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Theorem bramul 29504
Description: Linearity property of bra for multiplication. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
bramul ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 · 𝐶)) = (𝐵 · ((bra‘𝐴)‘𝐶)))

Proof of Theorem bramul
StepHypRef Expression
1 ax-his3 28640 . . 3 ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 · 𝐶) ·ih 𝐴) = (𝐵 · (𝐶 ·ih 𝐴)))
213comr 1105 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐵 · 𝐶) ·ih 𝐴) = (𝐵 · (𝐶 ·ih 𝐴)))
3 hvmulcl 28569 . . . 4 ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 · 𝐶) ∈ ℋ)
4 braval 29502 . . . 4 ((𝐴 ∈ ℋ ∧ (𝐵 · 𝐶) ∈ ℋ) → ((bra‘𝐴)‘(𝐵 · 𝐶)) = ((𝐵 · 𝐶) ·ih 𝐴))
53, 4sylan2 583 . . 3 ((𝐴 ∈ ℋ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → ((bra‘𝐴)‘(𝐵 · 𝐶)) = ((𝐵 · 𝐶) ·ih 𝐴))
653impb 1095 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 · 𝐶)) = ((𝐵 · 𝐶) ·ih 𝐴))
7 braval 29502 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·ih 𝐴))
873adant2 1111 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·ih 𝐴))
98oveq2d 6992 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 · ((bra‘𝐴)‘𝐶)) = (𝐵 · (𝐶 ·ih 𝐴)))
102, 6, 93eqtr4d 2825 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 · 𝐶)) = (𝐵 · ((bra‘𝐴)‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2050  cfv 6188  (class class class)co 6976  cc 10333   · cmul 10340  chba 28475   · csm 28477   ·ih csp 28478  bracbr 28512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-hilex 28555  ax-hfvmul 28561  ax-his3 28640
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-bra 29408
This theorem is referenced by:  bralnfn  29506
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