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Theorem adjsym 31852
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 3289 . . . 4 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦))
2 fveq2 6906 . . . . . . . 8 (𝑧 = 𝑦 → (𝑆𝑧) = (𝑆𝑦))
32oveq2d 7447 . . . . . . 7 (𝑧 = 𝑦 → (𝑥 ·ih (𝑆𝑧)) = (𝑥 ·ih (𝑆𝑦)))
4 oveq2 7439 . . . . . . 7 (𝑧 = 𝑦 → ((𝑇𝑥) ·ih 𝑧) = ((𝑇𝑥) ·ih 𝑦))
53, 4eqeq12d 2753 . . . . . 6 (𝑧 = 𝑦 → ((𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
65ralbidv 3178 . . . . 5 (𝑧 = 𝑦 → (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
76cbvralvw 3237 . . . 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 7438 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ·ih (𝑆𝑧)) = (𝑦 ·ih (𝑆𝑧)))
10 fveq2 6906 . . . . . . 7 (𝑥 = 𝑦 → (𝑇𝑥) = (𝑇𝑦))
1110oveq1d 7446 . . . . . 6 (𝑥 = 𝑦 → ((𝑇𝑥) ·ih 𝑧) = ((𝑇𝑦) ·ih 𝑧))
129, 11eqeq12d 2753 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧)))
1312cbvralvw 3237 . . . 4 (∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧))
1413ralbii 3093 . . 3 (∀𝑧 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑆𝑧)) = ((𝑇𝑥) ·ih 𝑧) ↔ ∀𝑧 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧))
15 fveq2 6906 . . . . . . 7 (𝑧 = 𝑥 → (𝑆𝑧) = (𝑆𝑥))
1615oveq2d 7447 . . . . . 6 (𝑧 = 𝑥 → (𝑦 ·ih (𝑆𝑧)) = (𝑦 ·ih (𝑆𝑥)))
17 oveq2 7439 . . . . . 6 (𝑧 = 𝑥 → ((𝑇𝑦) ·ih 𝑧) = ((𝑇𝑦) ·ih 𝑥))
1816, 17eqeq12d 2753 . . . . 5 (𝑧 = 𝑥 → ((𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧) ↔ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥)))
1918ralbidv 3178 . . . 4 (𝑧 = 𝑥 → (∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑧)) = ((𝑇𝑦) ·ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·ih (𝑆𝑥)) = ((𝑇𝑦) ·ih 𝑥)))
2019cbvralvw 3237 . . 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 7101 . . . . . . . . . . . 12 ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇𝑦) ∈ ℋ)
23 ax-his1 31101 . . . . . . . . . . . 12 (((𝑇𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑇𝑦))))
2422, 23sylan 580 . . . . . . . . . . 11 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑇𝑦))))
2524adantrl 716 . . . . . . . . . 10 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑇𝑦) ·ih 𝑥) = (∗‘(𝑥 ·ih (𝑇𝑦))))
26 ffvelcdm 7101 . . . . . . . . . . . 12 ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑆𝑥) ∈ ℋ)
27 ax-his1 31101 . . . . . . . . . . . 12 ((𝑦 ∈ ℋ ∧ (𝑆𝑥) ∈ ℋ) → (𝑦 ·ih (𝑆𝑥)) = (∗‘((𝑆𝑥) ·ih 𝑦)))
2826, 27sylan2 593 . . . . . . . . . . 11 ((𝑦 ∈ ℋ ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑆𝑥)) = (∗‘((𝑆𝑥) ·ih 𝑦)))
2928adantll 714 . . . . . . . . . 10 (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·ih (𝑆𝑥)) = (∗‘((𝑆𝑥) ·ih 𝑦)))
3025, 29eqeq12d 2753 . . . . . . . . 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 31099 . . . . . . . . . . 11 ((𝑥 ∈ ℋ ∧ (𝑇𝑦) ∈ ℋ) → (𝑥 ·ih (𝑇𝑦)) ∈ ℂ)
3322, 32sylan2 593 . . . . . . . . . 10 ((𝑥 ∈ ℋ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑇𝑦)) ∈ ℂ)
3433adantll 714 . . . . . . . . 9 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑇𝑦)) ∈ ℂ)
35 hicl 31099 . . . . . . . . . . 11 (((𝑆𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆𝑥) ·ih 𝑦) ∈ ℂ)
3626, 35sylan 580 . . . . . . . . . 10 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑆𝑥) ·ih 𝑦) ∈ ℂ)
3736adantrl 716 . . . . . . . . 9 (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆𝑥) ·ih 𝑦) ∈ ℂ)
38 cj11 15201 . . . . . . . . 9 (((𝑥 ·ih (𝑇𝑦)) ∈ ℂ ∧ ((𝑆𝑥) ·ih 𝑦) ∈ ℂ) → ((∗‘(𝑥 ·ih (𝑇𝑦))) = (∗‘((𝑆𝑥) ·ih 𝑦)) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦)))
3934, 37, 38syl2anc 584 . . . . . . . 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 660 . . . . . 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 2744 . . . . 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 3176 . . 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 3176 . 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 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  ccj 15135  chba 30938   ·ih csp 30941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-hfi 31098  ax-his1 31101
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-2 12329  df-cj 15138  df-re 15139  df-im 15140
This theorem is referenced by:  dfadj2  31904  adjval2  31910  cnlnadjeui  32096  cnlnssadj  32099  adjbdln  32102
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