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Theorem cnvadj 31817
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 6148 . . 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 1095 . . . . 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 7094 . . . . . . . . . . . . . . . . . 18 ((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑢𝑦) ∈ ℋ)
4 ax-his1 31007 . . . . . . . . . . . . . . . . . 18 (((𝑢𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑢𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑢𝑦))))
53, 4sylan 578 . . . . . . . . . . . . . . . . 17 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑢𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑢𝑦))))
65adantrl 714 . . . . . . . . . . . . . . . 16 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑢𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑢𝑦))))
7 ffvelcdm 7094 . . . . . . . . . . . . . . . . . 18 ((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑡𝑥) ∈ ℋ)
8 ax-his1 31007 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℋ ∧ (𝑡𝑥) ∈ ℋ) → (𝑦 ·ih (𝑡𝑥)) = (∗‘((𝑡𝑥) ·ih 𝑦)))
97, 8sylan2 591 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℋ ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑡𝑥)) = (∗‘((𝑡𝑥) ·ih 𝑦)))
109adantll 712 . . . . . . . . . . . . . . . 16 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑡𝑥)) = (∗‘((𝑡𝑥) ·ih 𝑦)))
116, 10eqeq12d 2741 . . . . . . . . . . . . . . 15 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥)) ↔ (∗‘(𝑥 ·ih (𝑢𝑦))) = (∗‘((𝑡𝑥) ·ih 𝑦))))
1211ancoms 457 . . . . . . . . . . . . . 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 31005 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℋ ∧ (𝑢𝑦) ∈ ℋ) → (𝑥 ·ih (𝑢𝑦)) ∈ ℂ)
143, 13sylan2 591 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑢𝑦)) ∈ ℂ)
1514adantll 712 . . . . . . . . . . . . . . 15 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑢𝑦)) ∈ ℂ)
16 hicl 31005 . . . . . . . . . . . . . . . . 17 (((𝑡𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑡𝑥) ·ih 𝑦) ∈ ℂ)
177, 16sylan 578 . . . . . . . . . . . . . . . 16 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑡𝑥) ·ih 𝑦) ∈ ℂ)
1817adantrl 714 . . . . . . . . . . . . . . 15 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑡𝑥) ·ih 𝑦) ∈ ℂ)
19 cj11 15162 . . . . . . . . . . . . . . 15 (((𝑥 ·ih (𝑢𝑦)) ∈ ℂ ∧ ((𝑡𝑥) ·ih 𝑦) ∈ ℂ) → ((∗‘(𝑥 ·ih (𝑢𝑦))) = (∗‘((𝑡𝑥) ·ih 𝑦)) ↔ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦)))
2015, 18, 19syl2anc 582 . . . . . . . . . . . . . 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 279 . . . . . . . . . . . . 13 (((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ ((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥))))
2221an4s 658 . . . . . . . . . . . 12 (((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ ((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥))))
2322anassrs 466 . . . . . . . . . . 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 2732 . . . . . . . . . . 11 (((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥)) ↔ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))
2523, 24bitrdi 286 . . . . . . . . . 10 ((((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦) ↔ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥)))
2625ralbidva 3165 . . . . . . . . 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 3165 . . . . . . . 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 3276 . . . . . . . 8 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))
2927, 28bitrdi 286 . . . . . . 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 573 . . . . . 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 1086 . . . . . 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 1086 . . . . . 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 302 . . . . 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 274 . . . 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 5219 . . 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 2753 . 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 31810 . . 3 adj = {⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦))}
3837cnveqi 5880 . 2 adj = {⟨𝑢, 𝑡⟩ ∣ (𝑢: ℋ⟶ ℋ ∧ 𝑡: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦))}
39 dfadj2 31810 . 2 adj = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))}
4036, 38, 393eqtr4i 2763 1 adj = adj
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  {copab 5214  ccnv 5680  wf 6549  cfv 6553  (class class class)co 7423  cc 11152  ccj 15096  chba 30844   ·ih csp 30847  adjcado 30880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745  ax-resscn 11211  ax-1cn 11212  ax-icn 11213  ax-addcl 11214  ax-addrcl 11215  ax-mulcl 11216  ax-mulrcl 11217  ax-mulcom 11218  ax-addass 11219  ax-mulass 11220  ax-distr 11221  ax-i2m1 11222  ax-1ne0 11223  ax-1rid 11224  ax-rnegex 11225  ax-rrecex 11226  ax-cnre 11227  ax-pre-lttri 11228  ax-pre-lttrn 11229  ax-pre-ltadd 11230  ax-pre-mulgt0 11231  ax-hfi 31004  ax-his1 31007
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-po 5593  df-so 5594  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-er 8733  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11296  df-mnf 11297  df-xr 11298  df-ltxr 11299  df-le 11300  df-sub 11492  df-neg 11493  df-div 11918  df-2 12322  df-cj 15099  df-re 15100  df-im 15101  df-adjh 31774
This theorem is referenced by:  funcnvadj  31818  adj1o  31819  adjbdlnb  32009
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