cnvadj - Hilbert Space Explorer

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Theorem cnvadj 31985

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 6094	. . 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 1104	. . . . 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 7026	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑢‘𝑦) ∈ ℋ)
4		ax-his1 31175	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢‘𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑢‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑢‘𝑦))))
5	3, 4	sylan 587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑢‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑢‘𝑦))))
6	5	adantrl 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑢‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑢‘𝑦))))
7		ffvelcdm 7026	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑡‘𝑥) ∈ ℋ)
8		ax-his1 31175	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℋ ∧ (𝑡‘𝑥) ∈ ℋ) → (𝑦 ·_ih (𝑡‘𝑥)) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)))
9	7, 8	sylan2 600	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℋ ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑡‘𝑥)) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)))
10	9	adantll 721	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑡‘𝑥)) = (∗‘((𝑡‘𝑥) ·_ih 𝑦)))
11	6, 10	eqeq12d 2757	. . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . 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 31173	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℋ ∧ (𝑢‘𝑦) ∈ ℋ) → (𝑥 ·_ih (𝑢‘𝑦)) ∈ ℂ)
14	3, 13	sylan2 600	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑢‘𝑦)) ∈ ℂ)
15	14	adantll 721	. . . . . . . . . . . . . . 15 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑢‘𝑦)) ∈ ℂ)
16		hicl 31173	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑡‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑡‘𝑥) ·_ih 𝑦) ∈ ℂ)
17	7, 16	sylan 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑡‘𝑥) ·_ih 𝑦) ∈ ℂ)
18	17	adantrl 723	. . . . . . . . . . . . . . 15 ⊢ (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑡‘𝑥) ·_ih 𝑦) ∈ ℂ)
19		cj11 15119	. . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . 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 282	. . . . . . . . . . . . 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 667	. . . . . . . . . . . 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 469	. . . . . . . . . . 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 2748	. . . . . . . . . . 11 ⊢ (((𝑢‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑡‘𝑥)) ↔ (𝑦 ·_ih (𝑡‘𝑥)) = ((𝑢‘𝑦) ·_ih 𝑥))
25	23, 24	bitrdi 289	. . . . . . . . . 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 3162	. . . . . . . . 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 3162	. . . . . . . 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 3269	. . . . . . . 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 289	. . . . . . 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 580	. . . . . 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 1095	. . . . . 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 1095	. . . . . 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 305	. . . . 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 277	. . . 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 5142	. . 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 2764	. 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 31978	. . 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 5819	. 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 31978	. 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 2774	1 ⊢ ^◡adj_ℎ = adj_ℎ

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ∀wral 3055 {copab 5137 ^◡ccnv 5620 ⟶wf 6485 ‘cfv 6489 (class class class)co 7360 ℂcc 11031 ∗ccj 15053 ℋchba 31012 ·_ih csp 31015 adj_ℎcado 31048

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-hfi 31172 ax-his1 31175

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-cj 15056 df-re 15057 df-im 15058 df-adjh 31942

This theorem is referenced by: funcnvadj 31986 adj1o 31987 adjbdlnb 32177

W3C validator