Theorem braaddt 9864

Description: Linearity property of bra for addition.

Assertion

Ref	Expression
braaddt	|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((bra` A)` (B +h C)) = (((bra` A)` B) + ((bra` A)` C)))

Proof of Theorem braaddt

Step	Hyp	Ref	Expression
1		ax-his2 8945	. . 3 |- ((B e. H~ /\ C e. H~ /\ A e. H~) -> ((B +h C) .ih A) = ((B .ih A) + (C .ih A)))
2	1	3comr 843	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((B +h C) .ih A) = ((B .ih A) + (C .ih A)))
3		bravalvalt 9863	. . . 4 |- ((A e. H~ /\ (B +h C) e. H~) -> ((bra` A)` (B +h C)) = ((B +h C) .ih A))
4		hvaddclt 8877	. . . 4 |- ((B e. H~ /\ C e. H~) -> (B +h C) e. H~)
5	3, 4	sylan2 453	. . 3 |- ((A e. H~ /\ (B e. H~ /\ C e. H~)) -> ((bra` A)` (B +h C)) = ((B +h C) .ih A))
6	5	3impb 831	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((bra` A)` (B +h C)) = ((B +h C) .ih A))
7		bravalvalt 9863	. . . 4 |- ((A e. H~ /\ B e. H~) -> ((bra` A)` B) = (B .ih A))
8	7	3adant3 801	. . 3 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((bra` A)` B) = (B .ih A))
9		bravalvalt 9863	. . . 4 |- ((A e. H~ /\ C e. H~) -> ((bra` A)` C) = (C .ih A))
10	9	3adant2 800	. . 3 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((bra` A)` C) = (C .ih A))
11	8, 10	opreq12d 3984	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> (((bra` A)` B) + ((bra` A)` C)) = ((B .ih A) + (C .ih A)))
12	2, 6, 11	3eqtr4d 1520	1 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((bra` A)` (B +h C)) = (((bra` A)` B) + ((bra` A)` C)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  ` cfv 3188  (class class class)co 3969   + caddc 5249  H~chil 8783   +h cva 8784   .ih csp 8788  bracbr 8820

This theorem is referenced by:  bralnfnt 9867

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-hilex 8864  ax-hfvadd 8865  ax-his2 8945

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-bra 9771

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