Theorem hoadddir 32065

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 11170	. . . . . . . 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 1125	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ))
4		homval 32002	. . . . . . 7 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 + 𝐵) ·ℎ (𝑇‘𝑥)))
5	4	3expa 1134	. . . . . 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 32002	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
8	7	3expa 1134	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
9	8	3adantl2 1184	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
10		homval 32002	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐵 ·op 𝑇)‘𝑥) = (𝐵 ·ℎ (𝑇‘𝑥)))
11	10	3expa 1134	. . . . . . . 8 ⊢ (((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐵 ·op 𝑇)‘𝑥) = (𝐵 ·ℎ (𝑇‘𝑥)))
12	11	3adantl1 1183	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐵 ·op 𝑇)‘𝑥) = (𝐵 ·ℎ (𝑇‘𝑥)))
13	9, 12	oveq12d 7418	. . . . . 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 7066	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
15		ax-hvdistr2 31270	. . . . . . . . . 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 1181	. . . . . . . . 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 1135	. . . . . . . 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 1364	. . . . . 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 2803	. . . . 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 2803	. . . 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 32020	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
23		homulcl 32020	. . . . . . 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 1368	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐵 ·op 𝑇): ℋ⟶ ℋ))
26		hosval 32001	. . . . . 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 1134	. . . . 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 2803	. . 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 32020	. . . 4 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇): ℋ⟶ ℋ)
32	1, 31	stoic3 1799	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇): ℋ⟶ ℋ)
33		hoaddcl 32019	. . . . 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 1368	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
36		hoeq 32021	. . 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 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  ℂcc 11086   + caddc 11091   ℋchba 31180   +_ℎ cva 31181   ·_ℎ csm 31182   +_op chos 31199   ·_op chot 31200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-addcl 11148  ax-hilex 31260  ax-hfvadd 31261  ax-hfvmul 31266  ax-hvdistr2 31270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-map 8814  df-hosum 31991  df-homul 31992

This theorem is referenced by:  ho2times  32080