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Theorem unop 29811
 Description: Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
unop ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵))

Proof of Theorem unop
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elunop 29768 . . . 4 (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
21simprbi 500 . . 3 (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦))
323ad2ant1 1130 . 2 ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦))
4 fveq2 6663 . . . . . 6 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
54oveq1d 7171 . . . . 5 (𝑥 = 𝐴 → ((𝑇𝑥) ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih (𝑇𝑦)))
6 oveq1 7163 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·ih 𝑦) = (𝐴 ·ih 𝑦))
75, 6eqeq12d 2774 . . . 4 (𝑥 = 𝐴 → (((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦) ↔ ((𝑇𝐴) ·ih (𝑇𝑦)) = (𝐴 ·ih 𝑦)))
8 fveq2 6663 . . . . . 6 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
98oveq2d 7172 . . . . 5 (𝑦 = 𝐵 → ((𝑇𝐴) ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih (𝑇𝐵)))
10 oveq2 7164 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·ih 𝑦) = (𝐴 ·ih 𝐵))
119, 10eqeq12d 2774 . . . 4 (𝑦 = 𝐵 → (((𝑇𝐴) ·ih (𝑇𝑦)) = (𝐴 ·ih 𝑦) ↔ ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵)))
127, 11rspc2v 3553 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦) → ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵)))
13123adant1 1127 . 2 ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦) → ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵)))
143, 13mpd 15 1 ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  –onto→wfo 6338  ‘cfv 6340  (class class class)co 7156   ℋchba 28815   ·ih csp 28818  UniOpcuo 28845 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-hilex 28895 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-unop 29739 This theorem is referenced by:  unopf1o  29812  unopnorm  29813  cnvunop  29814  unopadj  29815  counop  29817
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