Theorem homulass 30744

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 11135	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
2		homval 30683	. . . . . . . . 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 1110	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝑥 ∈ ℋ → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥))))
6	5	imp 407	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)))
7		homval 30683	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐵 ·op 𝑇)‘𝑥) = (𝐵 ·ℎ (𝑇‘𝑥)))
8	7	oveq2d 7373	. . . . . . . 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 1166	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
11		ffvelcdm 7032	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
12		ax-hvmulass 29949	. . . . . . . . . 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 437	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐵 ∈ ℂ → (𝑇: ℋ⟶ ℋ → (𝑥 ∈ ℋ → ((𝐴 · 𝐵) ·ℎ (𝑇‘𝑥)) = (𝐴 ·ℎ (𝐵 ·ℎ (𝑇‘𝑥)))))))
16	15	3imp1 1347	. . . . . 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 2779	. . . . 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 2779	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = (𝐴 ·ℎ ((𝐵 ·op 𝑇)‘𝑥)))
19		homulcl 30701	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐵 ·op 𝑇): ℋ⟶ ℋ)
20		homval 30683	. . . . . . . 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 1115	. . . . 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 407	. . . 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 2779	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥))
26	25	ralrimiva 3143	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ∀𝑥 ∈ ℋ (((𝐴 · 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 ·op (𝐵 ·op 𝑇))‘𝑥))
27		homulcl 30701	. . . 4 ⊢ (((𝐴 · 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇): ℋ⟶ ℋ)
28	1, 27	stoic3 1778	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇): ℋ⟶ ℋ)
29		homulcl 30701	. . . . 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 1115	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
32		hoeq 30702	. . 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 231	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  ℂcc 11049   · cmul 11056   ℋchba 29861   ·_ℎ csm 29863   ·_op chot 29881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-mulcl 11113  ax-hilex 29941  ax-hfvmul 29947  ax-hvmulass 29949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-homul 30673

This theorem is referenced by:  homul12  30747  honegneg  30748  leopmul  31076  nmopleid  31081  opsqrlem1  31082  opsqrlem6  31087