Theorem hoadddir 32134

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 11183	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
2	1	anim1i 626	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ))
3	2	3impa 1127	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ))
4		homval 32071	. . . . . . 7 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 + 𝐵) ·ℎ (𝑇‘𝑥)))
5	4	3expa 1136	. . . . . 6 ⊢ ((((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 + 𝐵) ·ℎ (𝑇‘𝑥)))
6	3, 5	sylan 591	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 + 𝐵) ·ℎ (𝑇‘𝑥)))
7		homval 32071	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
8	7	3expa 1136	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
9	8	3adantl2 1186	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
10		homval 32071	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐵 ·op 𝑇)‘𝑥) = (𝐵 ·ℎ (𝑇‘𝑥)))
11	10	3expa 1136	. . . . . . . 8 ⊢ (((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐵 ·op 𝑇)‘𝑥) = (𝐵 ·ℎ (𝑇‘𝑥)))
12	11	3adantl1 1185	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐵 ·op 𝑇)‘𝑥) = (𝐵 ·ℎ (𝑇‘𝑥)))
13	9, 12	oveq12d 7430	. . . . . 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 7078	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
15		ax-hvdistr2 31339	. . . . . . . . . 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 1183	. . . . . . . . 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 1137	. . . . . . . 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 436	. . . . . . 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 1366	. . . . . 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 2801	. . . . 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 2801	. . . 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 32089	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
23		homulcl 32089	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐵 ·op 𝑇): ℋ⟶ ℋ)
24	22, 23	anim12i 624	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ)) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐵 ·op 𝑇): ℋ⟶ ℋ))
25	24	3impdir 1370	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐵 ·op 𝑇): ℋ⟶ ℋ))
26		hosval 32070	. . . . . 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 1136	. . . . 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 591	. . . 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 2801	. . 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 3157	. 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 32089	. . . 4 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇): ℋ⟶ ℋ)
32	1, 31	stoic3 1806	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇): ℋ⟶ ℋ)
33		hoaddcl 32088	. . . . 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 607	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ)) → ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
35	34	3impdir 1370	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
36		hoeq 32090	. . 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 595	. 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 235	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇) = ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412  ℂcc 11099   + caddc 11104   ℋchba 31249   +_ℎ cva 31250   ·_ℎ csm 31251   +_op chos 31268   ·_op chot 31269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-addcl 11161  ax-hilex 31329  ax-hfvadd 31330  ax-hfvmul 31335  ax-hvdistr2 31339

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8827  df-hosum 32060  df-homul 32061

This theorem is referenced by:  ho2times  32149