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Theorem braadd 31465
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 30603 . . 3 ((𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹ ∧ 𝐴 ∈ β„‹) β†’ ((𝐡 +β„Ž 𝐢) Β·ih 𝐴) = ((𝐡 Β·ih 𝐴) + (𝐢 Β·ih 𝐴)))
213comr 1123 . 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 30532 . . . 4 ((𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐡 +β„Ž 𝐢) ∈ β„‹)
4 braval 31464 . . . 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 591 . . 3 ((𝐴 ∈ β„‹ ∧ (𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹)) β†’ ((braβ€˜π΄)β€˜(𝐡 +β„Ž 𝐢)) = ((𝐡 +β„Ž 𝐢) Β·ih 𝐴))
653impb 1113 . 2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜(𝐡 +β„Ž 𝐢)) = ((𝐡 +β„Ž 𝐢) Β·ih 𝐴))
7 braval 31464 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π΅) = (𝐡 Β·ih 𝐴))
873adant3 1130 . . 3 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π΅) = (𝐡 Β·ih 𝐴))
9 braval 31464 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜πΆ) = (𝐢 Β·ih 𝐴))
1093adant2 1129 . . 3 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜πΆ) = (𝐢 Β·ih 𝐴))
118, 10oveq12d 7429 . 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 2780 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  β€˜cfv 6542  (class class class)co 7411   + caddc 11115   β„‹chba 30439   +β„Ž cva 30440   Β·ih csp 30442  bracbr 30476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-hilex 30519  ax-hfvadd 30520  ax-his2 30603
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-bra 31370
This theorem is referenced by:  bralnfn  31468
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