Theorem xmulass 11422

Description: Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 11384 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 6225	. . . 4 |- ( x = A -> ( x xe B ) = ( A xe B ) )
2	1	oveq1d 6233	. . 3 |- ( x = A -> ( ( x xe B ) xe C ) = ( ( A xe B ) xe C ) )
3		oveq1 6225	. . 3 |- ( x = A -> ( x xe ( B xe C ) ) = ( A xe ( B xe C ) ) )
4	2, 3	eqeq12d 2418	. 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 6225	. . . 4 |- ( x = -e A -> ( x xe B ) = ( -e A xe B ) )
6	5	oveq1d 6233	. . 3 |- ( x = -e A -> ( ( x xe B ) xe C ) = ( ( -e A xe B ) xe C ) )
7		oveq1 6225	. . 3 |- ( x = -e A -> ( x xe ( B xe C ) ) = ( -e A xe ( B xe C ) ) )
8	6, 7	eqeq12d 2418	. 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 11408	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A xe B ) e. RR* )
10		xmulcl 11408	. . 3 |- ( ( ( A xe B ) e. RR* /\ C e. RR* ) -> ( ( A xe B ) xe C ) e. RR* )
11	9, 10	stoic3 1624	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A xe B ) xe C ) e. RR* )
12		simp1 994	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
13		xmulcl 11408	. . . 4 |- ( ( B e. RR* /\ C e. RR* ) -> ( B xe C ) e. RR* )
14	13	3adant1 1012	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B xe C ) e. RR* )
15		xmulcl 11408	. . 3 |- ( ( A e. RR* /\ ( B xe C ) e. RR* ) -> ( A xe ( B xe C ) ) e. RR* )
16	12, 14, 15	syl2anc 659	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A xe ( B xe C ) ) e. RR* )
17		oveq2 6226	. . . . 5 |- ( y = B -> ( x xe y ) = ( x xe B ) )
18	17	oveq1d 6233	. . . 4 |- ( y = B -> ( ( x xe y ) xe C ) = ( ( x xe B ) xe C ) )
19		oveq1 6225	. . . . 5 |- ( y = B -> ( y xe C ) = ( B xe C ) )
20	19	oveq2d 6234	. . . 4 |- ( y = B -> ( x xe ( y xe C ) ) = ( x xe ( B xe C ) ) )
21	18, 20	eqeq12d 2418	. . 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 6226	. . . . 5 |- ( y = -e B -> ( x xe y ) = ( x xe -e B ) )
23	22	oveq1d 6233	. . . 4 |- ( y = -e B -> ( ( x xe y ) xe C ) = ( ( x xe -e B ) xe C ) )
24		oveq1 6225	. . . . 5 |- ( y = -e B -> ( y xe C ) = ( -e B xe C ) )
25	24	oveq2d 6234	. . . 4 |- ( y = -e B -> ( x xe ( y xe C ) ) = ( x xe ( -e B xe C ) ) )
26	23, 25	eqeq12d 2418	. . 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 754	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> x e. RR* )
28		simpl2 998	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> B e. RR* )
29		xmulcl 11408	. . . . 5 |- ( ( x e. RR* /\ B e. RR* ) -> ( x xe B ) e. RR* )
30	27, 28, 29	syl2anc 659	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( x xe B ) e. RR* )
31		simpl3 999	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> C e. RR* )
32		xmulcl 11408	. . . 4 |- ( ( ( x xe B ) e. RR* /\ C e. RR* ) -> ( ( x xe B ) xe C ) e. RR* )
33	30, 31, 32	syl2anc 659	. . 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 463	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( B xe C ) e. RR* )
35		xmulcl 11408	. . . 4 |- ( ( x e. RR* /\ ( B xe C ) e. RR* ) -> ( x xe ( B xe C ) ) e. RR* )
36	27, 34, 35	syl2anc 659	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( x xe ( B xe C ) ) e. RR* )
37		oveq2 6226	. . . . 5 |- ( z = C -> ( ( x xe y ) xe z ) = ( ( x xe y ) xe C ) )
38		oveq2 6226	. . . . . 6 |- ( z = C -> ( y xe z ) = ( y xe C ) )
39	38	oveq2d 6234	. . . . 5 |- ( z = C -> ( x xe ( y xe z ) ) = ( x xe ( y xe C ) ) )
40	37, 39	eqeq12d 2418	. . . 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 6226	. . . . 5 |- ( z = -e C -> ( ( x xe y ) xe z ) = ( ( x xe y ) xe -e C ) )
42		oveq2 6226	. . . . . 6 |- ( z = -e C -> ( y xe z ) = ( y xe -e C ) )
43	42	oveq2d 6234	. . . . 5 |- ( z = -e C -> ( x xe ( y xe z ) ) = ( x xe ( y xe -e C ) ) )
44	41, 43	eqeq12d 2418	. . . 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 463	. . . . . 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 754	. . . . . 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 11408	. . . . . 6 |- ( ( x e. RR* /\ y e. RR* ) -> ( x xe y ) e. RR* )
48	45, 46, 47	syl2anc 659	. . . . 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 463	. . . . 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 11408	. . . . 5 |- ( ( ( x xe y ) e. RR* /\ C e. RR* ) -> ( ( x xe y ) xe C ) e. RR* )
51	48, 49, 50	syl2anc 659	. . . 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 11408	. . . . . 6 |- ( ( y e. RR* /\ C e. RR* ) -> ( y xe C ) e. RR* )
53	46, 49, 52	syl2anc 659	. . . . 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 11408	. . . . 5 |- ( ( x e. RR* /\ ( y xe C ) e. RR* ) -> ( x xe ( y xe C ) ) e. RR* )
55	45, 53, 54	syl2anc 659	. . . 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 11421	. . . . . 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 1194	. . . . 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 715	. . . 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 11402	. . . . . . . 8 |- ( ( x xe y ) e. RR* -> ( ( x xe y ) xe 0 ) = 0 )
60	48, 59	syl 16	. . . . . . 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 11402	. . . . . . . 8 |- ( x e. RR* -> ( x xe 0 ) = 0 )
62	45, 61	syl 16	. . . . . . 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 2440	. . . . . 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 11402	. . . . . . . 8 |- ( y e. RR* -> ( y xe 0 ) = 0 )
65	64	ad2antrl 725	. . . . . . 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 6234	. . . . . 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 2440	. . . . 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 6226	. . . . . 6 |- ( z = 0 -> ( ( x xe y ) xe z ) = ( ( x xe y ) xe 0 ) )
69		oveq2 6226	. . . . . . 7 |- ( z = 0 -> ( y xe z ) = ( y xe 0 ) )
70	69	oveq2d 6234	. . . . . 6 |- ( z = 0 -> ( x xe ( y xe z ) ) = ( x xe ( y xe 0 ) ) )
71	68, 70	eqeq12d 2418	. . . . 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 222	. . . 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 11405	. . . . 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 659	. . . 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 11405	. . . . . . 7 |- ( ( y e. RR* /\ C e. RR* ) -> ( y xe -e C ) = -e ( y xe C ) )
76	46, 49, 75	syl2anc 659	. . . . . 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 6234	. . . . 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 11405	. . . . . 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 659	. . . . 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 2437	. . . 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 11420	. . 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 11403	. . . . . . . 8 |- ( C e. RR* -> ( 0 xe C ) = 0 )
83	82	3ad2ant3 1017	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( 0 xe C ) = 0 )
84	83	adantr 463	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( 0 xe C ) = 0 )
85	61	ad2antrl 725	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( x xe 0 ) = 0 )
86	84, 85	eqtr4d 2440	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( x e. RR* /\ 0 < x ) ) -> ( 0 xe C ) = ( x xe 0 ) )
87	85	oveq1d 6233	. . . . 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 6234	. . . . 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 2447	. . . 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 6226	. . . . . 6 |- ( y = 0 -> ( x xe y ) = ( x xe 0 ) )
91	90	oveq1d 6233	. . . . 5 |- ( y = 0 -> ( ( x xe y ) xe C ) = ( ( x xe 0 ) xe C ) )
92		oveq1 6225	. . . . . 6 |- ( y = 0 -> ( y xe C ) = ( 0 xe C ) )
93	92	oveq2d 6234	. . . . 5 |- ( y = 0 -> ( x xe ( y xe C ) ) = ( x xe ( 0 xe C ) ) )
94	91, 93	eqeq12d 2418	. . . 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 222	. . 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 11405	. . . . . 6 |- ( ( x e. RR* /\ B e. RR* ) -> ( x xe -e B ) = -e ( x xe B ) )
97	27, 28, 96	syl2anc 659	. . . . 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 6233	. . . 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 11404	. . . . 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 659	. . . 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 2437	. . 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 11404	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* ) -> ( -e B xe C ) = -e ( B xe C ) )
103	28, 31, 102	syl2anc 659	. . . . 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 6234	. . . 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 11405	. . . . 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 659	. . . 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 2437	. . 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 11420	. 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 11403	. . . . . 6 |- ( B e. RR* -> ( 0 xe B ) = 0 )
110	109	3ad2ant2 1016	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( 0 xe B ) = 0 )
111	110	oveq1d 6233	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( 0 xe B ) xe C ) = ( 0 xe C ) )
112		xmul02 11403	. . . . 5 |- ( ( B xe C ) e. RR* -> ( 0 xe ( B xe C ) ) = 0 )
113	14, 112	syl 16	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( 0 xe ( B xe C ) ) = 0 )
114	83, 111, 113	3eqtr4d 2447	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( 0 xe B ) xe C ) = ( 0 xe ( B xe C ) ) )
115		oveq1 6225	. . . . 5 |- ( x = 0 -> ( x xe B ) = ( 0 xe B ) )
116	115	oveq1d 6233	. . . 4 |- ( x = 0 -> ( ( x xe B ) xe C ) = ( ( 0 xe B ) xe C ) )
117		oveq1 6225	. . . 4 |- ( x = 0 -> ( x xe ( B xe C ) ) = ( 0 xe ( B xe C ) ) )
118	116, 117	eqeq12d 2418	. . 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 222	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( x = 0 -> ( ( x xe B ) xe C ) = ( x xe ( B xe C ) ) ) )
120		xmulneg1 11404	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( -e A xe B ) = -e ( A xe B ) )
121	120	3adant3 1014	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( -e A xe B ) = -e ( A xe B ) )
122	121	oveq1d 6233	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( -e A xe B ) xe C ) = ( -e ( A xe B ) xe C ) )
123		xmulneg1 11404	. . . 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 1624	. . 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 2437	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( -e A xe B ) xe C ) = -e ( ( A xe B ) xe C ) )
126		xmulneg1 11404	. . 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 659	. 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 11420	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 367   /\ w3a 971   = wceq 1399   e. wcel 1836   class class class wbr 4384  (class class class)co 6218   0cc0 9425   RR*cxr 9560   < clt 9561   -ecxne 11258   xecxmu 11260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-po 4731  df-so 4732  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-1st 6721  df-2nd 6722  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-xneg 11261  df-xmul 11263

This theorem is referenced by:  xlemul1  11425  xrsmcmn  18577  nmoi2  21345  xmulcand  27804  xreceu  27805  xdivrec  27810  xrge0slmod  28023