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

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

Proof of Theorem cnvadj
Dummy variables 𝑢 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvopab 5775 . . 3 {⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦))} = {⟨𝑡, 𝑢⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦))}
2 3ancoma 1123 . . . . 5 ((𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦)) ↔ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦)))
3 ffvelrn 6606 . . . . . . . . . . . . . . . . . 18 ((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑢𝑦) ∈ ℋ)
4 ax-his1 28483 . . . . . . . . . . . . . . . . . 18 (((𝑢𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑢𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑢𝑦))))
53, 4sylan 575 . . . . . . . . . . . . . . . . 17 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑢𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑢𝑦))))
65adantrl 707 . . . . . . . . . . . . . . . 16 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑢𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑢𝑦))))
7 ffvelrn 6606 . . . . . . . . . . . . . . . . . 18 ((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑡𝑥) ∈ ℋ)
8 ax-his1 28483 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℋ ∧ (𝑡𝑥) ∈ ℋ) → (𝑦 ·ih (𝑡𝑥)) = (∗‘((𝑡𝑥) ·ih 𝑦)))
97, 8sylan2 586 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℋ ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑡𝑥)) = (∗‘((𝑡𝑥) ·ih 𝑦)))
109adantll 705 . . . . . . . . . . . . . . . 16 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑡𝑥)) = (∗‘((𝑡𝑥) ·ih 𝑦)))
116, 10eqeq12d 2840 . . . . . . . . . . . . . . 15 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥)) ↔ (∗‘(𝑥 ·ih (𝑢𝑦))) = (∗‘((𝑡𝑥) ·ih 𝑦))))
1211ancoms 452 . . . . . . . . . . . . . 14 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥)) ↔ (∗‘(𝑥 ·ih (𝑢𝑦))) = (∗‘((𝑡𝑥) ·ih 𝑦))))
13 hicl 28481 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℋ ∧ (𝑢𝑦) ∈ ℋ) → (𝑥 ·ih (𝑢𝑦)) ∈ ℂ)
143, 13sylan2 586 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑢𝑦)) ∈ ℂ)
1514adantll 705 . . . . . . . . . . . . . . 15 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑢𝑦)) ∈ ℂ)
16 hicl 28481 . . . . . . . . . . . . . . . . 17 (((𝑡𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑡𝑥) ·ih 𝑦) ∈ ℂ)
177, 16sylan 575 . . . . . . . . . . . . . . . 16 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑡𝑥) ·ih 𝑦) ∈ ℂ)
1817adantrl 707 . . . . . . . . . . . . . . 15 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑡𝑥) ·ih 𝑦) ∈ ℂ)
19 cj11 14279 . . . . . . . . . . . . . . 15 (((𝑥 ·ih (𝑢𝑦)) ∈ ℂ ∧ ((𝑡𝑥) ·ih 𝑦) ∈ ℂ) → ((∗‘(𝑥 ·ih (𝑢𝑦))) = (∗‘((𝑡𝑥) ·ih 𝑦)) ↔ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦)))
2015, 18, 19syl2anc 579 . . . . . . . . . . . . . 14 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((∗‘(𝑥 ·ih (𝑢𝑦))) = (∗‘((𝑡𝑥) ·ih 𝑦)) ↔ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦)))
2112, 20bitr2d 272 . . . . . . . . . . . . 13 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ ((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥))))
2221an4s 650 . . . . . . . . . . . 12 (((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ ((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥))))
2322anassrs 461 . . . . . . . . . . 11 ((((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ ((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥))))
24 eqcom 2832 . . . . . . . . . . 11 (((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥)) ↔ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))
2523, 24syl6bb 279 . . . . . . . . . 10 ((((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥)))
2625ralbidva 3194 . . . . . . . . 9 (((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥)))
2726ralbidva 3194 . . . . . . . 8 ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥)))
28 ralcom 3308 . . . . . . . 8 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))
2927, 28syl6bb 279 . . . . . . 7 ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥)))
3029pm5.32i 570 . . . . . 6 (((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦)) ↔ ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥)))
31 df-3an 1113 . . . . . 6 ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦)) ↔ ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦)))
32 df-3an 1113 . . . . . 6 ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥)) ↔ ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥)))
3330, 31, 323bitr4i 295 . . . . 5 ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦)) ↔ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥)))
342, 33bitri 267 . . . 4 ((𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦)) ↔ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥)))
3534opabbii 4940 . . 3 {⟨𝑡, 𝑢⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦))} = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))}
361, 35eqtri 2849 . 2 {⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦))} = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))}
37 dfadj2 29288 . . 3 adj = {⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦))}
3837cnveqi 5529 . 2 adj = {⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦))}
39 dfadj2 29288 . 2 adj = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))}
4036, 38, 393eqtr4i 2859 1 adj = adj
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∀wral 3117  {copab 4935  ◡ccnv 5341  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  ℂcc 10250  ∗ccj 14213   ℋchba 28320   ·ih csp 28323  adjℎcado 28356 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329  ax-hfi 28480  ax-his1 28483 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-po 5263  df-so 5264  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-div 11010  df-2 11414  df-cj 14216  df-re 14217  df-im 14218  df-adjh 29252 This theorem is referenced by:  funcnvadj  29296  adj1o  29297  adjbdlnb  29487
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