7745 . 2 (𝑇 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ ℋ)
4 fveq1 6083 . . . . 5 (𝑓 = 𝑆 → (𝑓𝑥) = (𝑆𝑥))
54oveq1d 6538 . . . 4 (𝑓 = 𝑆 → ((𝑓𝑥) + (𝑔𝑥)) = ((𝑆𝑥) + (𝑔𝑥)))
65mpteq2dv 4663 . . 3 (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑔𝑥))))
7 fveq1 6083 . . . . 5 (𝑔 = 𝑇 → (𝑔𝑥) = (𝑇𝑥))
87oveq2d 6539 . . . 4 (𝑔 = 𝑇 → ((𝑆𝑥) + (𝑔𝑥)) = ((𝑆𝑥) + (𝑇𝑥)))
98mpteq2dv 4663 . . 3 (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
10 df-hosum 27775 . . 3 +op = (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))
111mptex 6364 . . 3 (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))) ∈ V
126, 9, 10, 11ovmpt2 6668 . 2 ((𝑆 ∈ ( ℋ ↑𝑚 ℋ) ∧ 𝑇 ∈ ( ℋ ↑𝑚 ℋ)) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
132, 3, 12syl2anbr 495 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 382   = wceq 1474  wcel 1975  cmpt 4633  wf 5782  cfv 5786  (class class class)co 6523  𝑚 cmap 7717  chil 26962   + cva 26963   +op chos 26981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-hilex 27042
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-map 7719  df-hosum 27775
This theorem is referenced by:  hosval  27785  hoaddcl  27803
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