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Theorem braadd 30984
Description: Linearity property of bra for addition. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
braadd ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜(𝐡 +β„Ž 𝐢)) = (((braβ€˜π΄)β€˜π΅) + ((braβ€˜π΄)β€˜πΆ)))
Proof of Theorem braadd
StepHypRef Expression
1 ax-his2 30122 . . 3 ((𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹ ∧ 𝐴 ∈ β„‹) β†’ ((𝐡 +β„Ž 𝐢) Β·ih 𝐴) = ((𝐡 Β·ih 𝐴) + (𝐢 Β·ih 𝐴)))
213comr 1125 . 2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐡 +β„Ž 𝐢) Β·ih 𝐴) = ((𝐡 Β·ih 𝐴) + (𝐢 Β·ih 𝐴)))
3 hvaddcl 30051 . . . 4 ((𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐡 +β„Ž 𝐢) ∈ β„‹)
4 braval 30983 . . . 4 ((𝐴 ∈ β„‹ ∧ (𝐡 +β„Ž 𝐢) ∈ β„‹) β†’ ((braβ€˜π΄)β€˜(𝐡 +β„Ž 𝐢)) = ((𝐡 +β„Ž 𝐢) Β·ih 𝐴))
53, 4sylan2 593 . . 3 ((𝐴 ∈ β„‹ ∧ (𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹)) β†’ ((braβ€˜π΄)β€˜(𝐡 +β„Ž 𝐢)) = ((𝐡 +β„Ž 𝐢) Β·ih 𝐴))
653impb 1115 . 2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜(𝐡 +β„Ž 𝐢)) = ((𝐡 +β„Ž 𝐢) Β·ih 𝐴))
7 braval 30983 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π΅) = (𝐡 Β·ih 𝐴))
873adant3 1132 . . 3 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π΅) = (𝐡 Β·ih 𝐴))
9 braval 30983 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜πΆ) = (𝐢 Β·ih 𝐴))
1093adant2 1131 . . 3 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜πΆ) = (𝐢 Β·ih 𝐴))
118, 10oveq12d 7395 . 2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (((braβ€˜π΄)β€˜π΅) + ((braβ€˜π΄)β€˜πΆ)) = ((𝐡 Β·ih 𝐴) + (𝐢 Β·ih 𝐴)))
122, 6, 113eqtr4d 2781 1 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜(𝐡 +β„Ž 𝐢)) = (((braβ€˜π΄)β€˜π΅) + ((braβ€˜π΄)β€˜πΆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  β€˜cfv 6516  (class class class)co 7377   + caddc 11078   β„‹chba 29958   +β„Ž cva 29959   Β·ih csp 29961  bracbr 29995
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pr 5404  ax-hilex 30038  ax-hfvadd 30039  ax-his2 30122
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-bra 30889
This theorem is referenced by:  bralnfn  30987
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