Theorem homulass 32095

Description: Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
homulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇)))

Proof of Theorem homulass

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulcl 11184	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
2		homval 32034	. . . . . . . . 9 ⊢ (((𝐴 · 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)))
3	1, 2	syl3an1 1179	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)))
4	3	3expia 1137	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑇: ℋ⟶ ℋ) → (𝑥 ∈ ℋ → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥))))
5	4	3impa 1125	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝑥 ∈ ℋ → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥))))
6	5	imp 411	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)))
7		homval 32034	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐵 ·op 𝑇)‘𝑥) = (𝐵 ·ℎ (𝑇‘𝑥)))
8	7	oveq2d 7427	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
9	8	3expa 1134	. . . . . . 7 ⊢ (((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
10	9	3adantl1 1183	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
11		ffvelcdm 7077	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
12		ax-hvmulass 31300	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
13	11, 12	syl3an3 1181	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
14	13	3expa 1134	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
15	14	exp43 441	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐵 ∈ ℂ → (𝑇: ℋ⟶ ℋ → (𝑥 ∈ ℋ → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥)))))))
16	15	3imp1 1364	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
17	10, 16	eqtr4d 2807	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)))
18	6, 17	eqtr4d 2807	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)))
19		homulcl 32052	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐵 ·op 𝑇): ℋ⟶ ℋ)
20		homval 32034	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ·op 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)))
21	19, 20	syl3an2 1180	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)))
22	21	3expia 1137	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ)) → (𝑥 ∈ ℋ → ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥))))
23	22	3impb 1130	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝑥 ∈ ℋ → ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥))))
24	23	imp 411	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)))
25	18, 24	eqtr4d 2807	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥))
26	25	ralrimiva 3163	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ∀𝑥 ∈ ℋ (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥))
27		homulcl 32052	. . . 4 ⊢ (((𝐴 · 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇): ℋ⟶ ℋ)
28	1, 27	stoic3 1803	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇): ℋ⟶ ℋ)
29		homulcl 32052	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ·op 𝑇): ℋ⟶ ℋ) → (𝐴 ·op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
30	19, 29	sylan2 604	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ)) → (𝐴 ·op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
31	30	3impb 1130	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
32		hoeq 32053	. . 3 ⊢ ((((𝐴 · 𝐵) ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op (𝐵 ·op 𝑇)): ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) ↔ ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇))))
33	28, 31, 32	syl2anc 595	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥) ↔ ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇))))
34	26, 33	mpbid 235	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  ℂcc 11098   · cmul 11105   ℋchba 31212   ·_ℎ csm 31214   ·_op chot 31232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-mulcl 11162  ax-hilex 31292  ax-hfvmul 31298  ax-hvmulass 31300

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8826  df-homul 32024

This theorem is referenced by:  homul12  32098  honegneg  32099  leopmul  32427  nmopleid  32432  opsqrlem1  32433  opsqrlem6  32438