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Theorem cnvadj 29596
Description: The adjoint function equals its converse. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
cnvadj adj = adj

Proof of Theorem cnvadj
Dummy variables 𝑢 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvopab 5990 . . 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 1090 . . . . 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 6841 . . . . . . . . . . . . . . . . . 18 ((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑢𝑦) ∈ ℋ)
4 ax-his1 28786 . . . . . . . . . . . . . . . . . 18 (((𝑢𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑢𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑢𝑦))))
53, 4sylan 580 . . . . . . . . . . . . . . . . 17 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑢𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑢𝑦))))
65adantrl 712 . . . . . . . . . . . . . . . 16 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑢𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑢𝑦))))
7 ffvelrn 6841 . . . . . . . . . . . . . . . . . 18 ((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑡𝑥) ∈ ℋ)
8 ax-his1 28786 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℋ ∧ (𝑡𝑥) ∈ ℋ) → (𝑦 ·ih (𝑡𝑥)) = (∗‘((𝑡𝑥) ·ih 𝑦)))
97, 8sylan2 592 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℋ ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑡𝑥)) = (∗‘((𝑡𝑥) ·ih 𝑦)))
109adantll 710 . . . . . . . . . . . . . . . 16 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑡𝑥)) = (∗‘((𝑡𝑥) ·ih 𝑦)))
116, 10eqeq12d 2834 . . . . . . . . . . . . . . 15 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥)) ↔ (∗‘(𝑥 ·ih (𝑢𝑦))) = (∗‘((𝑡𝑥) ·ih 𝑦))))
1211ancoms 459 . . . . . . . . . . . . . 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 28784 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℋ ∧ (𝑢𝑦) ∈ ℋ) → (𝑥 ·ih (𝑢𝑦)) ∈ ℂ)
143, 13sylan2 592 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑢𝑦)) ∈ ℂ)
1514adantll 710 . . . . . . . . . . . . . . 15 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑢𝑦)) ∈ ℂ)
16 hicl 28784 . . . . . . . . . . . . . . . . 17 (((𝑡𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑡𝑥) ·ih 𝑦) ∈ ℂ)
177, 16sylan 580 . . . . . . . . . . . . . . . 16 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑡𝑥) ·ih 𝑦) ∈ ℂ)
1817adantrl 712 . . . . . . . . . . . . . . 15 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑡𝑥) ·ih 𝑦) ∈ ℂ)
19 cj11 14509 . . . . . . . . . . . . . . 15 (((𝑥 ·ih (𝑢𝑦)) ∈ ℂ ∧ ((𝑡𝑥) ·ih 𝑦) ∈ ℂ) → ((∗‘(𝑥 ·ih (𝑢𝑦))) = (∗‘((𝑡𝑥) ·ih 𝑦)) ↔ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦)))
2015, 18, 19syl2anc 584 . . . . . . . . . . . . . 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 281 . . . . . . . . . . . . 13 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ ((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥))))
2221an4s 656 . . . . . . . . . . . 12 (((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ ((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥))))
2322anassrs 468 . . . . . . . . . . 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 2825 . . . . . . . . . . 11 (((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥)) ↔ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))
2523, 24syl6bb 288 . . . . . . . . . 10 ((((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥)))
2625ralbidva 3193 . . . . . . . . 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 3193 . . . . . . . 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 3351 . . . . . . . 8 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))
2927, 28syl6bb 288 . . . . . . 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 575 . . . . . 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 1081 . . . . . 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 1081 . . . . . 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 304 . . . . 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 276 . . . 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 5124 . . 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 2841 . 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 29589 . . 3 adj = {⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦))}
3837cnveqi 5738 . 2 adj = {⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦))}
39 dfadj2 29589 . 2 adj = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))}
4036, 38, 393eqtr4i 2851 1 adj = adj
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  {copab 5119  ccnv 5547  wf 6344  cfv 6348  (class class class)co 7145  cc 10523  ccj 14443  chba 28623   ·ih csp 28626  adjcado 28659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-hfi 28783  ax-his1 28786
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-2 11688  df-cj 14446  df-re 14447  df-im 14448  df-adjh 29553
This theorem is referenced by:  funcnvadj  29597  adj1o  29598  adjbdlnb  29788
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