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Theorem hiassdi 28076
Description: Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hiassdi (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 · 𝐵) + 𝐶) ·ih 𝐷) = ((𝐴 · (𝐵 ·ih 𝐷)) + (𝐶 ·ih 𝐷)))

Proof of Theorem hiassdi
StepHypRef Expression
1 hvmulcl 27998 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 · 𝐵) ∈ ℋ)
2 ax-his2 28068 . . . 4 (((𝐴 · 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (((𝐴 · 𝐵) + 𝐶) ·ih 𝐷) = (((𝐴 · 𝐵) ·ih 𝐷) + (𝐶 ·ih 𝐷)))
323expb 1285 . . 3 (((𝐴 · 𝐵) ∈ ℋ ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 · 𝐵) + 𝐶) ·ih 𝐷) = (((𝐴 · 𝐵) ·ih 𝐷) + (𝐶 ·ih 𝐷)))
41, 3sylan 487 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 · 𝐵) + 𝐶) ·ih 𝐷) = (((𝐴 · 𝐵) ·ih 𝐷) + (𝐶 ·ih 𝐷)))
5 ax-his3 28069 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → ((𝐴 · 𝐵) ·ih 𝐷) = (𝐴 · (𝐵 ·ih 𝐷)))
653expa 1284 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ 𝐷 ∈ ℋ) → ((𝐴 · 𝐵) ·ih 𝐷) = (𝐴 · (𝐵 ·ih 𝐷)))
76adantrl 752 . . 3 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 · 𝐵) ·ih 𝐷) = (𝐴 · (𝐵 ·ih 𝐷)))
87oveq1d 6705 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 · 𝐵) ·ih 𝐷) + (𝐶 ·ih 𝐷)) = ((𝐴 · (𝐵 ·ih 𝐷)) + (𝐶 ·ih 𝐷)))
94, 8eqtrd 2685 1 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 · 𝐵) + 𝐶) ·ih 𝐷) = ((𝐴 · (𝐵 ·ih 𝐷)) + (𝐶 ·ih 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  (class class class)co 6690  cc 9972   + caddc 9977   · cmul 9979  chil 27904   + cva 27905   · csm 27906   ·ih csp 27907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-hfvmul 27990  ax-his2 28068  ax-his3 28069
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693
This theorem is referenced by:  unoplin  28907  hmoplin  28929
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