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Theorem adjval 28589
Description: Value of the adjoint function for 𝑇 in the domain of adj. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.)
Assertion
Ref Expression
adjval (𝑇 ∈ dom adj → (adj𝑇) = (𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
Distinct variable group:   𝑥,𝑢,𝑦,𝑇

Proof of Theorem adjval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 dmadjop 28587 . . . . 5 (𝑇 ∈ dom adj𝑇: ℋ⟶ ℋ)
21biantrurd 529 . . . 4 (𝑇 ∈ dom adj → ((𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))))
3 ax-hilex 27696 . . . . . 6 ℋ ∈ V
43, 3elmap 7831 . . . . 5 (𝑢 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑢: ℋ⟶ ℋ)
54anbi1i 730 . . . 4 ((𝑢 ∈ ( ℋ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ (𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
6 3anass 1040 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))))
72, 5, 63bitr4g 303 . . 3 (𝑇 ∈ dom adj → ((𝑢 ∈ ( ℋ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))))
87iotabidv 5834 . 2 (𝑇 ∈ dom adj → (℩𝑢(𝑢 ∈ ( ℋ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))) = (℩𝑢(𝑇: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))))
9 df-riota 6566 . . 3 (𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) = (℩𝑢(𝑢 ∈ ( ℋ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
109a1i 11 . 2 (𝑇 ∈ dom adj → (𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) = (℩𝑢(𝑢 ∈ ( ℋ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))))
11 dfadj2 28584 . . 3 adj = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦))}
12 feq1 5985 . . . 4 (𝑡 = 𝑇 → (𝑡: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
13 fveq1 6149 . . . . . . 7 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
1413oveq2d 6621 . . . . . 6 (𝑡 = 𝑇 → (𝑥 ·ih (𝑡𝑦)) = (𝑥 ·ih (𝑇𝑦)))
1514eqeq1d 2628 . . . . 5 (𝑡 = 𝑇 → ((𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
16152ralbidv 2988 . . . 4 (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
1712, 163anbi13d 1398 . . 3 (𝑡 = 𝑇 → ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))))
1811, 17fvopab5 6266 . 2 (𝑇 ∈ dom adj → (adj𝑇) = (℩𝑢(𝑇: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))))
198, 10, 183eqtr4rd 2671 1 (𝑇 ∈ dom adj → (adj𝑇) = (𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  dom cdm 5079  cio 5811  wf 5846  cfv 5850  crio 6565  (class class class)co 6605  𝑚 cmap 7803  chil 27616   ·ih csp 27619  adjcado 27652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-hilex 27696  ax-hfi 27776  ax-his1 27779
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-2 11024  df-cj 13768  df-re 13769  df-im 13770  df-adjh 28548
This theorem is referenced by:  adjval2  28590  adjbdln  28782
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