Theorem homco1 32460

Description: Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
homco1	⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) ∘ 𝑈) = (𝐴 ·op (𝑇 ∘ 𝑈)))

Proof of Theorem homco1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvco3 7016	. . . . . 6 ⊢ ((𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)))
2	1	3ad2antl3 1206	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)))
3		fvco3 7016	. . . . . . . 8 ⊢ ((𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 ∘ 𝑈)‘𝑥) = (𝑇‘(𝑈‘𝑥)))
4	3	3ad2antl3 1206	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇 ∘ 𝑈)‘𝑥) = (𝑇‘(𝑈‘𝑥)))
5	4	oveq2d 7467	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥))))
6		ffvelcdm 7113	. . . . . . . . . 10 ⊢ ((𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑈‘𝑥) ∈ ℋ)
7		homval 32400	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ (𝑈‘𝑥) ∈ ℋ) → ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥))))
8	6, 7	syl3an3 1183	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ (𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥))))
9	8	3expa 1136	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥))))
10	9	exp43 442	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝑇: ℋ⟶ ℋ → (𝑈: ℋ⟶ ℋ → (𝑥 ∈ ℋ → ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥)))))))
11	10	3imp1 1366	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥))))
12	5, 11	eqtr4d 2801	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥)) = ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)))
13	2, 12	eqtr4d 2801	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) ∘ 𝑈)‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥)))
14		fco 6764	. . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ)
15		homval 32400	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∘ 𝑈): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥)))
16	14, 15	syl3an2 1182	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥)))
17	16	3expia 1139	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → (𝑥 ∈ ℋ → ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥))))
18	17	3impb 1132	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑥 ∈ ℋ → ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥))))
19	18	imp 412	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥)))
20	13, 19	eqtr4d 2801	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥))
21	20	ralrimiva 3158	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ∀𝑥 ∈ ℋ (((𝐴 ·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥))
22		homulcl 32418	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
23		fco 6764	. . . 4 ⊢ (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) ∘ 𝑈): ℋ⟶ ℋ)
24	22, 23	stoic3 1809	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) ∘ 𝑈): ℋ⟶ ℋ)
25		homulcl 32418	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∘ 𝑈): ℋ⟶ ℋ) → (𝐴 ·op (𝑇 ∘ 𝑈)): ℋ⟶ ℋ)
26	14, 25	sylan2 605	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → (𝐴 ·op (𝑇 ∘ 𝑈)): ℋ⟶ ℋ)
27	26	3impb 1132	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 ∘ 𝑈)): ℋ⟶ ℋ)
28		hoeq 32419	. . 3 ⊢ ((((𝐴 ·op 𝑇) ∘ 𝑈): ℋ⟶ ℋ ∧ (𝐴 ·op (𝑇 ∘ 𝑈)): ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (((𝐴 ·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) ↔ ((𝐴 ·op 𝑇) ∘ 𝑈) = (𝐴 ·op (𝑇 ∘ 𝑈))))
29	24, 27, 28	syl2anc 596	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (((𝐴 ·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) ↔ ((𝐴 ·op 𝑇) ∘ 𝑈) = (𝐴 ·op (𝑇 ∘ 𝑈))))
30	21, 29	mpbid 235	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) ∘ 𝑈) = (𝐴 ·op (𝑇 ∘ 𝑈)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ∘ ccom 5662  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  ℂcc 11237   ℋchba 31578   ·_ℎ csm 31580   ·_op chot 31598

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-hilex 31658  ax-hfvmul 31664

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-map 8887  df-homul 32390

This theorem is used by:  opsqrlem1  32799