Theorem hodmval 31760

Description: Value of the difference 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
hodmval	⊢ ((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 hodmval

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

Step	Hyp	Ref	Expression
1		ax-hilex 31022	. . 3 ⊢ ℋ ∈ V
2	1, 1	elmap 8925	. 2 ⊢ (𝑆 ∈ ( ℋ ↑m ℋ) ↔ 𝑆: ℋ⟶ ℋ)
3	1, 1	elmap 8925	. 2 ⊢ (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
4		fveq1 6918	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
5	4	oveq1d 7460	. . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) −ℎ (𝑔‘𝑥)) = ((𝑆‘𝑥) −ℎ (𝑔‘𝑥)))
6	5	mpteq2dv 5271	. . 3 ⊢ (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑔‘𝑥))))
7		fveq1 6918	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
8	7	oveq2d 7461	. . . 4 ⊢ (𝑔 = 𝑇 → ((𝑆‘𝑥) −ℎ (𝑔‘𝑥)) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))
9	8	mpteq2dv 5271	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))))
10		df-hodif 31755	. . 3 ⊢ −op = (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −ℎ (𝑔‘𝑥))))
11	1	mptex 7258	. . 3 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) ∈ V
12	6, 9, 10, 11	ovmpo 7606	. 2 ⊢ ((𝑆 ∈ ( ℋ ↑m ℋ) ∧ 𝑇 ∈ ( ℋ ↑m ℋ)) → (𝑆 −op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))))
13	2, 3, 12	syl2anbr 598	1 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 −op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2103   ↦ cmpt 5252  ⟶wf 6568  ‘cfv 6572  (class class class)co 7445   ↑_m cmap 8880   ℋchba 30942   −_ℎ cmv 30948   −_op chod 30963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-hilex 31022

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-map 8882  df-hodif 31755

This theorem is referenced by:  hodval  31765  hosubcli  31792