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Theorem adjsym 29032
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 6500 . . . . . . . . . . . 12 ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇𝑦) ∈ ℋ)
2 ax-his1 28279 . . . . . . . . . . . 12 (((𝑇𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑇𝑦))))
31, 2sylan 569 . . . . . . . . . . 11 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑇𝑦))))
43adantrl 695 . . . . . . . . . 10 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑇𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑇𝑦))))
5 ffvelrn 6500 . . . . . . . . . . . 12 ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑆𝑥) ∈ ℋ)
6 ax-his1 28279 . . . . . . . . . . . 12 ((𝑦 ∈ ℋ ∧ (𝑆𝑥) ∈ ℋ) → (𝑦 ·ih (𝑆𝑥)) = (∗‘((𝑆𝑥) ·ih 𝑦)))
75, 6sylan2 580 . . . . . . . . . . 11 ((𝑦 ∈ ℋ ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑆𝑥)) = (∗‘((𝑆𝑥) ·ih 𝑦)))
87adantll 693 . . . . . . . . . 10 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑆𝑥)) = (∗‘((𝑆𝑥) ·ih 𝑦)))
94, 8eqeq12d 2786 . . . . . . . . 9 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥)) ↔ (∗‘(𝑥 ·ih (𝑇𝑦))) = (∗‘((𝑆𝑥) ·ih 𝑦))))
109ancoms 455 . . . . . . . 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 28277 . . . . . . . . . . 11 ((𝑥 ∈ ℋ ∧ (𝑇𝑦) ∈ ℋ) → (𝑥 ·ih (𝑇𝑦)) ∈ ℂ)
121, 11sylan2 580 . . . . . . . . . 10 ((𝑥 ∈ ℋ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑇𝑦)) ∈ ℂ)
1312adantll 693 . . . . . . . . 9 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑇𝑦)) ∈ ℂ)
14 hicl 28277 . . . . . . . . . . 11 (((𝑆𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆𝑥) ·ih 𝑦) ∈ ℂ)
155, 14sylan 569 . . . . . . . . . 10 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑆𝑥) ·ih 𝑦) ∈ ℂ)
1615adantrl 695 . . . . . . . . 9 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆𝑥) ·ih 𝑦) ∈ ℂ)
17 cj11 14110 . . . . . . . . 9 (((𝑥 ·ih (𝑇𝑦)) ∈ ℂ ∧ ((𝑆𝑥) ·ih 𝑦) ∈ ℂ) → ((∗‘(𝑥 ·ih (𝑇𝑦))) = (∗‘((𝑆𝑥) ·ih 𝑦)) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦)))
1813, 16, 17syl2anc 573 . . . . . . . 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 269 . . . . . . 7 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦) ↔ ((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥))))
2019an4s 639 . . . . . 6 (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦) ↔ ((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥))))
2120anassrs 458 . . . . 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 2778 . . . . 5 (((𝑇𝑦) ·ih 𝑥) = (𝑦 ·ih (𝑆𝑥)) ↔ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥))
2321, 22syl6bb 276 . . . 4 ((((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦) ↔ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥)))
2423ralbidva 3134 . . 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 3134 . 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 3246 . . . 4 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦))
27 fveq2 6332 . . . . . . . 8 (𝑧 = 𝑦 → (𝑆𝑧) = (𝑆𝑦))
2827oveq2d 6809 . . . . . . 7 (𝑧 = 𝑦 → (𝑥 ·ih (𝑆𝑧)) = (𝑥 ·ih (𝑆𝑦)))
29 oveq2 6801 . . . . . . 7 (𝑧 = 𝑦 → ((𝑇𝑥) ·ih 𝑧) = ((𝑇𝑥) ·ih 𝑦))
3028, 29eqeq12d 2786 . . . . . 6 (𝑧 = 𝑦 → ((𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
3130ralbidv 3135 . . . . 5 (𝑧 = 𝑦 → (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
3231cbvralv 3320 . . . 4 (∀𝑧 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦))
3326, 32bitr4i 267 . . 3 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑧 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧))
34 oveq1 6800 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ·ih (𝑆𝑧)) = (𝑦 ·ih (𝑆𝑧)))
35 fveq2 6332 . . . . . . 7 (𝑥 = 𝑦 → (𝑇𝑥) = (𝑇𝑦))
3635oveq1d 6808 . . . . . 6 (𝑥 = 𝑦 → ((𝑇𝑥) ·ih 𝑧) = ((𝑇𝑦) ·ih 𝑧))
3734, 36eqeq12d 2786 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧)))
3837cbvralv 3320 . . . 4 (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧))
3938ralbii 3129 . . 3 (∀𝑧 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑧 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧))
40 fveq2 6332 . . . . . . 7 (𝑧 = 𝑥 → (𝑆𝑧) = (𝑆𝑥))
4140oveq2d 6809 . . . . . 6 (𝑧 = 𝑥 → (𝑦 ·ih (𝑆𝑧)) = (𝑦 ·ih (𝑆𝑥)))
42 oveq2 6801 . . . . . 6 (𝑧 = 𝑥 → ((𝑇𝑦) ·ih 𝑧) = ((𝑇𝑦) ·ih 𝑥))
4341, 42eqeq12d 2786 . . . . 5 (𝑧 = 𝑥 → ((𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧) ↔ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥)))
4443ralbidv 3135 . . . 4 (𝑧 = 𝑥 → (∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥)))
4544cbvralv 3320 . . 3 (∀𝑧 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥))
4633, 39, 453bitri 286 . 2 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥))
4725, 46syl6rbbr 279 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 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wf 6027  cfv 6031  (class class class)co 6793  cc 10136  ccj 14044  chil 28116   ·ih csp 28119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-hfi 28276  ax-his1 28279
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-2 11281  df-cj 14047  df-re 14048  df-im 14049
This theorem is referenced by:  dfadj2  29084  adjval2  29090  cnlnadjeui  29276  cnlnssadj  29279  adjbdln  29282
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