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Theorem cnvadj 29661
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 1092 . . . . 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 6842 . . . . . . . . . . . . . . . . . 18 ((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑢𝑦) ∈ ℋ)
4 ax-his1 28851 . . . . . . . . . . . . . . . . . 18 (((𝑢𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑢𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑢𝑦))))
53, 4sylan 582 . . . . . . . . . . . . . . . . 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 ffvelrn 6842 . . . . . . . . . . . . . . . . . 18 ((𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑡𝑥) ∈ ℋ)
8 ax-his1 28851 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℋ ∧ (𝑡𝑥) ∈ ℋ) → (𝑦 ·ih (𝑡𝑥)) = (∗‘((𝑡𝑥) ·ih 𝑦)))
97, 8sylan2 594 . . . . . . . . . . . . . . . . 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 2835 . . . . . . . . . . . . . . 15 (((𝑢: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑡: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥)) ↔ (∗‘(𝑥 ·ih (𝑢𝑦))) = (∗‘((𝑡𝑥) ·ih 𝑦))))
1211ancoms 461 . . . . . . . . . . . . . 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 28849 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℋ ∧ (𝑢𝑦) ∈ ℋ) → (𝑥 ·ih (𝑢𝑦)) ∈ ℂ)
143, 13sylan2 594 . . . . . . . . . . . . . . . 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 28849 . . . . . . . . . . . . . . . . 17 (((𝑡𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑡𝑥) ·ih 𝑦) ∈ ℂ)
177, 16sylan 582 . . . . . . . . . . . . . . . 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 14513 . . . . . . . . . . . . . . 15 (((𝑥 ·ih (𝑢𝑦)) ∈ ℂ ∧ ((𝑡𝑥) ·ih 𝑦) ∈ ℂ) → ((∗‘(𝑥 ·ih (𝑢𝑦))) = (∗‘((𝑡𝑥) ·ih 𝑦)) ↔ (𝑥 ·ih (𝑢𝑦)) = ((𝑡𝑥) ·ih 𝑦)))
2015, 18, 19syl2anc 586 . . . . . . . . . . . . . 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 282 . . . . . . . . . . . . 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 470 . . . . . . . . . . 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 2826 . . . . . . . . . . 11 (((𝑢𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑡𝑥)) ↔ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))
2523, 24syl6bb 289 . . . . . . . . . 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 3352 . . . . . . . 8 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))
2927, 28syl6bb 289 . . . . . . 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 577 . . . . . 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 1083 . . . . . 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 1083 . . . . . 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 305 . . . . 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 277 . . . 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 2842 . 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 29654 . . 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 29654 . 2 adj = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑦 ·ih (𝑡𝑥)) = ((𝑢𝑦) ·ih 𝑥))}
4036, 38, 393eqtr4i 2852 1 adj = adj
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wral 3136  {copab 5119  ccnv 5547  wf 6344  cfv 6348  (class class class)co 7148  cc 10527  ccj 14447  chba 28688   ·ih csp 28691  adjcado 28724
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-hfi 28848  ax-his1 28851
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  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 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-2 11692  df-cj 14450  df-re 14451  df-im 14452  df-adjh 29618
This theorem is referenced by:  funcnvadj  29662  adj1o  29663  adjbdlnb  29853
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