Theorem homulass 31788

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 11218	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
2		homval 31727	. . . . . . . . 9 ⊢ (((𝐴 · 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)))
3	1, 2	syl3an1 1163	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)))
4	3	3expia 1121	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑇: ℋ⟶ ℋ) → (𝑥 ∈ ℋ → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥))))
5	4	3impa 1109	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝑥 ∈ ℋ → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥))))
6	5	imp 406	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)))
7		homval 31727	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐵 ·op 𝑇)‘𝑥) = (𝐵 ·ℎ (𝑇‘𝑥)))
8	7	oveq2d 7426	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
9	8	3expa 1118	. . . . . . 7 ⊢ (((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
10	9	3adantl1 1167	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
11		ffvelcdm 7076	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
12		ax-hvmulass 30993	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
13	11, 12	syl3an3 1165	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
14	13	3expa 1118	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
15	14	exp43 436	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐵 ∈ ℂ → (𝑇: ℋ⟶ ℋ → (𝑥 ∈ ℋ → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥)))))))
16	15	3imp1 1348	. . . . . 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 2774	. . . . 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 2774	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)))
19		homulcl 31745	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐵 ·op 𝑇): ℋ⟶ ℋ)
20		homval 31727	. . . . . . . 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 1164	. . . . . . 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 1121	. . . . . 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 1114	. . . . 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 406	. . . 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 2774	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥))
26	25	ralrimiva 3133	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ∀𝑥 ∈ ℋ (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥))
27		homulcl 31745	. . . 4 ⊢ (((𝐴 · 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇): ℋ⟶ ℋ)
28	1, 27	stoic3 1776	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇): ℋ⟶ ℋ)
29		homulcl 31745	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ·op 𝑇): ℋ⟶ ℋ) → (𝐴 ·op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
30	19, 29	sylan2 593	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ)) → (𝐴 ·op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
31	30	3impb 1114	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
32		hoeq 31746	. . 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 584	. 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 232	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℂcc 11132   · cmul 11139   ℋchba 30905   ·_ℎ csm 30907   ·_op chot 30925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-mulcl 11196  ax-hilex 30985  ax-hfvmul 30991  ax-hvmulass 30993

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8847  df-homul 31717

This theorem is referenced by:  homul12  31791  honegneg  31792  leopmul  32120  nmopleid  32125  opsqrlem1  32126  opsqrlem6  32131