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Theorem cnvadj 31921
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 6160 . . 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 1097 . . . . 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 ffvelcdm 7101 . . . . . . . . . . . . . . . . . 18 ((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑢𝑦) ∈ ℋ)
4 ax-his1 31111 . . . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . 16 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑢𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑢𝑦))))
7 ffvelcdm 7101 . . . . . . . . . . . . . . . . . 18 ((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑡𝑥) ∈ ℋ)
8 ax-his1 31111 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℋ ∧ (𝑡𝑥) ∈ ℋ) → (𝑦 ·ih (𝑡𝑥)) = (∗‘((𝑡𝑥) ·ih 𝑦)))
97, 8sylan2 593 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℋ ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑡𝑥)) = (∗‘((𝑡𝑥) ·ih 𝑦)))
109adantll 714 . . . . . . . . . . . . . . . 16 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑡𝑥)) = (∗‘((𝑡𝑥) ·ih 𝑦)))
116, 10eqeq12d 2751 . . . . . . . . . . . . . . 15 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥)) ↔ (∗‘(𝑥 ·ih (𝑢𝑦))) = (∗‘((𝑡𝑥) ·ih 𝑦))))
1211ancoms 458 . . . . . . . . . . . . . 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 31109 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℋ ∧ (𝑢𝑦) ∈ ℋ) → (𝑥 ·ih (𝑢𝑦)) ∈ ℂ)
143, 13sylan2 593 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑢𝑦)) ∈ ℂ)
1514adantll 714 . . . . . . . . . . . . . . 15 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑢𝑦)) ∈ ℂ)
16 hicl 31109 . . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . 15 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑡𝑥) ·ih 𝑦) ∈ ℂ)
19 cj11 15198 . . . . . . . . . . . . . . 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 280 . . . . . . . . . . . . 13 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ ((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥))))
2221an4s 660 . . . . . . . . . . . 12 (((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ ((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥))))
2322anassrs 467 . . . . . . . . . . 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 2742 . . . . . . . . . . 11 (((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥)) ↔ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))
2523, 24bitrdi 287 . . . . . . . . . 10 ((((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥)))
2625ralbidva 3174 . . . . . . . . 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 3174 . . . . . . . 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 3287 . . . . . . . 8 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))
2927, 28bitrdi 287 . . . . . . 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 574 . . . . . 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 1088 . . . . . 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 1088 . . . . . 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 303 . . . . 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 275 . . . 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 5215 . . 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 2763 . 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 31914 . . 3 adj = {⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦))}
3837cnveqi 5888 . 2 adj = {⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦))}
39 dfadj2 31914 . 2 adj = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))}
4036, 38, 393eqtr4i 2773 1 adj = adj
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  {copab 5210  ccnv 5688  wf 6559  cfv 6563  (class class class)co 7431  cc 11151  ccj 15132  chba 30948   ·ih csp 30951  adjcado 30984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-hfi 31108  ax-his1 31111
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-2 12327  df-cj 15135  df-re 15136  df-im 15137  df-adjh 31878
This theorem is referenced by:  funcnvadj  31922  adj1o  31923  adjbdlnb  32113
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