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Theorem adjsym 29297
Description: Symmetry property of an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adjsym ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝑆   𝑥,𝑇,𝑦

Proof of Theorem adjsym
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ffvelrn 6721 . . . . . . . . . . . 12 ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇𝑦) ∈ ℋ)
2 ax-his1 28546 . . . . . . . . . . . 12 (((𝑇𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑇𝑦))))
31, 2sylan 580 . . . . . . . . . . 11 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑇𝑦))))
43adantrl 712 . . . . . . . . . 10 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑇𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑇𝑦))))
5 ffvelrn 6721 . . . . . . . . . . . 12 ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑆𝑥) ∈ ℋ)
6 ax-his1 28546 . . . . . . . . . . . 12 ((𝑦 ∈ ℋ ∧ (𝑆𝑥) ∈ ℋ) → (𝑦 ·ih (𝑆𝑥)) = (∗‘((𝑆𝑥) ·ih 𝑦)))
75, 6sylan2 592 . . . . . . . . . . 11 ((𝑦 ∈ ℋ ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑆𝑥)) = (∗‘((𝑆𝑥) ·ih 𝑦)))
87adantll 710 . . . . . . . . . 10 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑆𝑥)) = (∗‘((𝑆𝑥) ·ih 𝑦)))
94, 8eqeq12d 2812 . . . . . . . . 9 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥)) ↔ (∗‘(𝑥 ·ih (𝑇𝑦))) = (∗‘((𝑆𝑥) ·ih 𝑦))))
109ancoms 459 . . . . . . . 8 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥)) ↔ (∗‘(𝑥 ·ih (𝑇𝑦))) = (∗‘((𝑆𝑥) ·ih 𝑦))))
11 hicl 28544 . . . . . . . . . . 11 ((𝑥 ∈ ℋ ∧ (𝑇𝑦) ∈ ℋ) → (𝑥 ·ih (𝑇𝑦)) ∈ ℂ)
121, 11sylan2 592 . . . . . . . . . 10 ((𝑥 ∈ ℋ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑇𝑦)) ∈ ℂ)
1312adantll 710 . . . . . . . . 9 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑇𝑦)) ∈ ℂ)
14 hicl 28544 . . . . . . . . . . 11 (((𝑆𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆𝑥) ·ih 𝑦) ∈ ℂ)
155, 14sylan 580 . . . . . . . . . 10 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑆𝑥) ·ih 𝑦) ∈ ℂ)
1615adantrl 712 . . . . . . . . 9 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆𝑥) ·ih 𝑦) ∈ ℂ)
17 cj11 14359 . . . . . . . . 9 (((𝑥 ·ih (𝑇𝑦)) ∈ ℂ ∧ ((𝑆𝑥) ·ih 𝑦) ∈ ℂ) → ((∗‘(𝑥 ·ih (𝑇𝑦))) = (∗‘((𝑆𝑥) ·ih 𝑦)) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦)))
1813, 16, 17syl2anc 584 . . . . . . . 8 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((∗‘(𝑥 ·ih (𝑇𝑦))) = (∗‘((𝑆𝑥) ·ih 𝑦)) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦)))
1910, 18bitr2d 281 . . . . . . 7 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦) ↔ ((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥))))
2019an4s 656 . . . . . 6 (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦) ↔ ((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥))))
2120anassrs 468 . . . . 5 ((((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦) ↔ ((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥))))
22 eqcom 2804 . . . . 5 (((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥)) ↔ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥))
2321, 22syl6bb 288 . . . 4 ((((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦) ↔ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥)))
2423ralbidva 3165 . . 3 (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦) ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥)))
2524ralbidva 3165 . 2 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥)))
26 ralcom 3317 . . . 4 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦))
27 fveq2 6545 . . . . . . . 8 (𝑧 = 𝑦 → (𝑆𝑧) = (𝑆𝑦))
2827oveq2d 7039 . . . . . . 7 (𝑧 = 𝑦 → (𝑥 ·ih (𝑆𝑧)) = (𝑥 ·ih (𝑆𝑦)))
29 oveq2 7031 . . . . . . 7 (𝑧 = 𝑦 → ((𝑇𝑥) ·ih 𝑧) = ((𝑇𝑥) ·ih 𝑦))
3028, 29eqeq12d 2812 . . . . . 6 (𝑧 = 𝑦 → ((𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
3130ralbidv 3166 . . . . 5 (𝑧 = 𝑦 → (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
3231cbvralv 3405 . . . 4 (∀𝑧 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦))
3326, 32bitr4i 279 . . 3 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑧 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧))
34 oveq1 7030 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ·ih (𝑆𝑧)) = (𝑦 ·ih (𝑆𝑧)))
35 fveq2 6545 . . . . . . 7 (𝑥 = 𝑦 → (𝑇𝑥) = (𝑇𝑦))
3635oveq1d 7038 . . . . . 6 (𝑥 = 𝑦 → ((𝑇𝑥) ·ih 𝑧) = ((𝑇𝑦) ·ih 𝑧))
3734, 36eqeq12d 2812 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧)))
3837cbvralv 3405 . . . 4 (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧))
3938ralbii 3134 . . 3 (∀𝑧 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑧 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧))
40 fveq2 6545 . . . . . . 7 (𝑧 = 𝑥 → (𝑆𝑧) = (𝑆𝑥))
4140oveq2d 7039 . . . . . 6 (𝑧 = 𝑥 → (𝑦 ·ih (𝑆𝑧)) = (𝑦 ·ih (𝑆𝑥)))
42 oveq2 7031 . . . . . 6 (𝑧 = 𝑥 → ((𝑇𝑦) ·ih 𝑧) = ((𝑇𝑦) ·ih 𝑥))
4341, 42eqeq12d 2812 . . . . 5 (𝑧 = 𝑥 → ((𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧) ↔ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥)))
4443ralbidv 3166 . . . 4 (𝑧 = 𝑥 → (∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥)))
4544cbvralv 3405 . . 3 (∀𝑧 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥))
4633, 39, 453bitri 298 . 2 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥))
4725, 46syl6rbbr 291 1 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wcel 2083  wral 3107  wf 6228  cfv 6232  (class class class)co 7023  cc 10388  ccj 14293  chba 28383   ·ih csp 28386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-hfi 28543  ax-his1 28546
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-po 5369  df-so 5370  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-2 11554  df-cj 14296  df-re 14297  df-im 14298
This theorem is referenced by:  dfadj2  29349  adjval2  29355  cnlnadjeui  29541  cnlnssadj  29544  adjbdln  29547
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