cnvadj - Hilbert Space Explorer

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Theorem cnvadj 32359

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 6135	. . 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 1115	. . . . 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 7077	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑢‘𝑦) ∈ ℋ)
4		ax-his1 31549	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢‘𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑢‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑢‘𝑦))))
5	3, 4	sylan 592	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑢‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑢‘𝑦))))
6	5	adantrl 729	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑢‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑢‘𝑦))))
7		ffvelcdm 7077	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑡‘𝑥) ∈ ℋ)
8		ax-his1 31549	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℋ ∧ (𝑡‘𝑥) ∈ ℋ) → (𝑦 ·_ih (𝑡‘𝑥)) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)))
9	7, 8	sylan2 605	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℋ ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑡‘𝑥)) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)))
10	9	adantll 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑡‘𝑥)) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)))
11	6, 10	eqeq12d 2778	. . . . . . . . . . . . . . 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 464	. . . . . . . . . . . . . 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 31547	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℋ ∧ (𝑢‘𝑦) ∈ ℋ) → (𝑥 ·_ih (𝑢‘𝑦)) ∈ ℂ)
14	3, 13	sylan2 605	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑢‘𝑦)) ∈ ℂ)
15	14	adantll 727	. . . . . . . . . . . . . . 15 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑢‘𝑦)) ∈ ℂ)
16		hicl 31547	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑡‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑡‘𝑥) ·_ih 𝑦) ∈ ℂ)
17	7, 16	sylan 592	. . . . . . . . . . . . . . . 16 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑡‘𝑥) ·_ih 𝑦) ∈ ℂ)
18	17	adantrl 729	. . . . . . . . . . . . . . 15 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑡‘𝑥) ·_ih 𝑦) ∈ ℂ)
19		cj11 15251	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 283	. . . . . . . . . . . . 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 673	. . . . . . . . . . . 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 473	. . . . . . . . . . 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 2769	. . . . . . . . . . 11 ⊢ (((𝑢‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑡‘𝑥)) ↔ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥))
25	23, 24	bitrdi 290	. . . . . . . . . 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 3185	. . . . . . . . 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 3185	. . . . . . . 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 3292	. . . . . . . 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 290	. . . . . . 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 585	. . . . . 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 1105	. . . . . 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 1105	. . . . . 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 306	. . . . 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 278	. . . 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 5176	. . 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 2785	. 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 32352	. . 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 5858	. 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 32352	. 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 2795	1 ⊢ ^◡adj_ℎ = adj_ℎ

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3078 {copab 5171 ^◡ccnv 5658 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 ℂcc 11125 ∗ccj 15185 ℋchba 31386 ·_ih csp 31389 adj_ℎcado 31422

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-hfi 31546 ax-his1 31549

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-cj 15188 df-re 15189 df-im 15190 df-adjh 32316

This theorem is used by: funcnvadj 32360 adj1o 32361 adjbdlnb 32551

W3C validator