Theorem hosmval 31662

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 30926	. . 3 ⊢ ℋ ∈ V
2	1, 1	elmap 8883	. 2 ⊢ (𝑆 ∈ ( ℋ ↑m ℋ) ↔ 𝑆: ℋ⟶ ℋ)
3	1, 1	elmap 8883	. 2 ⊢ (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
4		fveq1 6874	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
5	4	oveq1d 7418	. . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) +ℎ (𝑔‘𝑥)) = ((𝑆‘𝑥) +ℎ (𝑔‘𝑥)))
6	5	mpteq2dv 5215	. . 3 ⊢ (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑔‘𝑥))))
7		fveq1 6874	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
8	7	oveq2d 7419	. . . 4 ⊢ (𝑔 = 𝑇 → ((𝑆‘𝑥) +ℎ (𝑔‘𝑥)) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
9	8	mpteq2dv 5215	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
10		df-hosum 31657	. . 3 ⊢ +op = (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥))))
11	1	mptex 7214	. . 3 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) ∈ V
12	6, 9, 10, 11	ovmpo 7565	. 2 ⊢ ((𝑆 ∈ ( ℋ ↑m ℋ) ∧ 𝑇 ∈ ( ℋ ↑m ℋ)) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
13	2, 3, 12	syl2anbr 599	1 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ↦ cmpt 5201  ⟶wf 6526  ‘cfv 6530  (class class class)co 7403   ↑_m cmap 8838   ℋchba 30846   +_ℎ cva 30847   +_op chos 30865

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-hilex 30926

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-map 8840  df-hosum 31657

This theorem is referenced by:  hosval  31667  hoaddcl  31685