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Theorem adjsym 31518
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 ralcom 3285 . . . 4 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦))
2 fveq2 6891 . . . . . . . 8 (𝑧 = 𝑦 → (𝑆𝑧) = (𝑆𝑦))
32oveq2d 7428 . . . . . . 7 (𝑧 = 𝑦 → (𝑥 ·ih (𝑆𝑧)) = (𝑥 ·ih (𝑆𝑦)))
4 oveq2 7420 . . . . . . 7 (𝑧 = 𝑦 → ((𝑇𝑥) ·ih 𝑧) = ((𝑇𝑥) ·ih 𝑦))
53, 4eqeq12d 2747 . . . . . 6 (𝑧 = 𝑦 → ((𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
65ralbidv 3176 . . . . 5 (𝑧 = 𝑦 → (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
76cbvralvw 3233 . . . 4 (∀𝑧 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦))
81, 7bitr4i 278 . . 3 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑧 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧))
9 oveq1 7419 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ·ih (𝑆𝑧)) = (𝑦 ·ih (𝑆𝑧)))
10 fveq2 6891 . . . . . . 7 (𝑥 = 𝑦 → (𝑇𝑥) = (𝑇𝑦))
1110oveq1d 7427 . . . . . 6 (𝑥 = 𝑦 → ((𝑇𝑥) ·ih 𝑧) = ((𝑇𝑦) ·ih 𝑧))
129, 11eqeq12d 2747 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧)))
1312cbvralvw 3233 . . . 4 (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧))
1413ralbii 3092 . . 3 (∀𝑧 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑧 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧))
15 fveq2 6891 . . . . . . 7 (𝑧 = 𝑥 → (𝑆𝑧) = (𝑆𝑥))
1615oveq2d 7428 . . . . . 6 (𝑧 = 𝑥 → (𝑦 ·ih (𝑆𝑧)) = (𝑦 ·ih (𝑆𝑥)))
17 oveq2 7420 . . . . . 6 (𝑧 = 𝑥 → ((𝑇𝑦) ·ih 𝑧) = ((𝑇𝑦) ·ih 𝑥))
1816, 17eqeq12d 2747 . . . . 5 (𝑧 = 𝑥 → ((𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧) ↔ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥)))
1918ralbidv 3176 . . . 4 (𝑧 = 𝑥 → (∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥)))
2019cbvralvw 3233 . . 3 (∀𝑧 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥))
218, 14, 203bitri 297 . 2 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥))
22 ffvelcdm 7083 . . . . . . . . . . . 12 ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇𝑦) ∈ ℋ)
23 ax-his1 30767 . . . . . . . . . . . 12 (((𝑇𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑇𝑦))))
2422, 23sylan 579 . . . . . . . . . . 11 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑇𝑦))))
2524adantrl 713 . . . . . . . . . 10 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑇𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑇𝑦))))
26 ffvelcdm 7083 . . . . . . . . . . . 12 ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑆𝑥) ∈ ℋ)
27 ax-his1 30767 . . . . . . . . . . . 12 ((𝑦 ∈ ℋ ∧ (𝑆𝑥) ∈ ℋ) → (𝑦 ·ih (𝑆𝑥)) = (∗‘((𝑆𝑥) ·ih 𝑦)))
2826, 27sylan2 592 . . . . . . . . . . 11 ((𝑦 ∈ ℋ ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑆𝑥)) = (∗‘((𝑆𝑥) ·ih 𝑦)))
2928adantll 711 . . . . . . . . . 10 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑆𝑥)) = (∗‘((𝑆𝑥) ·ih 𝑦)))
3025, 29eqeq12d 2747 . . . . . . . . 9 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥)) ↔ (∗‘(𝑥 ·ih (𝑇𝑦))) = (∗‘((𝑆𝑥) ·ih 𝑦))))
3130ancoms 458 . . . . . . . 8 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥)) ↔ (∗‘(𝑥 ·ih (𝑇𝑦))) = (∗‘((𝑆𝑥) ·ih 𝑦))))
32 hicl 30765 . . . . . . . . . . 11 ((𝑥 ∈ ℋ ∧ (𝑇𝑦) ∈ ℋ) → (𝑥 ·ih (𝑇𝑦)) ∈ ℂ)
3322, 32sylan2 592 . . . . . . . . . 10 ((𝑥 ∈ ℋ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑇𝑦)) ∈ ℂ)
3433adantll 711 . . . . . . . . 9 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑇𝑦)) ∈ ℂ)
35 hicl 30765 . . . . . . . . . . 11 (((𝑆𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆𝑥) ·ih 𝑦) ∈ ℂ)
3626, 35sylan 579 . . . . . . . . . 10 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑆𝑥) ·ih 𝑦) ∈ ℂ)
3736adantrl 713 . . . . . . . . 9 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆𝑥) ·ih 𝑦) ∈ ℂ)
38 cj11 15116 . . . . . . . . 9 (((𝑥 ·ih (𝑇𝑦)) ∈ ℂ ∧ ((𝑆𝑥) ·ih 𝑦) ∈ ℂ) → ((∗‘(𝑥 ·ih (𝑇𝑦))) = (∗‘((𝑆𝑥) ·ih 𝑦)) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦)))
3934, 37, 38syl2anc 583 . . . . . . . 8 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((∗‘(𝑥 ·ih (𝑇𝑦))) = (∗‘((𝑆𝑥) ·ih 𝑦)) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦)))
4031, 39bitr2d 280 . . . . . . 7 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦) ↔ ((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥))))
4140an4s 657 . . . . . 6 (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦) ↔ ((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥))))
4241anassrs 467 . . . . 5 ((((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦) ↔ ((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥))))
43 eqcom 2738 . . . . 5 (((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥)) ↔ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥))
4442, 43bitrdi 287 . . . 4 ((((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦) ↔ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥)))
4544ralbidva 3174 . . 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)))
4645ralbidva 3174 . 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)))
4721, 46bitr4id 290 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 205  wa 395   = wceq 1540  wcel 2105  wral 3060  wf 6539  cfv 6543  (class class class)co 7412  cc 11114  ccj 15050  chba 30604   ·ih csp 30607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-hfi 30764  ax-his1 30767
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-2 12282  df-cj 15053  df-re 15054  df-im 15055
This theorem is referenced by:  dfadj2  31570  adjval2  31576  cnlnadjeui  31762  cnlnssadj  31765  adjbdln  31768
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