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

StepHypRef Expression
1 ax-his2 28876 . . 3 ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 + 𝐶) ·ih 𝐴) = ((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴)))
213comr 1122 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 + 𝐶) ·ih 𝐴) = ((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴)))
3 hvaddcl 28805 . . . 4 ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 + 𝐶) ∈ ℋ)
4 braval 29737 . . . 4 ((𝐴 ∈ ℋ ∧ (𝐵 + 𝐶) ∈ ℋ) → ((bra‘𝐴)‘(𝐵 + 𝐶)) = ((𝐵 + 𝐶) ·ih 𝐴))
53, 4sylan2 595 . . 3 ((𝐴 ∈ ℋ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → ((bra‘𝐴)‘(𝐵 + 𝐶)) = ((𝐵 + 𝐶) ·ih 𝐴))
653impb 1112 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 + 𝐶)) = ((𝐵 + 𝐶) ·ih 𝐴))
7 braval 29737 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴))
873adant3 1129 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴))
9 braval 29737 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·ih 𝐴))
1093adant2 1128 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·ih 𝐴))
118, 10oveq12d 7154 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶)) = ((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴)))
122, 6, 113eqtr4d 2843 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 + 𝐶)) = (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ‘cfv 6325  (class class class)co 7136   + caddc 10532   ℋchba 28712   +ℎ cva 28713   ·ih csp 28715  bracbr 28749 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 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-hilex 28792  ax-hfvadd 28793  ax-his2 28876 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-bra 29643 This theorem is referenced by:  bralnfn  29741
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