Theorem kbass5 29897

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 29731	. . . . . . . 8 ⊢ ((𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐶 ketbra 𝐷)‘𝑥) = ((𝑥 ·ih 𝐷) ·ℎ 𝐶))
2	1	3expa 1114	. . . . . . 7 ⊢ (((𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐶 ketbra 𝐷)‘𝑥) = ((𝑥 ·ih 𝐷) ·ℎ 𝐶))
3	2	adantll 712	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((𝐶 ketbra 𝐷)‘𝑥) = ((𝑥 ·ih 𝐷) ·ℎ 𝐶))
4	3	fveq2d 6674	. . . . 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 773	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → 𝐴 ∈ ℋ)
6		simpllr 774	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → 𝐵 ∈ ℋ)
7		simpr 487	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → 𝑥 ∈ ℋ)
8		simplrr 776	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → 𝐷 ∈ ℋ)
9		hicl 28857	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝑥 ·ih 𝐷) ∈ ℂ)
10	7, 8, 9	syl2anc 586	. . . . . . 7 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → (𝑥 ·ih 𝐷) ∈ ℂ)
11		simplrl 775	. . . . . . 7 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → 𝐶 ∈ ℋ)
12		hvmulcl 28790	. . . . . . 7 ⊢ (((𝑥 ·ih 𝐷) ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝑥 ·ih 𝐷) ·ℎ 𝐶) ∈ ℋ)
13	10, 11, 12	syl2anc 586	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·ih 𝐷) ·ℎ 𝐶) ∈ ℋ)
14		kbval 29731	. . . . . 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 1367	. . . . 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 2856	. . . 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 29730	. . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐶 ketbra 𝐷): ℋ⟶ ℋ)
18	17	adantl 484	. . . . 5 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (𝐶 ketbra 𝐷): ℋ⟶ ℋ)
19		fvco3 6760	. . . . 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 582	. . . 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 29731	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))
22	5, 6, 11, 21	syl3anc 1367	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴))
23	22	oveq2d 7172	. . . . 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 29730	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵): ℋ⟶ ℋ)
25	24	ffvelrnda 6851	. . . . . . . 8 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) ∈ ℋ)
26	25	adantrr 715	. . . . . . 7 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵)‘𝐶) ∈ ℋ)
27	26	adantr 483	. . . . . 6 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) ∈ ℋ)
28		kbval 29731	. . . . . 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 1367	. . . . 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 28861	. . . . . . . 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 1367	. . . . . . 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 7171	. . . . . 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 28857	. . . . . . . 8 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 ·ih 𝐵) ∈ ℂ)
34	11, 6, 33	syl2anc 586	. . . . . . 7 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) ∧ 𝑥 ∈ ℋ) → (𝐶 ·ih 𝐵) ∈ ℂ)
35		ax-hvmulass 28784	. . . . . . 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 1367	. . . . . 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 2856	. . . . 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 2866	. . . 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 2866	. . 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 3182	. 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 6531	. . . 4 ⊢ (((𝐴 ketbra 𝐵): ℋ⟶ ℋ ∧ (𝐶 ketbra 𝐷): ℋ⟶ ℋ) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)): ℋ⟶ ℋ)
42	24, 17, 41	syl2an 597	. . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)): ℋ⟶ ℋ)
43		kbop 29730	. . . . 5 ⊢ ((((𝐴 ketbra 𝐵)‘𝐶) ∈ ℋ ∧ 𝐷 ∈ ℋ) → (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷): ℋ⟶ ℋ)
44	25, 43	sylan 582	. . . 4 ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐶 ∈ ℋ) ∧ 𝐷 ∈ ℋ) → (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷): ℋ⟶ ℋ)
45	44	anasss 469	. . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷): ℋ⟶ ℋ)
46		ffn 6514	. . . 4 ⊢ (((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)): ℋ⟶ ℋ → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) Fn ℋ)
47		ffn 6514	. . . 4 ⊢ ((((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷): ℋ⟶ ℋ → (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷) Fn ℋ)
48		eqfnfv 6802	. . . 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 597	. . 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 586	. 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 259	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 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138   ∘ ccom 5559   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  ℂcc 10535   · cmul 10542   ℋchba 28696   ·_ℎ csm 28698   ·_ih csp 28699   ketbra ck 28734

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-hilex 28776  ax-hfvmul 28782  ax-hvmulass 28784  ax-hfi 28856  ax-his3 28861

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-kb 29628

This theorem is referenced by:  kbass6  29898