cnvadj - Hilbert Space Explorer

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Theorem cnvadj 28135

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 5452	. . 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 1038	. . . . 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		ffvelrn 6265	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑢‘𝑦) ∈ ℋ)
4		ax-his1 27323	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢‘𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑢‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑢‘𝑦))))
5	3, 4	sylan 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑢‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑢‘𝑦))))
6	5	adantrl 748	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑢‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑢‘𝑦))))
7		ffvelrn 6265	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑡‘𝑥) ∈ ℋ)
8		ax-his1 27323	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℋ ∧ (𝑡‘𝑥) ∈ ℋ) → (𝑦 ·_ih (𝑡‘𝑥)) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)))
9	7, 8	sylan2 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℋ ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑡‘𝑥)) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)))
10	9	adantll 746	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑡‘𝑥)) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)))
11	6, 10	eqeq12d 2625	. . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . 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 27321	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℋ ∧ (𝑢‘𝑦) ∈ ℋ) → (𝑥 ·_ih (𝑢‘𝑦)) ∈ ℂ)
14	3, 13	sylan2 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑢‘𝑦)) ∈ ℂ)
15	14	adantll 746	. . . . . . . . . . . . . . 15 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑢‘𝑦)) ∈ ℂ)
16		hicl 27321	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑡‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑡‘𝑥) ·_ih 𝑦) ∈ ℂ)
17	7, 16	sylan 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑡‘𝑥) ·_ih 𝑦) ∈ ℂ)
18	17	adantrl 748	. . . . . . . . . . . . . . 15 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑡‘𝑥) ·_ih 𝑦) ∈ ℂ)
19		cj11 13750	. . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . 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 268	. . . . . . . . . . . . 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 865	. . . . . . . . . . . 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 678	. . . . . . . . . . 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 2617	. . . . . . . . . . 11 ⊢ (((𝑢‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑡‘𝑥)) ↔ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥))
25	23, 24	syl6bb 275	. . . . . . . . . 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 2968	. . . . . . . . 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 2968	. . . . . . . 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 3079	. . . . . . . 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	syl6bb 275	. . . . . . 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 667	. . . . . 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 1033	. . . . . 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 1033	. . . . . 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 291	. . . . 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 263	. . . 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 4649	. . 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 2632	. 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 28128	. . 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 5219	. 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 28128	. 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 2642	1 ⊢ ^◡adj_ℎ = adj_ℎ

Colors of variables: wff setvar class

Syntax hints: ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {copab 4642 ^◡ccnv 5037 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ∗ccj 13684 ℋchil 27160 ·_ih csp 27163 adj_ℎcado 27196

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-hfi 27320 ax-his1 27323

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-cj 13687 df-re 13688 df-im 13689 df-adjh 28092

This theorem is referenced by: funcnvadj 28136 adj1o 28137 adjbdlnb 28327

W3C validator