Theorem hosmval 28722

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 ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))

Distinct variable groups:   𝑥,𝑆   𝑥,𝑇

Proof of Theorem hosmval

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 27984	. . 3 ⊢ ℋ ∈ V
2	1, 1	elmap 7928	. 2 ⊢ (𝑆 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑆: ℋ⟶ ℋ)
3	1, 1	elmap 7928	. 2 ⊢ (𝑇 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ ℋ)
4		fveq1 6228	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
5	4	oveq1d 6705	. . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) +ℎ (𝑔‘𝑥)) = ((𝑆‘𝑥) +ℎ (𝑔‘𝑥)))
6	5	mpteq2dv 4778	. . 3 ⊢ (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑔‘𝑥))))
7		fveq1 6228	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
8	7	oveq2d 6706	. . . 4 ⊢ (𝑔 = 𝑇 → ((𝑆‘𝑥) +ℎ (𝑔‘𝑥)) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
9	8	mpteq2dv 4778	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
10		df-hosum 28717	. . 3 ⊢ +op = (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥))))
11	1	mptex 6527	. . 3 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) ∈ V
12	6, 9, 10, 11	ovmpt2 6838	. 2 ⊢ ((𝑆 ∈ ( ℋ ↑𝑚 ℋ) ∧ 𝑇 ∈ ( ℋ ↑𝑚 ℋ)) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
13	2, 3, 12	syl2anbr 496	1 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ↦ cmpt 4762  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↑_𝑚 cmap 7899   ℋchil 27904   +_ℎ cva 27905   +_op chos 27923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-hilex 27984

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-hosum 28717

This theorem is referenced by:  hosval  28727  hoaddcl  28745