Theorem kbass5 30007

Description: Dirac bra-ket associative law ( ∣ 𝐴⟩⟨𝐵 ∣ )( ∣ 𝐶⟩⟨𝐷 ∣ ) = (( ∣ 𝐴⟩⟨𝐵 ∣ ) ∣ 𝐶⟩)⟨𝐷 ∣. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
kbass5	⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷))

Proof of Theorem kbass5

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kbval 29841	. . . . . . . 8 ⊢ ((𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐶 ketbra 𝐷)‘𝑥) = ((𝑥 ·ih 𝐷) ·ℎ 𝐶))
2	1	3expa 1115	. . . . . . 7 ⊢ (((𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐶 ketbra 𝐷)‘𝑥) = ((𝑥 ·ih 𝐷) ·ℎ 𝐶))
3	2	adantll 713	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((𝐶 ketbra 𝐷)‘𝑥) = ((𝑥 ·ih 𝐷) ·ℎ 𝐶))
4	3	fveq2d 6666	. . . . 5 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘((𝐶 ketbra 𝐷)‘𝑥)) = ((𝐴 ketbra 𝐵)‘((𝑥 ·ih 𝐷) ·ℎ 𝐶)))
5		simplll 774	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → 𝐴 ∈ ℋ)
6		simpllr 775	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → 𝐵 ∈ ℋ)
7		simpr 488	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → 𝑥 ∈ ℋ)
8		simplrr 777	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → 𝐷 ∈ ℋ)
9		hicl 28967	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝑥 ·ih 𝐷) ∈ ℂ)
10	7, 8, 9	syl2anc 587	. . . . . . 7 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → (𝑥 ·ih 𝐷) ∈ ℂ)
11		simplrl 776	. . . . . . 7 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → 𝐶 ∈ ℋ)
12		hvmulcl 28900	. . . . . . 7 ⊢ (((𝑥 ·ih 𝐷) ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝑥 ·ih 𝐷) ·ℎ 𝐶) ∈ ℋ)
13	10, 11, 12	syl2anc 587	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·ih 𝐷) ·ℎ 𝐶) ∈ ℋ)
14		kbval 29841	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ ((𝑥 ·ih 𝐷) ·ℎ 𝐶) ∈ ℋ) → ((𝐴 ketbra 𝐵)‘((𝑥 ·ih 𝐷) ·ℎ 𝐶)) = ((((𝑥 ·ih 𝐷) ·ℎ 𝐶) ·ih 𝐵) ·ℎ 𝐴))
15	5, 6, 13, 14	syl3anc 1368	. . . . 5 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘((𝑥 ·ih 𝐷) ·ℎ 𝐶)) = ((((𝑥 ·ih 𝐷) ·ℎ 𝐶) ·ih 𝐵) ·ℎ 𝐴))
16	4, 15	eqtrd 2793	. . . 4 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘((𝐶 ketbra 𝐷)‘𝑥)) = ((((𝑥 ·ih 𝐷) ·ℎ 𝐶) ·ih 𝐵) ·ℎ 𝐴))
17		kbop 29840	. . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐶 ketbra 𝐷): ℋ⟶ ℋ)
18	17	adantl 485	. . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (𝐶 ketbra 𝐷): ℋ⟶ ℋ)
19		fvco3 6755	. . . . 5 ⊢ (((𝐶 ketbra 𝐷): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷))‘𝑥) = ((𝐴 ketbra 𝐵)‘((𝐶 ketbra 𝐷)‘𝑥)))
20	18, 19	sylan 583	. . . 4 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → (((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷))‘𝑥) = ((𝐴 ketbra 𝐵)‘((𝐶 ketbra 𝐷)‘𝑥)))
21		kbval 29841	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))
22	5, 6, 11, 21	syl3anc 1368	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))
23	22	oveq2d 7171	. . . . 5 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·ih 𝐷) ·ℎ ((𝐴 ketbra 𝐵)‘𝐶)) = ((𝑥 ·ih 𝐷) ·ℎ ((𝐶 ·ih 𝐵) ·ℎ 𝐴)))
24		kbop 29840	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵): ℋ⟶ ℋ)
25	24	ffvelrnda 6847	. . . . . . . 8 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) ∈ ℋ)
26	25	adantrr 716	. . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵)‘𝐶) ∈ ℋ)
27	26	adantr 484	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) ∈ ℋ)
28		kbval 29841	. . . . . 6 ⊢ ((((𝐴 ketbra 𝐵)‘𝐶) ∈ ℋ ∧ 𝐷 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷)‘𝑥) = ((𝑥 ·ih 𝐷) ·ℎ ((𝐴 ketbra 𝐵)‘𝐶)))
29	27, 8, 7, 28	syl3anc 1368	. . . . 5 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷)‘𝑥) = ((𝑥 ·ih 𝐷) ·ℎ ((𝐴 ketbra 𝐵)‘𝐶)))
30		ax-his3 28971	. . . . . . . 8 ⊢ (((𝑥 ·ih 𝐷) ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((𝑥 ·ih 𝐷) ·ℎ 𝐶) ·ih 𝐵) = ((𝑥 ·ih 𝐷) · (𝐶 ·ih 𝐵)))
31	10, 11, 6, 30	syl3anc 1368	. . . . . . 7 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → (((𝑥 ·ih 𝐷) ·ℎ 𝐶) ·ih 𝐵) = ((𝑥 ·ih 𝐷) · (𝐶 ·ih 𝐵)))
32	31	oveq1d 7170	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((((𝑥 ·ih 𝐷) ·ℎ 𝐶) ·ih 𝐵) ·ℎ 𝐴) = (((𝑥 ·ih 𝐷) · (𝐶 ·ih 𝐵)) ·ℎ 𝐴))
33		hicl 28967	. . . . . . . 8 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 ·ih 𝐵) ∈ ℂ)
34	11, 6, 33	syl2anc 587	. . . . . . 7 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → (𝐶 ·ih 𝐵) ∈ ℂ)
35		ax-hvmulass 28894	. . . . . . 7 ⊢ (((𝑥 ·ih 𝐷) ∈ ℂ ∧ (𝐶 ·ih 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (((𝑥 ·ih 𝐷) · (𝐶 ·ih 𝐵)) ·ℎ 𝐴) = ((𝑥 ·ih 𝐷) ·ℎ ((𝐶 ·ih 𝐵) ·ℎ 𝐴)))
36	10, 34, 5, 35	syl3anc 1368	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → (((𝑥 ·ih 𝐷) · (𝐶 ·ih 𝐵)) ·ℎ 𝐴) = ((𝑥 ·ih 𝐷) ·ℎ ((𝐶 ·ih 𝐵) ·ℎ 𝐴)))
37	32, 36	eqtrd 2793	. . . . 5 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((((𝑥 ·ih 𝐷) ·ℎ 𝐶) ·ih 𝐵) ·ℎ 𝐴) = ((𝑥 ·ih 𝐷) ·ℎ ((𝐶 ·ih 𝐵) ·ℎ 𝐴)))
38	23, 29, 37	3eqtr4d 2803	. . . 4 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷)‘𝑥) = ((((𝑥 ·ih 𝐷) ·ℎ 𝐶) ·ih 𝐵) ·ℎ 𝐴))
39	16, 20, 38	3eqtr4d 2803	. . 3 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → (((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷))‘𝑥) = ((((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷)‘𝑥))
40	39	ralrimiva 3113	. 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ∀𝑥 ∈ ℋ (((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷))‘𝑥) = ((((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷)‘𝑥))
41		fco 6520	. . . 4 ⊢ (((𝐴 ketbra 𝐵): ℋ⟶ ℋ ∧ (𝐶 ketbra 𝐷): ℋ⟶ ℋ) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)): ℋ⟶ ℋ)
42	24, 17, 41	syl2an 598	. . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)): ℋ⟶ ℋ)
43		kbop 29840	. . . . 5 ⊢ ((((𝐴 ketbra 𝐵)‘𝐶) ∈ ℋ ∧ 𝐷 ∈ ℋ) → (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷): ℋ⟶ ℋ)
44	25, 43	sylan 583	. . . 4 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐶 ∈ ℋ) ∧ 𝐷 ∈ ℋ) → (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷): ℋ⟶ ℋ)
45	44	anasss 470	. . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷): ℋ⟶ ℋ)
46		ffn 6502	. . . 4 ⊢ (((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)): ℋ⟶ ℋ → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) Fn ℋ)
47		ffn 6502	. . . 4 ⊢ ((((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷): ℋ⟶ ℋ → (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷) Fn ℋ)
48		eqfnfv 6797	. . . 4 ⊢ ((((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) Fn ℋ ∧ (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷) Fn ℋ) → (((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷) ↔ ∀𝑥 ∈ ℋ (((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷))‘𝑥) = ((((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷)‘𝑥)))
49	46, 47, 48	syl2an 598	. . 3 ⊢ ((((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)): ℋ⟶ ℋ ∧ (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷): ℋ⟶ ℋ) → (((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷) ↔ ∀𝑥 ∈ ℋ (((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷))‘𝑥) = ((((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷)‘𝑥)))
50	42, 45, 49	syl2anc 587	. 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷) ↔ ∀𝑥 ∈ ℋ (((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷))‘𝑥) = ((((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷)‘𝑥)))
51	40, 50	mpbird 260	1 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ∘ ccom 5531   Fn wfn 6334  ⟶wf 6335  ‘cfv 6339  (class class class)co 7155  ℂcc 10578   · cmul 10585   ℋchba 28806   ·_ℎ csm 28808   ·_ih csp 28809   ketbra ck 28844

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-hilex 28886  ax-hfvmul 28892  ax-hvmulass 28894  ax-hfi 28966  ax-his3 28971

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-kb 29738

This theorem is referenced by:  kbass6  30008