xrge0adddir - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0adddir		Structured version Visualization version GIF version

Theorem xrge0adddir 33629

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 13568	. . . 4 ⊢ (0[,]+∞) ⊆ ℝ^*
2		simpl1 1210	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → 𝐴 ∈ (0[,]+∞))
3	1, 2	sselid 3907	. . 3 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → 𝐴 ∈ ℝ^*)
4		simpl2 1211	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → 𝐵 ∈ (0[,]+∞))
5	1, 4	sselid 3907	. . 3 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → 𝐵 ∈ ℝ^*)
6		rge0ssre 13594	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ
7		simpr 490	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → 𝐶 ∈ (0[,)+∞))
8	6, 7	sselid 3907	. . 3 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → 𝐶 ∈ ℝ)
9		xadddir 13433	. . 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 1398	. 2 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 ∈ (0[,)+∞)) → ((𝐴 +_e 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐶) +_e (𝐵 ·_e 𝐶)))
11		simpll1 1231	. . . . . . . 8 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 𝐴 ∈ (0[,]+∞))
12	1, 11	sselid 3907	. . . . . . 7 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 𝐴 ∈ ℝ^*)
13		simpll2 1232	. . . . . . . 8 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 𝐵 ∈ (0[,]+∞))
14	1, 13	sselid 3907	. . . . . . 7 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 𝐵 ∈ ℝ^*)
15	12, 14	xaddcld 13438	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐴 +_e 𝐵) ∈ ℝ^*)
16		simpr 490	. . . . . . 7 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 0 < 𝐴)
17		xrge0addgt0 33628	. . . . . . 7 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) ∧ 0 < 𝐴) → 0 < (𝐴 +_e 𝐵))
18	11, 13, 16, 17	syl21anc 851	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 0 < (𝐴 +_e 𝐵))
19		xmulpnf1 13411	. . . . . 6 ⊢ (((𝐴 +_e 𝐵) ∈ ℝ^* ∧ 0 < (𝐴 +_e 𝐵)) → ((𝐴 +_e 𝐵) ·_e +∞) = +∞)
20	15, 18, 19	syl2anc 596	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → ((𝐴 +_e 𝐵) ·_e +∞) = +∞)
21		oveq2 7432	. . . . . 6 ⊢ (𝐶 = +∞ → ((𝐴 +_e 𝐵) ·_e 𝐶) = ((𝐴 +_e 𝐵) ·_e +∞))
22	21	ad2antlr 740	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → ((𝐴 +_e 𝐵) ·_e 𝐶) = ((𝐴 +_e 𝐵) ·_e +∞))
23		simpll3 1233	. . . . . . . 8 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → 𝐶 ∈ (0[,]+∞))
24		ge0xmulcl 13601	. . . . . . . 8 ⊢ ((𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → (𝐵 ·_e 𝐶) ∈ (0[,]+∞))
25	13, 23, 24	syl2anc 596	. . . . . . 7 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐵 ·_e 𝐶) ∈ (0[,]+∞))
26	1, 25	sselid 3907	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐵 ·_e 𝐶) ∈ ℝ^*)
27		xrge0neqmnf 13590	. . . . . . 7 ⊢ ((𝐵 ·_e 𝐶) ∈ (0[,]+∞) → (𝐵 ·_e 𝐶) ≠ -∞)
28	25, 27	syl 18	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐵 ·_e 𝐶) ≠ -∞)
29		xaddpnf2 13364	. . . . . 6 ⊢ (((𝐵 ·_e 𝐶) ∈ ℝ^* ∧ (𝐵 ·_e 𝐶) ≠ -∞) → (+∞ +_e (𝐵 ·_e 𝐶)) = +∞)
30	26, 28, 29	syl2anc 596	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (+∞ +_e (𝐵 ·_e 𝐶)) = +∞)
31	20, 22, 30	3eqtr4d 2805	. . . 4 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → ((𝐴 +_e 𝐵) ·_e 𝐶) = (+∞ +_e (𝐵 ·_e 𝐶)))
32		oveq2 7432	. . . . . . 7 ⊢ (𝐶 = +∞ → (𝐴 ·_e 𝐶) = (𝐴 ·_e +∞))
33	32	ad2antlr 740	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐴 ·_e 𝐶) = (𝐴 ·_e +∞))
34		xmulpnf1 13411	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e +∞) = +∞)
35	12, 16, 34	syl2anc 596	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐴 ·_e +∞) = +∞)
36	33, 35	eqtrd 2795	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → (𝐴 ·_e 𝐶) = +∞)
37	36	oveq1d 7439	. . . 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 2798	. . 3 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 < 𝐴) → ((𝐴 +_e 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐶) +_e (𝐵 ·_e 𝐶)))
39		simpll3 1233	. . . . . . . 8 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → 𝐶 ∈ (0[,]+∞))
40	1, 39	sselid 3907	. . . . . . 7 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → 𝐶 ∈ ℝ^*)
41		xmul02 13405	. . . . . . 7 ⊢ (𝐶 ∈ ℝ^* → (0 ·_e 𝐶) = 0)
42	40, 41	syl 18	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → (0 ·_e 𝐶) = 0)
43	42	oveq1d 7439	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → ((0 ·_e 𝐶) +_e (𝐵 ·_e 𝐶)) = (0 +_e (𝐵 ·_e 𝐶)))
44		oveq1 7431	. . . . . . 7 ⊢ (0 = 𝐴 → (0 ·_e 𝐶) = (𝐴 ·_e 𝐶))
45	44	adantl 487	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → (0 ·_e 𝐶) = (𝐴 ·_e 𝐶))
46	45	oveq1d 7439	. . . . 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 1232	. . . . . . . 8 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → 𝐵 ∈ (0[,]+∞))
48	1, 47	sselid 3907	. . . . . . 7 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → 𝐵 ∈ ℝ^*)
49	48, 40	xmulcld 13439	. . . . . 6 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → (𝐵 ·_e 𝐶) ∈ ℝ^*)
50		xaddlid 13379	. . . . . 6 ⊢ ((𝐵 ·_e 𝐶) ∈ ℝ^* → (0 +_e (𝐵 ·_e 𝐶)) = (𝐵 ·_e 𝐶))
51	49, 50	syl 18	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → (0 +_e (𝐵 ·_e 𝐶)) = (𝐵 ·_e 𝐶))
52	43, 46, 51	3eqtr3d 2803	. . . 4 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → ((𝐴 ·_e 𝐶) +_e (𝐵 ·_e 𝐶)) = (𝐵 ·_e 𝐶))
53		xaddlid 13379	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (0 +_e 𝐵) = 𝐵)
54	48, 53	syl 18	. . . . 5 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → (0 +_e 𝐵) = 𝐵)
55	54	oveq1d 7439	. . . 4 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → ((0 +_e 𝐵) ·_e 𝐶) = (𝐵 ·_e 𝐶))
56		oveq1 7431	. . . . . 6 ⊢ (0 = 𝐴 → (0 +_e 𝐵) = (𝐴 +_e 𝐵))
57	56	oveq1d 7439	. . . . 5 ⊢ (0 = 𝐴 → ((0 +_e 𝐵) ·_e 𝐶) = ((𝐴 +_e 𝐵) ·_e 𝐶))
58	57	adantl 487	. . . 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 2802	. . 3 ⊢ ((((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) ∧ 0 = 𝐴) → ((𝐴 +_e 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐶) +_e (𝐵 ·_e 𝐶)))
60		0xr 11363	. . . . 5 ⊢ 0 ∈ ℝ^*
61	60	a1i 11	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → 0 ∈ ℝ^*)
62		simpl1 1210	. . . . 5 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → 𝐴 ∈ (0[,]+∞))
63	1, 62	sselid 3907	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → 𝐴 ∈ ℝ^*)
64		pnfxr 11370	. . . . . 6 ⊢ +∞ ∈ ℝ^*
65	64	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → +∞ ∈ ℝ^*)
66		iccgelb 13540	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐴 ∈ (0[,]+∞)) → 0 ≤ 𝐴)
67	61, 65, 62, 66	syl3anc 1398	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → 0 ≤ 𝐴)
68		xrleloe 13280	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
69	68	biimpa 482	. . . 4 ⊢ (((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) ∧ 0 ≤ 𝐴) → (0 < 𝐴 ∨ 0 = 𝐴))
70	61, 63, 67, 69	syl21anc 851	. . 3 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → (0 < 𝐴 ∨ 0 = 𝐴))
71	38, 59, 70	mpjaodan 973	. 2 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ∧ 𝐶 = +∞) → ((𝐴 +_e 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐶) +_e (𝐵 ·_e 𝐶)))
72		0lepnf 13269	. . . . 5 ⊢ 0 ≤ +∞
73		eliccelico 33419	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → (𝐶 ∈ (0[,]+∞) ↔ (𝐶 ∈ (0[,)+∞) ∨ 𝐶 = +∞)))
74	60, 64, 72, 73	mp3an 1490	. . . 4 ⊢ (𝐶 ∈ (0[,]+∞) ↔ (𝐶 ∈ (0[,)+∞) ∨ 𝐶 = +∞))
75	74	3anbi3i 1177	. . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) ↔ (𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ (𝐶 ∈ (0[,)+∞) ∨ 𝐶 = +∞)))
76	75	simp3bi 1165	. 2 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → (𝐶 ∈ (0[,)+∞) ∨ 𝐶 = +∞))
77	10, 71, 76	mpjaodan 973	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 (class class class)co 7424 ℝcr 11206 0cc0 11207 +∞cpnf 11347 -∞cmnf 11348 ℝ^*cxr 11349 < clt 11350 ≤ cle 11351 +_e cxad 13246 ·_e cxmu 13247 [,)cico 13485 [,]cicc 13486

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8005 df-2nd 8006 df-er 8724 df-en 8981 df-dom 8982 df-sdom 8983 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ico 13489 df-icc 13490

This theorem is used by: xrge0adddi 33630 xrge0slmod 33959 esummulc1 34763

W3C validator