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Theorem bramul 28654
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 27790 . . 3 ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 · 𝐶) ·ih 𝐴) = (𝐵 · (𝐶 ·ih 𝐴)))
213comr 1270 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐵 · 𝐶) ·ih 𝐴) = (𝐵 · (𝐶 ·ih 𝐴)))
3 hvmulcl 27719 . . . 4 ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 · 𝐶) ∈ ℋ)
4 braval 28652 . . . 4 ((𝐴 ∈ ℋ ∧ (𝐵 · 𝐶) ∈ ℋ) → ((bra‘𝐴)‘(𝐵 · 𝐶)) = ((𝐵 · 𝐶) ·ih 𝐴))
53, 4sylan2 491 . . 3 ((𝐴 ∈ ℋ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → ((bra‘𝐴)‘(𝐵 · 𝐶)) = ((𝐵 · 𝐶) ·ih 𝐴))
653impb 1257 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 · 𝐶)) = ((𝐵 · 𝐶) ·ih 𝐴))
7 braval 28652 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·ih 𝐴))
873adant2 1078 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·ih 𝐴))
98oveq2d 6620 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 · ((bra‘𝐴)‘𝐶)) = (𝐵 · (𝐶 ·ih 𝐴)))
102, 6, 93eqtr4d 2665 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 384  w3a 1036   = wceq 1480  wcel 1987  cfv 5847  (class class class)co 6604  cc 9878   · cmul 9885  chil 27625   · csm 27627   ·ih csp 27628  bracbr 27662
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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-hilex 27705  ax-hfvmul 27711  ax-his3 27790
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-bra 28558
This theorem is referenced by:  bralnfn  28656
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