hoadddir - Hilbert Space Explorer

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Theorem hoadddir 31656

Description: Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hoadddir	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·_op 𝑇) = ((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇)))

Proof of Theorem hoadddir Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		addcl 11218	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
2	1	anim1i 613	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ))
3	2	3impa 1107	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ))
4		homval 31593	. . . . . . 7 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·_op 𝑇)‘𝑥) = ((𝐴 + 𝐵) ·_ℎ (𝑇‘𝑥)))
5	4	3expa 1115	. . . . . 6 ⊢ ((((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·_op 𝑇)‘𝑥) = ((𝐴 + 𝐵) ·_ℎ (𝑇‘𝑥)))
6	3, 5	sylan 578	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·_op 𝑇)‘𝑥) = ((𝐴 + 𝐵) ·_ℎ (𝑇‘𝑥)))
7		homval 31593	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·_op 𝑇)‘𝑥) = (𝐴 ·_ℎ (𝑇‘𝑥)))
8	7	3expa 1115	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·_op 𝑇)‘𝑥) = (𝐴 ·_ℎ (𝑇‘𝑥)))
9	8	3adantl2 1164	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·_op 𝑇)‘𝑥) = (𝐴 ·_ℎ (𝑇‘𝑥)))
10		homval 31593	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐵 ·_op 𝑇)‘𝑥) = (𝐵 ·_ℎ (𝑇‘𝑥)))
11	10	3expa 1115	. . . . . . . 8 ⊢ (((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐵 ·_op 𝑇)‘𝑥) = (𝐵 ·_ℎ (𝑇‘𝑥)))
12	11	3adantl1 1163	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐵 ·_op 𝑇)‘𝑥) = (𝐵 ·_ℎ (𝑇‘𝑥)))
13	9, 12	oveq12d 7433	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·_op 𝑇)‘𝑥) +_ℎ ((𝐵 ·_op 𝑇)‘𝑥)) = ((𝐴 ·_ℎ (𝑇‘𝑥)) +_ℎ (𝐵 ·_ℎ (𝑇‘𝑥))))
14		ffvelcdm 7085	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
15		ax-hvdistr2 30861	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝐴 + 𝐵) ·_ℎ (𝑇‘𝑥)) = ((𝐴 ·_ℎ (𝑇‘𝑥)) +_ℎ (𝐵 ·_ℎ (𝑇‘𝑥))))
16	14, 15	syl3an3 1162	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝐴 + 𝐵) ·_ℎ (𝑇‘𝑥)) = ((𝐴 ·_ℎ (𝑇‘𝑥)) +_ℎ (𝐵 ·_ℎ (𝑇‘𝑥))))
17	16	3exp 1116	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐵 ∈ ℂ → ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 + 𝐵) ·_ℎ (𝑇‘𝑥)) = ((𝐴 ·_ℎ (𝑇‘𝑥)) +_ℎ (𝐵 ·_ℎ (𝑇‘𝑥))))))
18	17	exp4a 430	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐵 ∈ ℂ → (𝑇: ℋ⟶ ℋ → (𝑥 ∈ ℋ → ((𝐴 + 𝐵) ·_ℎ (𝑇‘𝑥)) = ((𝐴 ·_ℎ (𝑇‘𝑥)) +_ℎ (𝐵 ·_ℎ (𝑇‘𝑥)))))))
19	18	3imp1 1344	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 + 𝐵) ·_ℎ (𝑇‘𝑥)) = ((𝐴 ·_ℎ (𝑇‘𝑥)) +_ℎ (𝐵 ·_ℎ (𝑇‘𝑥))))
20	13, 19	eqtr4d 2768	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·_op 𝑇)‘𝑥) +_ℎ ((𝐵 ·_op 𝑇)‘𝑥)) = ((𝐴 + 𝐵) ·_ℎ (𝑇‘𝑥)))
21	6, 20	eqtr4d 2768	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·_op 𝑇)‘𝑥) = (((𝐴 ·_op 𝑇)‘𝑥) +_ℎ ((𝐵 ·_op 𝑇)‘𝑥)))
22		homulcl 31611	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·_op 𝑇): ℋ⟶ ℋ)
23		homulcl 31611	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐵 ·_op 𝑇): ℋ⟶ ℋ)
24	22, 23	anim12i 611	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ)) → ((𝐴 ·_op 𝑇): ℋ⟶ ℋ ∧ (𝐵 ·_op 𝑇): ℋ⟶ ℋ))
25	24	3impdir 1348	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·_op 𝑇): ℋ⟶ ℋ ∧ (𝐵 ·_op 𝑇): ℋ⟶ ℋ))
26		hosval 31592	. . . . . 6 ⊢ (((𝐴 ·_op 𝑇): ℋ⟶ ℋ ∧ (𝐵 ·_op 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇))‘𝑥) = (((𝐴 ·_op 𝑇)‘𝑥) +_ℎ ((𝐵 ·_op 𝑇)‘𝑥)))
27	26	3expa 1115	. . . . 5 ⊢ ((((𝐴 ·_op 𝑇): ℋ⟶ ℋ ∧ (𝐵 ·_op 𝑇): ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇))‘𝑥) = (((𝐴 ·_op 𝑇)‘𝑥) +_ℎ ((𝐵 ·_op 𝑇)‘𝑥)))
28	25, 27	sylan 578	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇))‘𝑥) = (((𝐴 ·_op 𝑇)‘𝑥) +_ℎ ((𝐵 ·_op 𝑇)‘𝑥)))
29	21, 28	eqtr4d 2768	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·_op 𝑇)‘𝑥) = (((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇))‘𝑥))
30	29	ralrimiva 3136	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ∀𝑥 ∈ ℋ (((𝐴 + 𝐵) ·_op 𝑇)‘𝑥) = (((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇))‘𝑥))
31		homulcl 31611	. . . 4 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·_op 𝑇): ℋ⟶ ℋ)
32	1, 31	stoic3 1770	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·_op 𝑇): ℋ⟶ ℋ)
33		hoaddcl 31610	. . . . 5 ⊢ (((𝐴 ·_op 𝑇): ℋ⟶ ℋ ∧ (𝐵 ·_op 𝑇): ℋ⟶ ℋ) → ((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇)): ℋ⟶ ℋ)
34	22, 23, 33	syl2an 594	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ)) → ((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇)): ℋ⟶ ℋ)
35	34	3impdir 1348	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇)): ℋ⟶ ℋ)
36		hoeq 31612	. . 3 ⊢ ((((𝐴 + 𝐵) ·_op 𝑇): ℋ⟶ ℋ ∧ ((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇)): ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (((𝐴 + 𝐵) ·_op 𝑇)‘𝑥) = (((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇))‘𝑥) ↔ ((𝐴 + 𝐵) ·_op 𝑇) = ((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇))))
37	32, 35, 36	syl2anc 582	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (((𝐴 + 𝐵) ·_op 𝑇)‘𝑥) = (((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇))‘𝑥) ↔ ((𝐴 + 𝐵) ·_op 𝑇) = ((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇))))
38	30, 37	mpbid 231	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·_op 𝑇) = ((𝐴 ·_op 𝑇) +_op (𝐵 ·_op 𝑇)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ⟶wf 6538 ‘cfv 6542 (class class class)co 7415 ℂcc 11134 + caddc 11139 ℋchba 30771 +_ℎ cva 30772 ·_ℎ csm 30773 +_op chos 30790 ·_op chot 30791

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-addcl 11196 ax-hilex 30851 ax-hfvadd 30852 ax-hfvmul 30857 ax-hvdistr2 30861

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7418 df-oprab 7419 df-mpo 7420 df-map 8843 df-hosum 31582 df-homul 31583

This theorem is referenced by: ho2times 31671

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