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Theorem hiassdi 27135
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 27057 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 · 𝐵) ∈ ℋ)
2 ax-his2 27127 . . . 4 (((𝐴 · 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (((𝐴 · 𝐵) + 𝐶) ·ih 𝐷) = (((𝐴 · 𝐵) ·ih 𝐷) + (𝐶 ·ih 𝐷)))
323expb 1257 . . 3 (((𝐴 · 𝐵) ∈ ℋ ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 · 𝐵) + 𝐶) ·ih 𝐷) = (((𝐴 · 𝐵) ·ih 𝐷) + (𝐶 ·ih 𝐷)))
41, 3sylan 486 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 · 𝐵) + 𝐶) ·ih 𝐷) = (((𝐴 · 𝐵) ·ih 𝐷) + (𝐶 ·ih 𝐷)))
5 ax-his3 27128 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → ((𝐴 · 𝐵) ·ih 𝐷) = (𝐴 · (𝐵 ·ih 𝐷)))
653expa 1256 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ 𝐷 ∈ ℋ) → ((𝐴 · 𝐵) ·ih 𝐷) = (𝐴 · (𝐵 ·ih 𝐷)))
76adantrl 747 . . 3 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 · 𝐵) ·ih 𝐷) = (𝐴 · (𝐵 ·ih 𝐷)))
87oveq1d 6539 . 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 2640 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 382   = wceq 1474  wcel 1976  (class class class)co 6524  cc 9787   + caddc 9792   · cmul 9794  chil 26963   + cva 26964   · csm 26965   ·ih csp 26966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825  ax-hfvmul 27049  ax-his2 27127  ax-his3 27128
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-fv 5795  df-ov 6527
This theorem is referenced by:  unoplin  27966  hmoplin  27988
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