xrge0adddir - Mathbox for Thierry Arnoux

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Theorem xrge0adddir 30683

Description: Right-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)

**Assertion**
Ref	Expression
xrge0adddir	⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)))

**Proof of Theorem xrge0adddir**
Step	Hyp	Ref	Expression
1		iccssxr 12822	. . . 4 ⊢ (0[,]+∞) ⊆ ℝ^*
2		simpl1 1187	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → 𝐴 ∈ (0[,]+∞))
3	1, 2	sseldi 3968	. . 3 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → 𝐴 ∈ ℝ^*)
4		simpl2 1188	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → 𝐵 ∈ (0[,]+∞))
5	1, 4	sseldi 3968	. . 3 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → 𝐵 ∈ ℝ^*)
6		rge0ssre 12847	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ
7		simpr 487	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → 𝐶 ∈ (0[,)+∞))
8	6, 7	sseldi 3968	. . 3 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → 𝐶 ∈ ℝ)
9		xadddir 12692	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)))
10	3, 5, 8, 9	syl3anc 1367	. 2 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)))
11		simpll1 1208	. . . . . . . 8 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 𝐴 ∈ (0[,]+∞))
12	1, 11	sseldi 3968	. . . . . . 7 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 𝐴 ∈ ℝ^*)
13		simpll2 1209	. . . . . . . 8 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 𝐵 ∈ (0[,]+∞))
14	1, 13	sseldi 3968	. . . . . . 7 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 𝐵 ∈ ℝ^*)
15	12, 14	xaddcld 12697	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)
16		simpr 487	. . . . . . 7 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 0 < 𝐴)
17		xrge0addgt0 30682	. . . . . . 7 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) ∧ 0 < 𝐴) → 0 < (𝐴 +_𝑒 𝐵))
18	11, 13, 16, 17	syl21anc 835	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 0 < (𝐴 +_𝑒 𝐵))
19		xmulpnf1 12670	. . . . . 6 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ 0 < (𝐴 +_𝑒 𝐵)) → ((𝐴 +_𝑒 𝐵) ·_e +∞) = +∞)
20	15, 18, 19	syl2anc 586	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → ((𝐴 +_𝑒 𝐵) ·_e +∞) = +∞)
21		oveq2 7167	. . . . . 6 ⊢ (𝐶 = +∞ → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = ((𝐴 +_𝑒 𝐵) ·_e +∞))
22	21	ad2antlr 725	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = ((𝐴 +_𝑒 𝐵) ·_e +∞))
23		simpll3 1210	. . . . . . . 8 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 𝐶 ∈ (0[,]+∞))
24		ge0xmulcl 12854	. . . . . . . 8 ⊢ ((𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → (𝐵 ·_e 𝐶) ∈ (0[,]+∞))
25	13, 23, 24	syl2anc 586	. . . . . . 7 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐵 ·_e 𝐶) ∈ (0[,]+∞))
26	1, 25	sseldi 3968	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐵 ·_e 𝐶) ∈ ℝ^*)
27		xrge0neqmnf 12843	. . . . . . 7 ⊢ ((𝐵 ·_e 𝐶) ∈ (0[,]+∞) → (𝐵 ·_e 𝐶) ≠ -∞)
28	25, 27	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐵 ·_e 𝐶) ≠ -∞)
29		xaddpnf2 12623	. . . . . 6 ⊢ (((𝐵 ·_e 𝐶) ∈ ℝ^* ∧ (𝐵 ·_e 𝐶) ≠ -∞) → (+∞ +_𝑒 (𝐵 ·_e 𝐶)) = +∞)
30	26, 28, 29	syl2anc 586	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (+∞ +_𝑒 (𝐵 ·_e 𝐶)) = +∞)
31	20, 22, 30	3eqtr4d 2869	. . . 4 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = (+∞ +_𝑒 (𝐵 ·_e 𝐶)))
32		oveq2 7167	. . . . . . 7 ⊢ (𝐶 = +∞ → (𝐴 ·_e 𝐶) = (𝐴 ·_e +∞))
33	32	ad2antlr 725	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐴 ·_e 𝐶) = (𝐴 ·_e +∞))
34		xmulpnf1 12670	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e +∞) = +∞)
35	12, 16, 34	syl2anc 586	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐴 ·_e +∞) = +∞)
36	33, 35	eqtrd 2859	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐴 ·_e 𝐶) = +∞)
37	36	oveq1d 7174	. . . 4 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → ((𝐴 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)) = (+∞ +_𝑒 (𝐵 ·_e 𝐶)))
38	31, 37	eqtr4d 2862	. . 3 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)))
39		simpll3 1210	. . . . . . . 8 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → 𝐶 ∈ (0[,]+∞))
40	1, 39	sseldi 3968	. . . . . . 7 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → 𝐶 ∈ ℝ^*)
41		xmul02 12664	. . . . . . 7 ⊢ (𝐶 ∈ ℝ^* → (0 ·_e 𝐶) = 0)
42	40, 41	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → (0 ·_e 𝐶) = 0)
43	42	oveq1d 7174	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → ((0 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)) = (0 +_𝑒 (𝐵 ·_e 𝐶)))
44		oveq1 7166	. . . . . . 7 ⊢ (0 = 𝐴 → (0 ·_e 𝐶) = (𝐴 ·_e 𝐶))
45	44	adantl 484	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → (0 ·_e 𝐶) = (𝐴 ·_e 𝐶))
46	45	oveq1d 7174	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → ((0 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)) = ((𝐴 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)))
47		simpll2 1209	. . . . . . . 8 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → 𝐵 ∈ (0[,]+∞))
48	1, 47	sseldi 3968	. . . . . . 7 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → 𝐵 ∈ ℝ^*)
49	48, 40	xmulcld 12698	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → (𝐵 ·_e 𝐶) ∈ ℝ^*)
50		xaddid2 12638	. . . . . 6 ⊢ ((𝐵 ·_e 𝐶) ∈ ℝ^* → (0 +_𝑒 (𝐵 ·_e 𝐶)) = (𝐵 ·_e 𝐶))
51	49, 50	syl 17	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → (0 +_𝑒 (𝐵 ·_e 𝐶)) = (𝐵 ·_e 𝐶))
52	43, 46, 51	3eqtr3d 2867	. . . 4 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → ((𝐴 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)) = (𝐵 ·_e 𝐶))
53		xaddid2 12638	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (0 +_𝑒 𝐵) = 𝐵)
54	48, 53	syl 17	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → (0 +_𝑒 𝐵) = 𝐵)
55	54	oveq1d 7174	. . . 4 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → ((0 +_𝑒 𝐵) ·_e 𝐶) = (𝐵 ·_e 𝐶))
56		oveq1 7166	. . . . . 6 ⊢ (0 = 𝐴 → (0 +_𝑒 𝐵) = (𝐴 +_𝑒 𝐵))
57	56	oveq1d 7174	. . . . 5 ⊢ (0 = 𝐴 → ((0 +_𝑒 𝐵) ·_e 𝐶) = ((𝐴 +_𝑒 𝐵) ·_e 𝐶))
58	57	adantl 484	. . . 4 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → ((0 +_𝑒 𝐵) ·_e 𝐶) = ((𝐴 +_𝑒 𝐵) ·_e 𝐶))
59	52, 55, 58	3eqtr2rd 2866	. . 3 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)))
60		0xr 10691	. . . . 5 ⊢ 0 ∈ ℝ^*
61	60	a1i 11	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → 0 ∈ ℝ^*)
62		simpl1 1187	. . . . 5 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → 𝐴 ∈ (0[,]+∞))
63	1, 62	sseldi 3968	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → 𝐴 ∈ ℝ^*)
64		pnfxr 10698	. . . . . 6 ⊢ +∞ ∈ ℝ^*
65	64	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → +∞ ∈ ℝ^*)
66		iccgelb 12796	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐴 ∈ (0[,]+∞)) → 0 ≤ 𝐴)
67	61, 65, 62, 66	syl3anc 1367	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → 0 ≤ 𝐴)
68		xrleloe 12540	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
69	68	biimpa 479	. . . 4 ⊢ (((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) ∧ 0 ≤ 𝐴) → (0 < 𝐴 ∨ 0 = 𝐴))
70	61, 63, 67, 69	syl21anc 835	. . 3 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → (0 < 𝐴 ∨ 0 = 𝐴))
71	38, 59, 70	mpjaodan 955	. 2 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)))
72		0lepnf 12530	. . . . 5 ⊢ 0 ≤ +∞
73		eliccelico 30503	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → (𝐶 ∈ (0[,]+∞) ↔ (𝐶 ∈ (0[,)+∞) ∨ 𝐶 = +∞)))
74	60, 64, 72, 73	mp3an 1457	. . . 4 ⊢ (𝐶 ∈ (0[,]+∞) ↔ (𝐶 ∈ (0[,)+∞) ∨ 𝐶 = +∞))
75	74	3anbi3i 1155	. . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ↔ (𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ (𝐶 ∈ (0[,)+∞) ∨ 𝐶 = +∞)))
76	75	simp3bi 1143	. 2 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → (𝐶 ∈ (0[,)+∞) ∨ 𝐶 = +∞))
77	10, 71, 76	mpjaodan 955	1 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 class class class wbr 5069 (class class class)co 7159 ℝcr 10539 0cc0 10540 +∞cpnf 10675 -∞cmnf 10676 ℝ^*cxr 10677 < clt 10678 ≤ cle 10679 +_𝑒 cxad 12508 ·_e cxmu 12509 [,)cico 12743 [,]cicc 12744

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ico 12747 df-icc 12748

This theorem is referenced by: xrge0adddi 30684 xrge0slmod 30921 esummulc1 31344

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