cnvadj - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > cnvadj		Structured version Visualization version GIF version

Theorem cnvadj 31951

Description: The adjoint function equals its converse. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
cnvadj	⊢ ^◡adj_ℎ = adj_ℎ

Proof of Theorem cnvadj Dummy variables 𝑢 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvopab 6089	. . 3 ⊢ ^◡{⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))} = {⟨𝑡, 𝑢⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))}
2		3ancoma 1098	. . . . 5 ⊢ ((𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)) ↔ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)))
3		ffvelcdm 7022	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑢‘𝑦) ∈ ℋ)
4		ax-his1 31141	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢‘𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑢‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑢‘𝑦))))
5	3, 4	sylan 581	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑢‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑢‘𝑦))))
6	5	adantrl 717	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑢‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑢‘𝑦))))
7		ffvelcdm 7022	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑡‘𝑥) ∈ ℋ)
8		ax-his1 31141	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℋ ∧ (𝑡‘𝑥) ∈ ℋ) → (𝑦 ·_ih (𝑡‘𝑥)) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)))
9	7, 8	sylan2 594	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℋ ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑡‘𝑥)) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)))
10	9	adantll 715	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑡‘𝑥)) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)))
11	6, 10	eqeq12d 2751	. . . . . . . . . . . . . . 15 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (((𝑢‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑡‘𝑥)) ↔ (∗‘(𝑥 ·_ih (𝑢‘𝑦))) = (∗‘((𝑡‘𝑥) ·_ih 𝑦))))
12	11	ancoms 458	. . . . . . . . . . . . . 14 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑢‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑡‘𝑥)) ↔ (∗‘(𝑥 ·_ih (𝑢‘𝑦))) = (∗‘((𝑡‘𝑥) ·_ih 𝑦))))
13		hicl 31139	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℋ ∧ (𝑢‘𝑦) ∈ ℋ) → (𝑥 ·_ih (𝑢‘𝑦)) ∈ ℂ)
14	3, 13	sylan2 594	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑢‘𝑦)) ∈ ℂ)
15	14	adantll 715	. . . . . . . . . . . . . . 15 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑢‘𝑦)) ∈ ℂ)
16		hicl 31139	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑡‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑡‘𝑥) ·_ih 𝑦) ∈ ℂ)
17	7, 16	sylan 581	. . . . . . . . . . . . . . . 16 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑡‘𝑥) ·_ih 𝑦) ∈ ℂ)
18	17	adantrl 717	. . . . . . . . . . . . . . 15 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑡‘𝑥) ·_ih 𝑦) ∈ ℂ)
19		cj11 15113	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ·_ih (𝑢‘𝑦)) ∈ ℂ ∧ ((𝑡‘𝑥) ·_ih 𝑦) ∈ ℂ) → ((∗‘(𝑥 ·_ih (𝑢‘𝑦))) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)) ↔ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)))
20	15, 18, 19	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((∗‘(𝑥 ·_ih (𝑢‘𝑦))) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)) ↔ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)))
21	12, 20	bitr2d 280	. . . . . . . . . . . . 13 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ ((𝑢‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑡‘𝑥))))
22	21	an4s 661	. . . . . . . . . . . 12 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ ((𝑢‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑡‘𝑥))))
23	22	anassrs 467	. . . . . . . . . . 11 ⊢ ((((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ ((𝑢‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑡‘𝑥))))
24		eqcom 2742	. . . . . . . . . . 11 ⊢ (((𝑢‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑡‘𝑥)) ↔ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥))
25	23, 24	bitrdi 287	. . . . . . . . . 10 ⊢ ((((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥)))
26	25	ralbidva 3156	. . . . . . . . 9 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥)))
27	26	ralbidva 3156	. . . . . . . 8 ⊢ ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥)))
28		ralcom 3263	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥))
29	27, 28	bitrdi 287	. . . . . . 7 ⊢ ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥)))
30	29	pm5.32i 574	. . . . . 6 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)) ↔ ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥)))
31		df-3an 1089	. . . . . 6 ⊢ ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)) ↔ ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)))
32		df-3an 1089	. . . . . 6 ⊢ ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥)) ↔ ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥)))
33	30, 31, 32	3bitr4i 303	. . . . 5 ⊢ ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)) ↔ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥)))
34	2, 33	bitri 275	. . . 4 ⊢ ((𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)) ↔ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥)))
35	34	opabbii 5141	. . 3 ⊢ {⟨𝑡, 𝑢⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))} = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥))}
36	1, 35	eqtri 2758	. 2 ⊢ ^◡{⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))} = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥))}
37		dfadj2 31944	. . 3 ⊢ adj_ℎ = {⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))}
38	37	cnveqi 5818	. 2 ⊢ ^◡adj_ℎ = ^◡{⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑢‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦))}
39		dfadj2 31944	. 2 ⊢ adj_ℎ = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥))}
40	36, 38, 39	3eqtr4i 2768	1 ⊢ ^◡adj_ℎ = adj_ℎ

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3049 {copab 5136 ^◡ccnv 5619 ⟶wf 6483 ‘cfv 6487 (class class class)co 7356 ℂcc 11025 ∗ccj 15047 ℋchba 30978 ·_ih csp 30981 adj_ℎcado 31014

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-hfi 31138 ax-his1 31141

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 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-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-2nd 7932 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-er 8632 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-cj 15050 df-re 15051 df-im 15052 df-adjh 31908

This theorem is referenced by: funcnvadj 31952 adj1o 31953 adjbdlnb 32143

W3C validator