Theorem hosmval 31806

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 31070	. . 3 ⊢ ℋ ∈ V
2	1, 1	elmap 8819	. 2 ⊢ (𝑆 ∈ ( ℋ ↑m ℋ) ↔ 𝑆: ℋ⟶ ℋ)
3	1, 1	elmap 8819	. 2 ⊢ (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
4		fveq1 6839	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
5	4	oveq1d 7382	. . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) +ℎ (𝑔‘𝑥)) = ((𝑆‘𝑥) +ℎ (𝑔‘𝑥)))
6	5	mpteq2dv 5179	. . 3 ⊢ (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑔‘𝑥))))
7		fveq1 6839	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
8	7	oveq2d 7383	. . . 4 ⊢ (𝑔 = 𝑇 → ((𝑆‘𝑥) +ℎ (𝑔‘𝑥)) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
9	8	mpteq2dv 5179	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
10		df-hosum 31801	. . 3 ⊢ +op = (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥))))
11	1	mptex 7178	. . 3 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) ∈ V
12	6, 9, 10, 11	ovmpo 7527	. 2 ⊢ ((𝑆 ∈ ( ℋ ↑m ℋ) ∧ 𝑇 ∈ ( ℋ ↑m ℋ)) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
13	2, 3, 12	syl2anbr 600	1 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ↦ cmpt 5166  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773   ℋchba 30990   +_ℎ cva 30991   +_op chos 31009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-hilex 31070

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-map 8775  df-hosum 31801

This theorem is referenced by:  hosval  31811  hoaddcl  31829