Theorem hvadd12t 8904

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 8871	. . . 4 |- ((A e. H~ /\ B e. H~) -> (A +h B) = (B +h A))
2	1	opreq1d 3975	. . 3 |- ((A e. H~ /\ B e. H~) -> ((A +h B) +h C) = ((B +h A) +h C))
3	2	3adant3 799	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h B) +h C) = ((B +h A) +h C))
4		ax-hvass 8872	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h B) +h C) = (A +h (B +h C)))
5		ax-hvass 8872	. . 3 |- ((B e. H~ /\ A e. H~ /\ C e. H~) -> ((B +h A) +h C) = (B +h (A +h C)))
6	5	3com12 837	. 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 1515	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 775   = wceq 956   e. wcel 958  (class class class)co 3963  H~chil 8788   +h cva 8789

This theorem is referenced by:  hvaddsub12t 8907

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-hvcom 8871  ax-hvass 8872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965

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