Theorem hvadd12t 8899

Description: Commutative/associative law.

Assertion

Ref	Expression
hvadd12t	|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> (A +h (B +h C)) = (B +h (A +h C)))

Proof of Theorem hvadd12t

Step	Hyp	Ref	Expression
1		ax-hvcom 8866	. . . 4 |- ((A e. H~ /\ B e. H~) -> (A +h B) = (B +h A))
2	1	opreq1d 3981	. . 3 |- ((A e. H~ /\ B e. H~) -> ((A +h B) +h C) = ((B +h A) +h C))
3	2	3adant3 801	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h B) +h C) = ((B +h A) +h C))
4		ax-hvass 8867	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h B) +h C) = (A +h (B +h C)))
5		ax-hvass 8867	. . 3 |- ((B e. H~ /\ A e. H~ /\ C e. H~) -> ((B +h A) +h C) = (B +h (A +h C)))
6	5	3com12 839	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((B +h A) +h C) = (B +h (A +h C)))
7	3, 4, 6	3eqtr3d 1518	1 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> (A +h (B +h C)) = (B +h (A +h C)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  (class class class)co 3969  H~chil 8783   +h cva 8784

This theorem is referenced by:  hvaddsub12t 8902

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-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-sep 2708  ax-pow 2748  ax-pr 2785  ax-hvcom 8866  ax-hvass 8867

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-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-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971

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