Theorem homco1 30451

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 6923	. . . . . 6 ⊢ ((𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)))
2	1	3ad2antl3 1186	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)))
3		fvco3 6923	. . . . . . . 8 ⊢ ((𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 ∘ 𝑈)‘𝑥) = (𝑇‘(𝑈‘𝑥)))
4	3	3ad2antl3 1186	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇 ∘ 𝑈)‘𝑥) = (𝑇‘(𝑈‘𝑥)))
5	4	oveq2d 7353	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥))))
6		ffvelcdm 7015	. . . . . . . . . 10 ⊢ ((𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑈‘𝑥) ∈ ℋ)
7		homval 30391	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ (𝑈‘𝑥) ∈ ℋ) → ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥))))
8	6, 7	syl3an3 1164	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ (𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥))))
9	8	3expa 1117	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥))))
10	9	exp43 437	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝑇: ℋ⟶ ℋ → (𝑈: ℋ⟶ ℋ → (𝑥 ∈ ℋ → ((𝐴 ·op 𝑇)‘(𝑈‘𝑥)) = (𝐴 ·ℎ (𝑇‘(𝑈‘𝑥)))))))
11	10	3imp1 1346	. . . . . 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 2779	. . . . 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 2779	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) ∘ 𝑈)‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥)))
14		fco 6675	. . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ)
15		homval 30391	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∘ 𝑈): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥)))
16	14, 15	syl3an2 1163	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥)))
17	16	3expia 1120	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → (𝑥 ∈ ℋ → ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥))))
18	17	3impb 1114	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑥 ∈ ℋ → ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 ∘ 𝑈)‘𝑥))))
19	18	imp 407	. . . 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 2779	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥))
21	20	ralrimiva 3139	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ∀𝑥 ∈ ℋ (((𝐴 ·op 𝑇) ∘ 𝑈)‘𝑥) = ((𝐴 ·op (𝑇 ∘ 𝑈))‘𝑥))
22		homulcl 30409	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
23		fco 6675	. . . 4 ⊢ (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) ∘ 𝑈): ℋ⟶ ℋ)
24	22, 23	stoic3 1777	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) ∘ 𝑈): ℋ⟶ ℋ)
25		homulcl 30409	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∘ 𝑈): ℋ⟶ ℋ) → (𝐴 ·op (𝑇 ∘ 𝑈)): ℋ⟶ ℋ)
26	14, 25	sylan2 593	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → (𝐴 ·op (𝑇 ∘ 𝑈)): ℋ⟶ ℋ)
27	26	3impb 1114	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 ∘ 𝑈)): ℋ⟶ ℋ)
28		hoeq 30410	. . 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 584	. 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 231	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) ∘ 𝑈) = (𝐴 ·op (𝑇 ∘ 𝑈)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3061   ∘ ccom 5624  ⟶wf 6475  ‘cfv 6479  (class class class)co 7337  ℂcc 10970   ℋchba 29569   ·_ℎ csm 29571   ·_op chot 29589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-hilex 29649  ax-hfvmul 29655

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-map 8688  df-homul 30381

This theorem is referenced by:  opsqrlem1  30790