cnvadj - Hilbert Space Explorer

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Theorem cnvadj 32494

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 6130	. . 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 7081	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑢‘𝑦) ∈ ℋ)
4		ax-his1 31684	. . . . . . . . . . . . . . . . . 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 7081	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑡‘𝑥) ∈ ℋ)
8		ax-his1 31684	. . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . 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 31682	. . . . . . . . . . . . . . . . 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 31682	. . . . . . . . . . . . . . . . 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 15329	. . . . . . . . . . . . . . 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 2767	. . . . . . . . . . 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 3183	. . . . . . . . 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 3183	. . . . . . . 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 3290	. . . . . . . 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 5171	. . 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 2783	. 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 32487	. . 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 5851	. 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 32487	. 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 2793	1 ⊢ ^◡adj_ℎ = adj_ℎ

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 {copab 5166 ^◡ccnv 5649 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ∗ccj 15263 ℋchba 31521 ·_ih csp 31524 adj_ℎcado 31557

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-hfi 31681 ax-his1 31684

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-cj 15266 df-re 15267 df-im 15268 df-adjh 32451

This theorem is used by: funcnvadj 32495 adj1o 32496 adjbdlnb 32686

W3C validator