Theorem xmulass 12117

Description: Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 12079 which has to avoid the "undefined" combinations +oo +e -oo and -oo +e +oo. The equivalent "undefined" expression here would be 0 xe +oo, but since this is defined to equal 0 any zeroes in the expression make the whole thing evaluate to zero (on both sides), thus establishing the identity in this case. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xmulass	|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )

Proof of Theorem xmulass

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

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . 4 |- ( x = A -> ( x xe B ) = ( A xe B ) )
2	1	oveq1d 6665	. . 3 |- ( x = A -> ( ( x xe B ) xe C ) = ( ( A xe B ) xe C ) )
3		oveq1 6657	. . 3 |- ( x = A -> ( x xe ( B xe C ) ) = ( A xe ( B xe C ) ) )
4	2, 3	eqeq12d 2637	. 2 |- ( x = A -> ( ( ( x xe B ) xe C ) = ( x xe ( B xe C ) ) <-> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) ) )
5		oveq1 6657	. . . 4 |- ( x = -e A -> ( x xe B ) = ( -e A xe B ) )
6	5	oveq1d 6665	. . 3 |- ( x = -e A -> ( ( x xe B ) xe C ) = ( ( -e A xe B ) xe C ) )
7		oveq1 6657	. . 3 |- ( x = -e A -> ( x xe ( B xe C ) ) = ( -e A xe ( B xe C ) ) )
8	6, 7	eqeq12d 2637	. 2 |- ( x = -e A -> ( ( ( x xe B ) xe C ) = ( x xe ( B xe C ) ) <-> ( ( -e A xe B ) xe C ) = ( -e A xe ( B xe C ) ) ) )
9		xmulcl 12103	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A xe B ) e. RR* )
10		xmulcl 12103	. . 3 |- ( ( ( A xe B ) e. RR* /\ C e. RR* ) -> ( ( A xe B ) xe C ) e. RR* )
11	9, 10	stoic3 1701	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A xe B ) xe C ) e. RR* )
12		simp1 1061	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
13		xmulcl 12103	. . . 4 |- ( ( B e. RR* /\ C e. RR* ) -> ( B xe C ) e. RR* )
14	13	3adant1 1079	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B xe C ) e. RR* )
15		xmulcl 12103	. . 3 |- ( ( A e. RR* /\ ( B xe C ) e. RR* ) -> ( A xe ( B xe C ) ) e. RR* )
16	12, 14, 15	syl2anc 693	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A xe ( B xe C ) ) e. RR* )
17		oveq2 6658	. . . . 5 |- ( y = B -> ( x xe y ) = ( x xe B ) )
18	17	oveq1d 6665	. . . 4 |- ( y = B -> ( ( x xe y ) xe C ) = ( ( x xe B ) xe C ) )
19		oveq1 6657	. . . . 5 |- ( y = B -> ( y xe C ) = ( B xe C ) )
20	19	oveq2d 6666	. . . 4 |- ( y = B -> ( x xe ( y xe C ) ) = ( x xe ( B xe C ) ) )
21	18, 20	eqeq12d 2637	. . 3 |- ( y = B -> ( ( ( x xe y ) xe C ) = ( x xe ( y xe C ) ) <-> ( ( x xe B ) xe C ) = ( x xe ( B xe C ) ) ) )
22		oveq2 6658	. . . . 5 |- ( y = -e B -> ( x xe y ) = ( x xe -e B ) )
23	22	oveq1d 6665	. . . 4 |- ( y = -e B -> ( ( x xe y ) xe C ) = ( ( x xe -e B ) xe C ) )
24		oveq1 6657	. . . . 5 |- ( y = -e B -> ( y xe C ) = ( -e B xe C ) )
25	24	oveq2d 6666	. . . 4 |- ( y = -e B -> ( x xe ( y xe C ) ) = ( x xe ( -e B xe C ) ) )
26	23, 25	eqeq12d 2637	. . 3 |- ( y = -e B -> ( ( ( x xe y ) xe C ) = ( x xe ( y xe C ) ) <-> ( ( x xe -e B ) xe C ) = ( x xe ( -e B xe C ) ) ) )
27		simprl 794	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> x e. RR* )
28		simpl2 1065	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> B e. RR* )
29		xmulcl 12103	. . . . 5 |- ( ( x e. RR* /\ B e. RR* ) -> ( x xe B ) e. RR* )
30	27, 28, 29	syl2anc 693	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( x xe B ) e. RR* )
31		simpl3 1066	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> C e. RR* )
32		xmulcl 12103	. . . 4 |- ( ( ( x xe B ) e. RR* /\ C e. RR* ) -> ( ( x xe B ) xe C ) e. RR* )
33	30, 31, 32	syl2anc 693	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( ( x xe B ) xe C ) e. RR* )
34	14	adantr 481	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( B xe C ) e. RR* )
35		xmulcl 12103	. . . 4 |- ( ( x e. RR* /\ ( B xe C ) e. RR* ) -> ( x xe ( B xe C ) ) e. RR* )
36	27, 34, 35	syl2anc 693	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( x xe ( B xe C ) ) e. RR* )
37		oveq2 6658	. . . . 5 |- ( z = C -> ( ( x xe y ) xe z ) = ( ( x xe y ) xe C ) )
38		oveq2 6658	. . . . . 6 |- ( z = C -> ( y xe z ) = ( y xe C ) )
39	38	oveq2d 6666	. . . . 5 |- ( z = C -> ( x xe ( y xe z ) ) = ( x xe ( y xe C ) ) )
40	37, 39	eqeq12d 2637	. . . 4 |- ( z = C -> ( ( ( x xe y ) xe z ) = ( x xe ( y xe z ) ) <-> ( ( x xe y ) xe C ) = ( x xe ( y xe C ) ) ) )
41		oveq2 6658	. . . . 5 |- ( z = -e C -> ( ( x xe y ) xe z ) = ( ( x xe y ) xe -e C ) )
42		oveq2 6658	. . . . . 6 |- ( z = -e C -> ( y xe z ) = ( y xe -e C ) )
43	42	oveq2d 6666	. . . . 5 |- ( z = -e C -> ( x xe ( y xe z ) ) = ( x xe ( y xe -e C ) ) )
44	41, 43	eqeq12d 2637	. . . 4 |- ( z = -e C -> ( ( ( x xe y ) xe z ) = ( x xe ( y xe z ) ) <-> ( ( x xe y ) xe -e C ) = ( x xe ( y xe -e C ) ) ) )
45	27	adantr 481	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> x e. RR* )
46		simprl 794	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> y e. RR* )
47		xmulcl 12103	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> ( x xe y ) e. RR* )
48	45, 46, 47	syl2anc 693	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( x xe y ) e. RR* )
49	31	adantr 481	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> C e. RR* )
50		xmulcl 12103	. . . . 5 |- ( ( ( x xe y ) e. RR* /\ C e. RR* ) -> ( ( x xe y ) xe C ) e. RR* )
51	48, 49, 50	syl2anc 693	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( ( x xe y ) xe C ) e. RR* )
52		xmulcl 12103	. . . . . 6 |- ( ( y e. RR* /\ C e. RR* ) -> ( y xe C ) e. RR* )
53	46, 49, 52	syl2anc 693	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( y xe C ) e. RR* )
54		xmulcl 12103	. . . . 5 |- ( ( x e. RR* /\ ( y xe C ) e. RR* ) -> ( x xe ( y xe C ) ) e. RR* )
55	45, 53, 54	syl2anc 693	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( x xe ( y xe C ) ) e. RR* )
56		xmulasslem3 12116	. . . . . 6 |- ( ( ( x e. RR* /\ 0 < x ) /\ ( y e. RR* /\ 0 < y ) /\ ( z e. RR* /\ 0 < z ) ) -> ( ( x xe y ) xe z ) = ( x xe ( y xe z ) ) )
57	56	3expa 1265	. . . . 5 |- ( ( ( ( x e. RR* /\ 0 < x ) /\ ( y e. RR* /\ 0 < y ) ) /\ ( z e. RR* /\ 0 < z ) ) -> ( ( x xe y ) xe z ) = ( x xe ( y xe z ) ) )
58	57	adantlll 754	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) /\ ( z e. RR* /\ 0 < z ) ) -> ( ( x xe y ) xe z ) = ( x xe ( y xe z ) ) )
59		xmul01 12097	. . . . . . . 8 |- ( ( x xe y ) e. RR* -> ( ( x xe y ) xe 0 ) = 0 )
60	48, 59	syl 17	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( ( x xe y ) xe 0 ) = 0 )
61		xmul01 12097	. . . . . . . 8 |- ( x e. RR* -> ( x xe 0 ) = 0 )
62	45, 61	syl 17	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( x xe 0 ) = 0 )
63	60, 62	eqtr4d 2659	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( ( x xe y ) xe 0 ) = ( x xe 0 ) )
64		xmul01 12097	. . . . . . . 8 |- ( y e. RR* -> ( y xe 0 ) = 0 )
65	64	ad2antrl 764	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( y xe 0 ) = 0 )
66	65	oveq2d 6666	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( x xe ( y xe 0 ) ) = ( x xe 0 ) )
67	63, 66	eqtr4d 2659	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( ( x xe y ) xe 0 ) = ( x xe ( y xe 0 ) ) )
68		oveq2 6658	. . . . . 6 |- ( z = 0 -> ( ( x xe y ) xe z ) = ( ( x xe y ) xe 0 ) )
69		oveq2 6658	. . . . . . 7 |- ( z = 0 -> ( y xe z ) = ( y xe 0 ) )
70	69	oveq2d 6666	. . . . . 6 |- ( z = 0 -> ( x xe ( y xe z ) ) = ( x xe ( y xe 0 ) ) )
71	68, 70	eqeq12d 2637	. . . . 5 |- ( z = 0 -> ( ( ( x xe y ) xe z ) = ( x xe ( y xe z ) ) <-> ( ( x xe y ) xe 0 ) = ( x xe ( y xe 0 ) ) ) )
72	67, 71	syl5ibrcom 237	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( z = 0 -> ( ( x xe y ) xe z ) = ( x xe ( y xe z ) ) ) )
73		xmulneg2 12100	. . . . 5 |- ( ( ( x xe y ) e. RR* /\ C e. RR* ) -> ( ( x xe y ) xe -e C ) = -e ( ( x xe y ) xe C ) )
74	48, 49, 73	syl2anc 693	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( ( x xe y ) xe -e C ) = -e ( ( x xe y ) xe C ) )
75		xmulneg2 12100	. . . . . . 7 |- ( ( y e. RR* /\ C e. RR* ) -> ( y xe -e C ) = -e ( y xe C ) )
76	46, 49, 75	syl2anc 693	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( y xe -e C ) = -e ( y xe C ) )
77	76	oveq2d 6666	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( x xe ( y xe -e C ) ) = ( x xe -e ( y xe C ) ) )
78		xmulneg2 12100	. . . . . 6 |- ( ( x e. RR* /\ ( y xe C ) e. RR* ) -> ( x xe -e ( y xe C ) ) = -e ( x xe ( y xe C ) ) )
79	45, 53, 78	syl2anc 693	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( x xe -e ( y xe C ) ) = -e ( x xe ( y xe C ) ) )
80	77, 79	eqtrd 2656	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( x xe ( y xe -e C ) ) = -e ( x xe ( y xe C ) ) )
81	40, 44, 51, 55, 49, 58, 72, 74, 80	xmulasslem 12115	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) /\ ( y e. RR* /\ 0 < y ) ) -> ( ( x xe y ) xe C ) = ( x xe ( y xe C ) ) )
82		xmul02 12098	. . . . . . . 8 |- ( C e. RR* -> ( 0 xe C ) = 0 )
83	82	3ad2ant3 1084	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( 0 xe C ) = 0 )
84	83	adantr 481	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( 0 xe C ) = 0 )
85	61	ad2antrl 764	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( x xe 0 ) = 0 )
86	84, 85	eqtr4d 2659	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( 0 xe C ) = ( x xe 0 ) )
87	85	oveq1d 6665	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( ( x xe 0 ) xe C ) = ( 0 xe C ) )
88	84	oveq2d 6666	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( x xe ( 0 xe C ) ) = ( x xe 0 ) )
89	86, 87, 88	3eqtr4d 2666	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( ( x xe 0 ) xe C ) = ( x xe ( 0 xe C ) ) )
90		oveq2 6658	. . . . . 6 |- ( y = 0 -> ( x xe y ) = ( x xe 0 ) )
91	90	oveq1d 6665	. . . . 5 |- ( y = 0 -> ( ( x xe y ) xe C ) = ( ( x xe 0 ) xe C ) )
92		oveq1 6657	. . . . . 6 |- ( y = 0 -> ( y xe C ) = ( 0 xe C ) )
93	92	oveq2d 6666	. . . . 5 |- ( y = 0 -> ( x xe ( y xe C ) ) = ( x xe ( 0 xe C ) ) )
94	91, 93	eqeq12d 2637	. . . 4 |- ( y = 0 -> ( ( ( x xe y ) xe C ) = ( x xe ( y xe C ) ) <-> ( ( x xe 0 ) xe C ) = ( x xe ( 0 xe C ) ) ) )
95	89, 94	syl5ibrcom 237	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( y = 0 -> ( ( x xe y ) xe C ) = ( x xe ( y xe C ) ) ) )
96		xmulneg2 12100	. . . . . 6 |- ( ( x e. RR* /\ B e. RR* ) -> ( x xe -e B ) = -e ( x xe B ) )
97	27, 28, 96	syl2anc 693	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( x xe -e B ) = -e ( x xe B ) )
98	97	oveq1d 6665	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( ( x xe -e B ) xe C ) = ( -e ( x xe B ) xe C ) )
99		xmulneg1 12099	. . . . 5 |- ( ( ( x xe B ) e. RR* /\ C e. RR* ) -> ( -e ( x xe B ) xe C ) = -e ( ( x xe B ) xe C ) )
100	30, 31, 99	syl2anc 693	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( -e ( x xe B ) xe C ) = -e ( ( x xe B ) xe C ) )
101	98, 100	eqtrd 2656	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( ( x xe -e B ) xe C ) = -e ( ( x xe B ) xe C ) )
102		xmulneg1 12099	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> ( -e B xe C ) = -e ( B xe C ) )
103	28, 31, 102	syl2anc 693	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( -e B xe C ) = -e ( B xe C ) )
104	103	oveq2d 6666	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( x xe ( -e B xe C ) ) = ( x xe -e ( B xe C ) ) )
105		xmulneg2 12100	. . . . 5 |- ( ( x e. RR* /\ ( B xe C ) e. RR* ) -> ( x xe -e ( B xe C ) ) = -e ( x xe ( B xe C ) ) )
106	27, 34, 105	syl2anc 693	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( x xe -e ( B xe C ) ) = -e ( x xe ( B xe C ) ) )
107	104, 106	eqtrd 2656	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( x xe ( -e B xe C ) ) = -e ( x xe ( B xe C ) ) )
108	21, 26, 33, 36, 28, 81, 95, 101, 107	xmulasslem 12115	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( ( x xe B ) xe C ) = ( x xe ( B xe C ) ) )
109		xmul02 12098	. . . . . 6 |- ( B e. RR* -> ( 0 xe B ) = 0 )
110	109	3ad2ant2 1083	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( 0 xe B ) = 0 )
111	110	oveq1d 6665	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( 0 xe B ) xe C ) = ( 0 xe C ) )
112		xmul02 12098	. . . . 5 |- ( ( B xe C ) e. RR* -> ( 0 xe ( B xe C ) ) = 0 )
113	14, 112	syl 17	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( 0 xe ( B xe C ) ) = 0 )
114	83, 111, 113	3eqtr4d 2666	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( 0 xe B ) xe C ) = ( 0 xe ( B xe C ) ) )
115		oveq1 6657	. . . . 5 |- ( x = 0 -> ( x xe B ) = ( 0 xe B ) )
116	115	oveq1d 6665	. . . 4 |- ( x = 0 -> ( ( x xe B ) xe C ) = ( ( 0 xe B ) xe C ) )
117		oveq1 6657	. . . 4 |- ( x = 0 -> ( x xe ( B xe C ) ) = ( 0 xe ( B xe C ) ) )
118	116, 117	eqeq12d 2637	. . 3 |- ( x = 0 -> ( ( ( x xe B ) xe C ) = ( x xe ( B xe C ) ) <-> ( ( 0 xe B ) xe C ) = ( 0 xe ( B xe C ) ) ) )
119	114, 118	syl5ibrcom 237	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x = 0 -> ( ( x xe B ) xe C ) = ( x xe ( B xe C ) ) ) )
120		xmulneg1 12099	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( -e A xe B ) = -e ( A xe B ) )
121	120	3adant3 1081	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -e A xe B ) = -e ( A xe B ) )
122	121	oveq1d 6665	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( -e A xe B ) xe C ) = ( -e ( A xe B ) xe C ) )
123		xmulneg1 12099	. . . 4 |- ( ( ( A xe B ) e. RR* /\ C e. RR* ) -> ( -e ( A xe B ) xe C ) = -e ( ( A xe B ) xe C ) )
124	9, 123	stoic3 1701	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -e ( A xe B ) xe C ) = -e ( ( A xe B ) xe C ) )
125	122, 124	eqtrd 2656	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( -e A xe B ) xe C ) = -e ( ( A xe B ) xe C ) )
126		xmulneg1 12099	. . 3 |- ( ( A e. RR* /\ ( B xe C ) e. RR* ) -> ( -e A xe ( B xe C ) ) = -e ( A xe ( B xe C ) ) )
127	12, 14, 126	syl2anc 693	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -e A xe ( B xe C ) ) = -e ( A xe ( B xe C ) ) )
128	4, 8, 11, 16, 12, 108, 119, 125, 127	xmulasslem 12115	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A xe B ) xe C ) = ( A xe ( B xe C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   0cc0 9936   RR*cxr 10073   < clt 10074   -ecxne 11943   xecxmu 11945

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-xneg 11946  df-xmul 11948

This theorem is referenced by:  xlemul1  12120  xrsmcmn  19769  nmoi2  22534  xmulcand  29629  xreceu  29630  xdivrec  29635  xrge0slmod  29844