Theorem hvadd12 27892

Description: Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
hvadd12	|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A +h ( B +h C ) ) = ( B +h ( A +h C ) ) )

Proof of Theorem hvadd12

Step	Hyp	Ref	Expression
1		ax-hvcom 27858	. . . 4 |- ( ( A e. ~H /\ B e. ~H ) -> ( A +h B ) = ( B +h A ) )
2	1	oveq1d 6665	. . 3 |- ( ( A e. ~H /\ B e. ~H ) -> ( ( A +h B ) +h C ) = ( ( B +h A ) +h C ) )
3	2	3adant3 1081	. 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h B ) +h C ) = ( ( B +h A ) +h C ) )
4		ax-hvass 27859	. 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h B ) +h C ) = ( A +h ( B +h C ) ) )
5		ax-hvass 27859	. . 3 |- ( ( B e. ~H /\ A e. ~H /\ C e. ~H ) -> ( ( B +h A ) +h C ) = ( B +h ( A +h C ) ) )
6	5	3com12 1269	. 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 2664	1 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A +h ( B +h C ) ) = ( B +h ( A +h C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990  (class class class)co 6650   ~Hchil 27776   +h cva 27777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-hvcom 27858  ax-hvass 27859

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  hvaddsub12  27895