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Theorem braadd 29113
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)))

Proof of Theorem braadd
StepHypRef Expression
1 ax-his2 28249 . . 3 ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 + 𝐶) ·ih 𝐴) = ((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴)))
213comr 1120 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 + 𝐶) ·ih 𝐴) = ((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴)))
3 hvaddcl 28178 . . . 4 ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 + 𝐶) ∈ ℋ)
4 braval 29112 . . . 4 ((𝐴 ∈ ℋ ∧ (𝐵 + 𝐶) ∈ ℋ) → ((bra‘𝐴)‘(𝐵 + 𝐶)) = ((𝐵 + 𝐶) ·ih 𝐴))
53, 4sylan2 492 . . 3 ((𝐴 ∈ ℋ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → ((bra‘𝐴)‘(𝐵 + 𝐶)) = ((𝐵 + 𝐶) ·ih 𝐴))
653impb 1108 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 + 𝐶)) = ((𝐵 + 𝐶) ·ih 𝐴))
7 braval 29112 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴))
873adant3 1127 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴))
9 braval 29112 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·ih 𝐴))
1093adant2 1126 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·ih 𝐴))
118, 10oveq12d 6831 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶)) = ((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴)))
122, 6, 113eqtr4d 2804 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 383  w3a 1072   = wceq 1632  wcel 2139  cfv 6049  (class class class)co 6813   + caddc 10131  chil 28085   + cva 28086   ·ih csp 28088  bracbr 28122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-hilex 28165  ax-hfvadd 28166  ax-his2 28249
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-bra 29018
This theorem is referenced by:  bralnfn  29116
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