Theorem xmulass 13206

Description: Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 13168 which has to avoid the "undefined" combinations +∞ +_𝑒 -∞ and -∞ +_𝑒 +∞. The equivalent "undefined" expression here would be 0 ·_e +∞, 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 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ·e 𝐵) ·e 𝐶) = (𝐴 ·e (𝐵 ·e 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 7364	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ·e 𝐵) = (𝐴 ·e 𝐵))
2	1	oveq1d 7372	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ·e 𝐵) ·e 𝐶) = ((𝐴 ·e 𝐵) ·e 𝐶))
3		oveq1 7364	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·e (𝐵 ·e 𝐶)) = (𝐴 ·e (𝐵 ·e 𝐶)))
4	2, 3	eqeq12d 2752	. 2 ⊢ (𝑥 = 𝐴 → (((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶)) ↔ ((𝐴 ·e 𝐵) ·e 𝐶) = (𝐴 ·e (𝐵 ·e 𝐶))))
5		oveq1 7364	. . . 4 ⊢ (𝑥 = -𝑒𝐴 → (𝑥 ·e 𝐵) = (-𝑒𝐴 ·e 𝐵))
6	5	oveq1d 7372	. . 3 ⊢ (𝑥 = -𝑒𝐴 → ((𝑥 ·e 𝐵) ·e 𝐶) = ((-𝑒𝐴 ·e 𝐵) ·e 𝐶))
7		oveq1 7364	. . 3 ⊢ (𝑥 = -𝑒𝐴 → (𝑥 ·e (𝐵 ·e 𝐶)) = (-𝑒𝐴 ·e (𝐵 ·e 𝐶)))
8	6, 7	eqeq12d 2752	. 2 ⊢ (𝑥 = -𝑒𝐴 → (((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶)) ↔ ((-𝑒𝐴 ·e 𝐵) ·e 𝐶) = (-𝑒𝐴 ·e (𝐵 ·e 𝐶))))
9		xmulcl 13192	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) ∈ ℝ*)
10		xmulcl 13192	. . 3 ⊢ (((𝐴 ·e 𝐵) ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ·e 𝐵) ·e 𝐶) ∈ ℝ*)
11	9, 10	stoic3 1778	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ·e 𝐵) ·e 𝐶) ∈ ℝ*)
12		simp1 1136	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → 𝐴 ∈ ℝ*)
13		xmulcl 13192	. . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 ·e 𝐶) ∈ ℝ*)
14	13	3adant1 1130	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 ·e 𝐶) ∈ ℝ*)
15		xmulcl 13192	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐵 ·e 𝐶) ∈ ℝ*) → (𝐴 ·e (𝐵 ·e 𝐶)) ∈ ℝ*)
16	12, 14, 15	syl2anc 584	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ·e (𝐵 ·e 𝐶)) ∈ ℝ*)
17		oveq2 7365	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑥 ·e 𝑦) = (𝑥 ·e 𝐵))
18	17	oveq1d 7372	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝑥 ·e 𝑦) ·e 𝐶) = ((𝑥 ·e 𝐵) ·e 𝐶))
19		oveq1 7364	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ·e 𝐶) = (𝐵 ·e 𝐶))
20	19	oveq2d 7373	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑥 ·e (𝑦 ·e 𝐶)) = (𝑥 ·e (𝐵 ·e 𝐶)))
21	18, 20	eqeq12d 2752	. . 3 ⊢ (𝑦 = 𝐵 → (((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶)) ↔ ((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶))))
22		oveq2 7365	. . . . 5 ⊢ (𝑦 = -𝑒𝐵 → (𝑥 ·e 𝑦) = (𝑥 ·e -𝑒𝐵))
23	22	oveq1d 7372	. . . 4 ⊢ (𝑦 = -𝑒𝐵 → ((𝑥 ·e 𝑦) ·e 𝐶) = ((𝑥 ·e -𝑒𝐵) ·e 𝐶))
24		oveq1 7364	. . . . 5 ⊢ (𝑦 = -𝑒𝐵 → (𝑦 ·e 𝐶) = (-𝑒𝐵 ·e 𝐶))
25	24	oveq2d 7373	. . . 4 ⊢ (𝑦 = -𝑒𝐵 → (𝑥 ·e (𝑦 ·e 𝐶)) = (𝑥 ·e (-𝑒𝐵 ·e 𝐶)))
26	23, 25	eqeq12d 2752	. . 3 ⊢ (𝑦 = -𝑒𝐵 → (((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶)) ↔ ((𝑥 ·e -𝑒𝐵) ·e 𝐶) = (𝑥 ·e (-𝑒𝐵 ·e 𝐶))))
27		simprl 769	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝑥 ∈ ℝ*)
28		simpl2 1192	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝐵 ∈ ℝ*)
29		xmulcl 13192	. . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑥 ·e 𝐵) ∈ ℝ*)
30	27, 28, 29	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e 𝐵) ∈ ℝ*)
31		simpl3 1193	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝐶 ∈ ℝ*)
32		xmulcl 13192	. . . 4 ⊢ (((𝑥 ·e 𝐵) ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝑥 ·e 𝐵) ·e 𝐶) ∈ ℝ*)
33	30, 31, 32	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e 𝐵) ·e 𝐶) ∈ ℝ*)
34	14	adantr 481	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝐵 ·e 𝐶) ∈ ℝ*)
35		xmulcl 13192	. . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ (𝐵 ·e 𝐶) ∈ ℝ*) → (𝑥 ·e (𝐵 ·e 𝐶)) ∈ ℝ*)
36	27, 34, 35	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e (𝐵 ·e 𝐶)) ∈ ℝ*)
37		oveq2 7365	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝑥 ·e 𝑦) ·e 𝑧) = ((𝑥 ·e 𝑦) ·e 𝐶))
38		oveq2 7365	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝑦 ·e 𝑧) = (𝑦 ·e 𝐶))
39	38	oveq2d 7373	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑥 ·e (𝑦 ·e 𝑧)) = (𝑥 ·e (𝑦 ·e 𝐶)))
40	37, 39	eqeq12d 2752	. . . 4 ⊢ (𝑧 = 𝐶 → (((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)) ↔ ((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶))))
41		oveq2 7365	. . . . 5 ⊢ (𝑧 = -𝑒𝐶 → ((𝑥 ·e 𝑦) ·e 𝑧) = ((𝑥 ·e 𝑦) ·e -𝑒𝐶))
42		oveq2 7365	. . . . . 6 ⊢ (𝑧 = -𝑒𝐶 → (𝑦 ·e 𝑧) = (𝑦 ·e -𝑒𝐶))
43	42	oveq2d 7373	. . . . 5 ⊢ (𝑧 = -𝑒𝐶 → (𝑥 ·e (𝑦 ·e 𝑧)) = (𝑥 ·e (𝑦 ·e -𝑒𝐶)))
44	41, 43	eqeq12d 2752	. . . 4 ⊢ (𝑧 = -𝑒𝐶 → (((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)) ↔ ((𝑥 ·e 𝑦) ·e -𝑒𝐶) = (𝑥 ·e (𝑦 ·e -𝑒𝐶))))
45	27	adantr 481	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → 𝑥 ∈ ℝ*)
46		simprl 769	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → 𝑦 ∈ ℝ*)
47		xmulcl 13192	. . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) ∈ ℝ*)
48	45, 46, 47	syl2anc 584	. . . . 5 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e 𝑦) ∈ ℝ*)
49	31	adantr 481	. . . . 5 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → 𝐶 ∈ ℝ*)
50		xmulcl 13192	. . . . 5 ⊢ (((𝑥 ·e 𝑦) ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝑥 ·e 𝑦) ·e 𝐶) ∈ ℝ*)
51	48, 49, 50	syl2anc 584	. . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e 𝐶) ∈ ℝ*)
52		xmulcl 13192	. . . . . 6 ⊢ ((𝑦 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝑦 ·e 𝐶) ∈ ℝ*)
53	46, 49, 52	syl2anc 584	. . . . 5 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑦 ·e 𝐶) ∈ ℝ*)
54		xmulcl 13192	. . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ·e 𝐶) ∈ ℝ*) → (𝑥 ·e (𝑦 ·e 𝐶)) ∈ ℝ*)
55	45, 53, 54	syl2anc 584	. . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e (𝑦 ·e 𝐶)) ∈ ℝ*)
56		xmulasslem3 13205	. . . . 5 ⊢ (((𝑥 ∈ ℝ* ∧ 0 < 𝑥) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦) ∧ (𝑧 ∈ ℝ* ∧ 0 < 𝑧)) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)))
57	56	ad4ant234 1175	. . . 4 ⊢ (((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) ∧ (𝑧 ∈ ℝ* ∧ 0 < 𝑧)) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)))
58		xmul01 13186	. . . . . . . 8 ⊢ ((𝑥 ·e 𝑦) ∈ ℝ* → ((𝑥 ·e 𝑦) ·e 0) = 0)
59	48, 58	syl 17	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e 0) = 0)
60		xmul01 13186	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ* → (𝑥 ·e 0) = 0)
61	45, 60	syl 17	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e 0) = 0)
62	59, 61	eqtr4d 2779	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e 0) = (𝑥 ·e 0))
63		xmul01 13186	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ* → (𝑦 ·e 0) = 0)
64	63	ad2antrl 726	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑦 ·e 0) = 0)
65	64	oveq2d 7373	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e (𝑦 ·e 0)) = (𝑥 ·e 0))
66	62, 65	eqtr4d 2779	. . . . 5 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e 0) = (𝑥 ·e (𝑦 ·e 0)))
67		oveq2 7365	. . . . . 6 ⊢ (𝑧 = 0 → ((𝑥 ·e 𝑦) ·e 𝑧) = ((𝑥 ·e 𝑦) ·e 0))
68		oveq2 7365	. . . . . . 7 ⊢ (𝑧 = 0 → (𝑦 ·e 𝑧) = (𝑦 ·e 0))
69	68	oveq2d 7373	. . . . . 6 ⊢ (𝑧 = 0 → (𝑥 ·e (𝑦 ·e 𝑧)) = (𝑥 ·e (𝑦 ·e 0)))
70	67, 69	eqeq12d 2752	. . . . 5 ⊢ (𝑧 = 0 → (((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)) ↔ ((𝑥 ·e 𝑦) ·e 0) = (𝑥 ·e (𝑦 ·e 0))))
71	66, 70	syl5ibrcom 246	. . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑧 = 0 → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧))))
72		xmulneg2 13189	. . . . 5 ⊢ (((𝑥 ·e 𝑦) ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝑥 ·e 𝑦) ·e -𝑒𝐶) = -𝑒((𝑥 ·e 𝑦) ·e 𝐶))
73	48, 49, 72	syl2anc 584	. . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e -𝑒𝐶) = -𝑒((𝑥 ·e 𝑦) ·e 𝐶))
74		xmulneg2 13189	. . . . . . 7 ⊢ ((𝑦 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝑦 ·e -𝑒𝐶) = -𝑒(𝑦 ·e 𝐶))
75	46, 49, 74	syl2anc 584	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑦 ·e -𝑒𝐶) = -𝑒(𝑦 ·e 𝐶))
76	75	oveq2d 7373	. . . . 5 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e (𝑦 ·e -𝑒𝐶)) = (𝑥 ·e -𝑒(𝑦 ·e 𝐶)))
77		xmulneg2 13189	. . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ (𝑦 ·e 𝐶) ∈ ℝ*) → (𝑥 ·e -𝑒(𝑦 ·e 𝐶)) = -𝑒(𝑥 ·e (𝑦 ·e 𝐶)))
78	45, 53, 77	syl2anc 584	. . . . 5 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e -𝑒(𝑦 ·e 𝐶)) = -𝑒(𝑥 ·e (𝑦 ·e 𝐶)))
79	76, 78	eqtrd 2776	. . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e (𝑦 ·e -𝑒𝐶)) = -𝑒(𝑥 ·e (𝑦 ·e 𝐶)))
80	40, 44, 51, 55, 49, 57, 71, 73, 79	xmulasslem 13204	. . 3 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶)))
81		xmul02 13187	. . . . . . . 8 ⊢ (𝐶 ∈ ℝ* → (0 ·e 𝐶) = 0)
82	81	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (0 ·e 𝐶) = 0)
83	82	adantr 481	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (0 ·e 𝐶) = 0)
84	60	ad2antrl 726	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e 0) = 0)
85	83, 84	eqtr4d 2779	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (0 ·e 𝐶) = (𝑥 ·e 0))
86	84	oveq1d 7372	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e 0) ·e 𝐶) = (0 ·e 𝐶))
87	83	oveq2d 7373	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e (0 ·e 𝐶)) = (𝑥 ·e 0))
88	85, 86, 87	3eqtr4d 2786	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e 0) ·e 𝐶) = (𝑥 ·e (0 ·e 𝐶)))
89		oveq2 7365	. . . . . 6 ⊢ (𝑦 = 0 → (𝑥 ·e 𝑦) = (𝑥 ·e 0))
90	89	oveq1d 7372	. . . . 5 ⊢ (𝑦 = 0 → ((𝑥 ·e 𝑦) ·e 𝐶) = ((𝑥 ·e 0) ·e 𝐶))
91		oveq1 7364	. . . . . 6 ⊢ (𝑦 = 0 → (𝑦 ·e 𝐶) = (0 ·e 𝐶))
92	91	oveq2d 7373	. . . . 5 ⊢ (𝑦 = 0 → (𝑥 ·e (𝑦 ·e 𝐶)) = (𝑥 ·e (0 ·e 𝐶)))
93	90, 92	eqeq12d 2752	. . . 4 ⊢ (𝑦 = 0 → (((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶)) ↔ ((𝑥 ·e 0) ·e 𝐶) = (𝑥 ·e (0 ·e 𝐶))))
94	88, 93	syl5ibrcom 246	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑦 = 0 → ((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶))))
95		xmulneg2 13189	. . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑥 ·e -𝑒𝐵) = -𝑒(𝑥 ·e 𝐵))
96	27, 28, 95	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e -𝑒𝐵) = -𝑒(𝑥 ·e 𝐵))
97	96	oveq1d 7372	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e -𝑒𝐵) ·e 𝐶) = (-𝑒(𝑥 ·e 𝐵) ·e 𝐶))
98		xmulneg1 13188	. . . . 5 ⊢ (((𝑥 ·e 𝐵) ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (-𝑒(𝑥 ·e 𝐵) ·e 𝐶) = -𝑒((𝑥 ·e 𝐵) ·e 𝐶))
99	30, 31, 98	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (-𝑒(𝑥 ·e 𝐵) ·e 𝐶) = -𝑒((𝑥 ·e 𝐵) ·e 𝐶))
100	97, 99	eqtrd 2776	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e -𝑒𝐵) ·e 𝐶) = -𝑒((𝑥 ·e 𝐵) ·e 𝐶))
101		xmulneg1 13188	. . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (-𝑒𝐵 ·e 𝐶) = -𝑒(𝐵 ·e 𝐶))
102	28, 31, 101	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (-𝑒𝐵 ·e 𝐶) = -𝑒(𝐵 ·e 𝐶))
103	102	oveq2d 7373	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e (-𝑒𝐵 ·e 𝐶)) = (𝑥 ·e -𝑒(𝐵 ·e 𝐶)))
104		xmulneg2 13189	. . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ (𝐵 ·e 𝐶) ∈ ℝ*) → (𝑥 ·e -𝑒(𝐵 ·e 𝐶)) = -𝑒(𝑥 ·e (𝐵 ·e 𝐶)))
105	27, 34, 104	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e -𝑒(𝐵 ·e 𝐶)) = -𝑒(𝑥 ·e (𝐵 ·e 𝐶)))
106	103, 105	eqtrd 2776	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e (-𝑒𝐵 ·e 𝐶)) = -𝑒(𝑥 ·e (𝐵 ·e 𝐶)))
107	21, 26, 33, 36, 28, 80, 94, 100, 106	xmulasslem 13204	. 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶)))
108		xmul02 13187	. . . . . 6 ⊢ (𝐵 ∈ ℝ* → (0 ·e 𝐵) = 0)
109	108	3ad2ant2 1134	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (0 ·e 𝐵) = 0)
110	109	oveq1d 7372	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((0 ·e 𝐵) ·e 𝐶) = (0 ·e 𝐶))
111		xmul02 13187	. . . . 5 ⊢ ((𝐵 ·e 𝐶) ∈ ℝ* → (0 ·e (𝐵 ·e 𝐶)) = 0)
112	14, 111	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (0 ·e (𝐵 ·e 𝐶)) = 0)
113	82, 110, 112	3eqtr4d 2786	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((0 ·e 𝐵) ·e 𝐶) = (0 ·e (𝐵 ·e 𝐶)))
114		oveq1 7364	. . . . 5 ⊢ (𝑥 = 0 → (𝑥 ·e 𝐵) = (0 ·e 𝐵))
115	114	oveq1d 7372	. . . 4 ⊢ (𝑥 = 0 → ((𝑥 ·e 𝐵) ·e 𝐶) = ((0 ·e 𝐵) ·e 𝐶))
116		oveq1 7364	. . . 4 ⊢ (𝑥 = 0 → (𝑥 ·e (𝐵 ·e 𝐶)) = (0 ·e (𝐵 ·e 𝐶)))
117	115, 116	eqeq12d 2752	. . 3 ⊢ (𝑥 = 0 → (((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶)) ↔ ((0 ·e 𝐵) ·e 𝐶) = (0 ·e (𝐵 ·e 𝐶))))
118	113, 117	syl5ibrcom 246	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝑥 = 0 → ((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶))))
119		xmulneg1 13188	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = -𝑒(𝐴 ·e 𝐵))
120	119	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = -𝑒(𝐴 ·e 𝐵))
121	120	oveq1d 7372	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((-𝑒𝐴 ·e 𝐵) ·e 𝐶) = (-𝑒(𝐴 ·e 𝐵) ·e 𝐶))
122		xmulneg1 13188	. . . 4 ⊢ (((𝐴 ·e 𝐵) ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (-𝑒(𝐴 ·e 𝐵) ·e 𝐶) = -𝑒((𝐴 ·e 𝐵) ·e 𝐶))
123	9, 122	stoic3 1778	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (-𝑒(𝐴 ·e 𝐵) ·e 𝐶) = -𝑒((𝐴 ·e 𝐵) ·e 𝐶))
124	121, 123	eqtrd 2776	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((-𝑒𝐴 ·e 𝐵) ·e 𝐶) = -𝑒((𝐴 ·e 𝐵) ·e 𝐶))
125		xmulneg1 13188	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐵 ·e 𝐶) ∈ ℝ*) → (-𝑒𝐴 ·e (𝐵 ·e 𝐶)) = -𝑒(𝐴 ·e (𝐵 ·e 𝐶)))
126	12, 14, 125	syl2anc 584	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (-𝑒𝐴 ·e (𝐵 ·e 𝐶)) = -𝑒(𝐴 ·e (𝐵 ·e 𝐶)))
127	4, 8, 11, 16, 12, 107, 118, 124, 126	xmulasslem 13204	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ·e 𝐵) ·e 𝐶) = (𝐴 ·e (𝐵 ·e 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5105  (class class class)co 7357  0cc0 11051  ℝ^*cxr 11188   < clt 11189  -_𝑒cxne 13030   ·_e cxmu 13032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-xneg 13033  df-xmul 13035

This theorem is referenced by:  xlemul1  13209  xrsmcmn  20820  nmoi2  24094  xmulcand  31777  xreceu  31778  xdivrec  31783  xrge0slmod  32140