Theorem addassnq 9780

Description: Addition of positive fractions is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
addassnq	|- ( ( A +Q B ) +Q C ) = ( A +Q ( B +Q C ) )

Proof of Theorem addassnq

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addasspi 9717	. . . . . . . 8 |- ( ( ( ( 1st ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( 1st ` B ) .N ( 2nd ` A ) ) .N ( 2nd ` C ) ) ) +N ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) = ( ( ( 1st ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( ( 1st ` B ) .N ( 2nd ` A ) ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) )
2		ovex 6678	. . . . . . . . . . 11 |- ( ( 1st ` A ) .N ( 2nd ` B ) ) e. _V
3		ovex 6678	. . . . . . . . . . 11 |- ( ( 1st ` B ) .N ( 2nd ` A ) ) e. _V
4		fvex 6201	. . . . . . . . . . 11 |- ( 2nd ` C ) e. _V
5		mulcompi 9718	. . . . . . . . . . 11 |- ( x .N y ) = ( y .N x )
6		distrpi 9720	. . . . . . . . . . 11 |- ( x .N ( y +N z ) ) = ( ( x .N y ) +N ( x .N z ) )
7	2, 3, 4, 5, 6	caovdir 6868	. . . . . . . . . 10 |- ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) .N ( 2nd ` C ) ) = ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( 2nd ` C ) ) +N ( ( ( 1st ` B ) .N ( 2nd ` A ) ) .N ( 2nd ` C ) ) )
8		mulasspi 9719	. . . . . . . . . . 11 |- ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( 2nd ` C ) ) = ( ( 1st ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) )
9	8	oveq1i 6660	. . . . . . . . . 10 |- ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( 2nd ` C ) ) +N ( ( ( 1st ` B ) .N ( 2nd ` A ) ) .N ( 2nd ` C ) ) ) = ( ( ( 1st ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( 1st ` B ) .N ( 2nd ` A ) ) .N ( 2nd ` C ) ) )
10	7, 9	eqtri 2644	. . . . . . . . 9 |- ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) .N ( 2nd ` C ) ) = ( ( ( 1st ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( 1st ` B ) .N ( 2nd ` A ) ) .N ( 2nd ` C ) ) )
11	10	oveq1i 6660	. . . . . . . 8 |- ( ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) = ( ( ( ( 1st ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( 1st ` B ) .N ( 2nd ` A ) ) .N ( 2nd ` C ) ) ) +N ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) )
12		ovex 6678	. . . . . . . . . . 11 |- ( ( 1st ` B ) .N ( 2nd ` C ) ) e. _V
13		ovex 6678	. . . . . . . . . . 11 |- ( ( 1st ` C ) .N ( 2nd ` B ) ) e. _V
14		fvex 6201	. . . . . . . . . . 11 |- ( 2nd ` A ) e. _V
15	12, 13, 14, 5, 6	caovdir 6868	. . . . . . . . . 10 |- ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) .N ( 2nd ` A ) ) = ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) .N ( 2nd ` A ) ) +N ( ( ( 1st ` C ) .N ( 2nd ` B ) ) .N ( 2nd ` A ) ) )
16		fvex 6201	. . . . . . . . . . . 12 |- ( 1st ` B ) e. _V
17		mulasspi 9719	. . . . . . . . . . . 12 |- ( ( x .N y ) .N z ) = ( x .N ( y .N z ) )
18	16, 4, 14, 5, 17	caov32 6861	. . . . . . . . . . 11 |- ( ( ( 1st ` B ) .N ( 2nd ` C ) ) .N ( 2nd ` A ) ) = ( ( ( 1st ` B ) .N ( 2nd ` A ) ) .N ( 2nd ` C ) )
19		mulasspi 9719	. . . . . . . . . . . 12 |- ( ( ( 1st ` C ) .N ( 2nd ` B ) ) .N ( 2nd ` A ) ) = ( ( 1st ` C ) .N ( ( 2nd ` B ) .N ( 2nd ` A ) ) )
20		mulcompi 9718	. . . . . . . . . . . . 13 |- ( ( 2nd ` B ) .N ( 2nd ` A ) ) = ( ( 2nd ` A ) .N ( 2nd ` B ) )
21	20	oveq2i 6661	. . . . . . . . . . . 12 |- ( ( 1st ` C ) .N ( ( 2nd ` B ) .N ( 2nd ` A ) ) ) = ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) )
22	19, 21	eqtri 2644	. . . . . . . . . . 11 |- ( ( ( 1st ` C ) .N ( 2nd ` B ) ) .N ( 2nd ` A ) ) = ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) )
23	18, 22	oveq12i 6662	. . . . . . . . . 10 |- ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) .N ( 2nd ` A ) ) +N ( ( ( 1st ` C ) .N ( 2nd ` B ) ) .N ( 2nd ` A ) ) ) = ( ( ( ( 1st ` B ) .N ( 2nd ` A ) ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) )
24	15, 23	eqtri 2644	. . . . . . . . 9 |- ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) .N ( 2nd ` A ) ) = ( ( ( ( 1st ` B ) .N ( 2nd ` A ) ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) )
25	24	oveq2i 6661	. . . . . . . 8 |- ( ( ( 1st ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) .N ( 2nd ` A ) ) ) = ( ( ( 1st ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( ( 1st ` B ) .N ( 2nd ` A ) ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) )
26	1, 11, 25	3eqtr4i 2654	. . . . . . 7 |- ( ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) = ( ( ( 1st ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) .N ( 2nd ` A ) ) )
27		mulasspi 9719	. . . . . . 7 |- ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( 2nd ` C ) ) = ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) )
28	26, 27	opeq12i 4407	. . . . . 6 |- <. ( ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( 2nd ` C ) ) >. = <. ( ( ( 1st ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >.
29		elpqn 9747	. . . . . . . . . 10 |- ( A e. Q. -> A e. ( N. X. N. ) )
30	29	3ad2ant1 1082	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A e. ( N. X. N. ) )
31		elpqn 9747	. . . . . . . . . 10 |- ( B e. Q. -> B e. ( N. X. N. ) )
32	31	3ad2ant2 1083	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> B e. ( N. X. N. ) )
33		addpipq2 9758	. . . . . . . . 9 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A +pQ B ) = <. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )
34	30, 32, 33	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A +pQ B ) = <. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )
35		relxp 5227	. . . . . . . . 9 |- Rel ( N. X. N. )
36		elpqn 9747	. . . . . . . . . 10 |- ( C e. Q. -> C e. ( N. X. N. ) )
37	36	3ad2ant3 1084	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> C e. ( N. X. N. ) )
38		1st2nd 7214	. . . . . . . . 9 |- ( ( Rel ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> C = <. ( 1st ` C ) , ( 2nd ` C ) >. )
39	35, 37, 38	sylancr 695	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> C = <. ( 1st ` C ) , ( 2nd ` C ) >. )
40	34, 39	oveq12d 6668	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A +pQ B ) +pQ C ) = ( <. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. +pQ <. ( 1st ` C ) , ( 2nd ` C ) >. ) )
41		xp1st 7198	. . . . . . . . . . 11 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
42	30, 41	syl 17	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 1st ` A ) e. N. )
43		xp2nd 7199	. . . . . . . . . . 11 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
44	32, 43	syl 17	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 2nd ` B ) e. N. )
45		mulclpi 9715	. . . . . . . . . 10 |- ( ( ( 1st ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. )
46	42, 44, 45	syl2anc 693	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. )
47		xp1st 7198	. . . . . . . . . . 11 |- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. )
48	32, 47	syl 17	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 1st ` B ) e. N. )
49		xp2nd 7199	. . . . . . . . . . 11 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
50	30, 49	syl 17	. . . . . . . . . 10 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 2nd ` A ) e. N. )
51		mulclpi 9715	. . . . . . . . . 10 |- ( ( ( 1st ` B ) e. N. /\ ( 2nd ` A ) e. N. ) -> ( ( 1st ` B ) .N ( 2nd ` A ) ) e. N. )
52	48, 50, 51	syl2anc 693	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` B ) .N ( 2nd ` A ) ) e. N. )
53		addclpi 9714	. . . . . . . . 9 |- ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. /\ ( ( 1st ` B ) .N ( 2nd ` A ) ) e. N. ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) e. N. )
54	46, 52, 53	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) e. N. )
55		mulclpi 9715	. . . . . . . . 9 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
56	50, 44, 55	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
57		xp1st 7198	. . . . . . . . 9 |- ( C e. ( N. X. N. ) -> ( 1st ` C ) e. N. )
58	37, 57	syl 17	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 1st ` C ) e. N. )
59		xp2nd 7199	. . . . . . . . 9 |- ( C e. ( N. X. N. ) -> ( 2nd ` C ) e. N. )
60	37, 59	syl 17	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( 2nd ` C ) e. N. )
61		addpipq 9759	. . . . . . . 8 |- ( ( ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) e. N. /\ ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. ) /\ ( ( 1st ` C ) e. N. /\ ( 2nd ` C ) e. N. ) ) -> ( <. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. +pQ <. ( 1st ` C ) , ( 2nd ` C ) >. ) = <. ( ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( 2nd ` C ) ) >. )
62	54, 56, 58, 60, 61	syl22anc 1327	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( <. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. +pQ <. ( 1st ` C ) , ( 2nd ` C ) >. ) = <. ( ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( 2nd ` C ) ) >. )
63	40, 62	eqtrd 2656	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A +pQ B ) +pQ C ) = <. ( ( ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( ( 2nd ` A ) .N ( 2nd ` B ) ) ) ) , ( ( ( 2nd ` A ) .N ( 2nd ` B ) ) .N ( 2nd ` C ) ) >. )
64		1st2nd 7214	. . . . . . . . 9 |- ( ( Rel ( N. X. N. ) /\ A e. ( N. X. N. ) ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
65	35, 30, 64	sylancr 695	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
66		addpipq2 9758	. . . . . . . . 9 |- ( ( B e. ( N. X. N. ) /\ C e. ( N. X. N. ) ) -> ( B +pQ C ) = <. ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. )
67	32, 37, 66	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( B +pQ C ) = <. ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. )
68	65, 67	oveq12d 6668	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A +pQ ( B +pQ C ) ) = ( <. ( 1st ` A ) , ( 2nd ` A ) >. +pQ <. ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. ) )
69		mulclpi 9715	. . . . . . . . . 10 |- ( ( ( 1st ` B ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 1st ` B ) .N ( 2nd ` C ) ) e. N. )
70	48, 60, 69	syl2anc 693	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` B ) .N ( 2nd ` C ) ) e. N. )
71		mulclpi 9715	. . . . . . . . . 10 |- ( ( ( 1st ` C ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 1st ` C ) .N ( 2nd ` B ) ) e. N. )
72	58, 44, 71	syl2anc 693	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 1st ` C ) .N ( 2nd ` B ) ) e. N. )
73		addclpi 9714	. . . . . . . . 9 |- ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) e. N. /\ ( ( 1st ` C ) .N ( 2nd ` B ) ) e. N. ) -> ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) e. N. )
74	70, 72, 73	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) e. N. )
75		mulclpi 9715	. . . . . . . . 9 |- ( ( ( 2nd ` B ) e. N. /\ ( 2nd ` C ) e. N. ) -> ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. )
76	44, 60, 75	syl2anc 693	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. )
77		addpipq 9759	. . . . . . . 8 |- ( ( ( ( 1st ` A ) e. N. /\ ( 2nd ` A ) e. N. ) /\ ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) e. N. /\ ( ( 2nd ` B ) .N ( 2nd ` C ) ) e. N. ) ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. +pQ <. ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. ) = <. ( ( ( 1st ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >. )
78	42, 50, 74, 76, 77	syl22anc 1327	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. +pQ <. ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) , ( ( 2nd ` B ) .N ( 2nd ` C ) ) >. ) = <. ( ( ( 1st ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >. )
79	68, 78	eqtrd 2656	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A +pQ ( B +pQ C ) ) = <. ( ( ( 1st ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) +N ( ( ( ( 1st ` B ) .N ( 2nd ` C ) ) +N ( ( 1st ` C ) .N ( 2nd ` B ) ) ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( ( 2nd ` B ) .N ( 2nd ` C ) ) ) >. )
80	28, 63, 79	3eqtr4a 2682	. . . . 5 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A +pQ B ) +pQ C ) = ( A +pQ ( B +pQ C ) ) )
81	80	fveq2d 6195	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( /Q ` ( ( A +pQ B ) +pQ C ) ) = ( /Q ` ( A +pQ ( B +pQ C ) ) ) )
82		adderpq 9778	. . . 4 |- ( ( /Q ` ( A +pQ B ) ) +Q ( /Q ` C ) ) = ( /Q ` ( ( A +pQ B ) +pQ C ) )
83		adderpq 9778	. . . 4 |- ( ( /Q ` A ) +Q ( /Q ` ( B +pQ C ) ) ) = ( /Q ` ( A +pQ ( B +pQ C ) ) )
84	81, 82, 83	3eqtr4g 2681	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( /Q ` ( A +pQ B ) ) +Q ( /Q ` C ) ) = ( ( /Q ` A ) +Q ( /Q ` ( B +pQ C ) ) ) )
85		addpqnq 9760	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) = ( /Q ` ( A +pQ B ) ) )
86	85	3adant3 1081	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A +Q B ) = ( /Q ` ( A +pQ B ) ) )
87		nqerid 9755	. . . . . 6 |- ( C e. Q. -> ( /Q ` C ) = C )
88	87	eqcomd 2628	. . . . 5 |- ( C e. Q. -> C = ( /Q ` C ) )
89	88	3ad2ant3 1084	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> C = ( /Q ` C ) )
90	86, 89	oveq12d 6668	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A +Q B ) +Q C ) = ( ( /Q ` ( A +pQ B ) ) +Q ( /Q ` C ) ) )
91		nqerid 9755	. . . . . 6 |- ( A e. Q. -> ( /Q ` A ) = A )
92	91	eqcomd 2628	. . . . 5 |- ( A e. Q. -> A = ( /Q ` A ) )
93	92	3ad2ant1 1082	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> A = ( /Q ` A ) )
94		addpqnq 9760	. . . . 5 |- ( ( B e. Q. /\ C e. Q. ) -> ( B +Q C ) = ( /Q ` ( B +pQ C ) ) )
95	94	3adant1 1079	. . . 4 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( B +Q C ) = ( /Q ` ( B +pQ C ) ) )
96	93, 95	oveq12d 6668	. . 3 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A +Q ( B +Q C ) ) = ( ( /Q ` A ) +Q ( /Q ` ( B +pQ C ) ) ) )
97	84, 90, 96	3eqtr4d 2666	. 2 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A +Q B ) +Q C ) = ( A +Q ( B +Q C ) ) )
98		addnqf 9770	. . . 4 |- +Q : ( Q. X. Q. ) --> Q.
99	98	fdmi 6052	. . 3 |- dom +Q = ( Q. X. Q. )
100		0nnq 9746	. . 3 |- -. (/) e. Q.
101	99, 100	ndmovass 6822	. 2 |- ( -. ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( ( A +Q B ) +Q C ) = ( A +Q ( B +Q C ) ) )
102	97, 101	pm2.61i 176	1 |- ( ( A +Q B ) +Q C ) = ( A +Q ( B +Q C ) )

Syntax hints:   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   N.cnpi 9666   +N cpli 9667   .N cmi 9668   +pQ cplpq 9670   Q.cnq 9674   /Qcerq 9676   +Q cplq 9677

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-1nq 9738

This theorem is referenced by:  ltaddnq  9796  addasspr  9844  prlem934  9855  ltexprlem7  9864