Theorem braaddt 9785

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 8871	. . 3 |- ((B e. H~ /\ C e. H~ /\ A e. H~) -> ((B +h C) .ih A) = ((B .ih A) + (C .ih A)))
2	1	3comr 839	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((B +h C) .ih A) = ((B .ih A) + (C .ih A)))
3		bravalvalt 9784	. . . 4 |- ((A e. H~ /\ (B +h C) e. H~) -> ((bra` A)` (B +h C)) = ((B +h C) .ih A))
4		hvaddclt 8803	. . . 4 |- ((B e. H~ /\ C e. H~) -> (B +h C) e. H~)
5	3, 4	sylan2 451	. . 3 |- ((A e. H~ /\ (B e. H~ /\ C e. H~)) -> ((bra` A)` (B +h C)) = ((B +h C) .ih A))
6	5	3impb 827	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((bra` A)` (B +h C)) = ((B +h C) .ih A))
7		bravalvalt 9784	. . . 4 |- ((A e. H~ /\ B e. H~) -> ((bra` A)` B) = (B .ih A))
8	7	3adant3 797	. . 3 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((bra` A)` B) = (B .ih A))
9		bravalvalt 9784	. . . 4 |- ((A e. H~ /\ C e. H~) -> ((bra` A)` C) = (C .ih A))
10	9	3adant2 796	. . 3 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((bra` A)` C) = (C .ih A))
11	8, 10	opreq12d 3963	. 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 1509	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 773   = wceq 953   e. wcel 955  ` cfv 3172  (class class class)co 3948   + caddc 5209  H~chil 8727   +h cva 8728   .ih csp 8732  bracbr 8764

This theorem is referenced by:  bralnfnt 9788

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-hilex 8790  ax-hfvadd 8791  ax-his2 8871

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-bra 9693

Copyright terms: Public domain