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Theorem xadddi 13311

Description: Distributive property for extended real addition and multiplication. Like xaddass 13265, this has an unusual domain of correctness due to counterexamples like (+∞ · (2 − 1)) = -∞ ≠ ((+∞ · 2) − (+∞ · 1)) = (+∞ − +∞) = 0. In this theorem we show that if the multiplier is real then everything works as expected. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xadddi	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))

**Proof of Theorem xadddi**
Step	Hyp	Ref	Expression
1		xadddilem 13310	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 < 𝐴) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
2		simpl2 1193	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 0 = 𝐴) → 𝐵 ∈ ℝ^)
3		simpl3 1194	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 0 = 𝐴) → 𝐶 ∈ ℝ^)
4		xaddcl 13255	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
5	2, 3, 4	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 0 = 𝐴) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
6		xmul02 13284	. . . . 5 ⊢ ((𝐵 +_𝑒 𝐶) ∈ ℝ^* → (0 ·_e (𝐵 +_𝑒 𝐶)) = 0)
7	5, 6	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e (𝐵 +_𝑒 𝐶)) = 0)
8		0xr 11282	. . . . 5 ⊢ 0 ∈ ℝ^*
9		xaddrid 13257	. . . . 5 ⊢ (0 ∈ ℝ^* → (0 +_𝑒 0) = 0)
10	8, 9	ax-mp 5	. . . 4 ⊢ (0 +_𝑒 0) = 0
11	7, 10	eqtr4di 2788	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e (𝐵 +_𝑒 𝐶)) = (0 +_𝑒 0))
12		simpr 484	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → 0 = 𝐴)
13	12	oveq1d 7420	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e (𝐵 +_𝑒 𝐶)) = (𝐴 ·_e (𝐵 +_𝑒 𝐶)))
14		xmul02 13284	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (0 ·_e 𝐵) = 0)
15	2, 14	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e 𝐵) = 0)
16	12	oveq1d 7420	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e 𝐵) = (𝐴 ·_e 𝐵))
17	15, 16	eqtr3d 2772	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → 0 = (𝐴 ·_e 𝐵))
18		xmul02 13284	. . . . . 6 ⊢ (𝐶 ∈ ℝ^* → (0 ·_e 𝐶) = 0)
19	3, 18	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e 𝐶) = 0)
20	12	oveq1d 7420	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e 𝐶) = (𝐴 ·_e 𝐶))
21	19, 20	eqtr3d 2772	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → 0 = (𝐴 ·_e 𝐶))
22	17, 21	oveq12d 7423	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 +_𝑒 0) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
23	11, 13, 22	3eqtr3d 2778	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
24		simp1 1136	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐴 ∈ ℝ)
25	24	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
26		rexneg 13227	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
27		renegcl 11546	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
28	26, 27	eqeltrd 2834	. . . . . 6 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ)
29	25, 28	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → -_𝑒𝐴 ∈ ℝ)
30		simpl2 1193	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → 𝐵 ∈ ℝ^)
31		simpl3 1194	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → 𝐶 ∈ ℝ^)
32	24	rexrd 11285	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
33		xlt0neg1 13235	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → (𝐴 < 0 ↔ 0 < -_𝑒𝐴))
34	32, 33	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 0 ↔ 0 < -_𝑒𝐴))
35	34	biimpa 476	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → 0 < -_𝑒𝐴)
36		xadddilem 13310	. . . . 5 ⊢ (((-_𝑒𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 < -_𝑒𝐴) → (-_𝑒𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((-_𝑒𝐴 ·_e 𝐵) +_𝑒 (-_𝑒𝐴 ·_e 𝐶)))
37	29, 30, 31, 35, 36	syl31anc 1375	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((-_𝑒𝐴 ·_e 𝐵) +_𝑒 (-_𝑒𝐴 ·_e 𝐶)))
38	32	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → 𝐴 ∈ ℝ^)
39	30, 31, 4	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
40		xmulneg1 13285	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 +_𝑒 𝐶) ∈ ℝ^*) → (-_𝑒𝐴 ·_e (𝐵 +_𝑒 𝐶)) = -_𝑒(𝐴 ·_e (𝐵 +_𝑒 𝐶)))
41	38, 39, 40	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e (𝐵 +_𝑒 𝐶)) = -_𝑒(𝐴 ·_e (𝐵 +_𝑒 𝐶)))
42		xmulneg1 13285	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 ·_e 𝐵) = -_𝑒(𝐴 ·_e 𝐵))
43	38, 30, 42	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e 𝐵) = -_𝑒(𝐴 ·_e 𝐵))
44		xmulneg1 13285	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (-_𝑒𝐴 ·_e 𝐶) = -_𝑒(𝐴 ·_e 𝐶))
45	38, 31, 44	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e 𝐶) = -_𝑒(𝐴 ·_e 𝐶))
46	43, 45	oveq12d 7423	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → ((-_𝑒𝐴 ·_e 𝐵) +_𝑒 (-_𝑒𝐴 ·_e 𝐶)) = (-_𝑒(𝐴 ·_e 𝐵) +_𝑒 -_𝑒(𝐴 ·_e 𝐶)))
47		xmulcl 13289	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ·_e 𝐵) ∈ ℝ^)
48	38, 30, 47	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → (𝐴 ·_e 𝐵) ∈ ℝ^)
49		xmulcl 13289	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 ·_e 𝐶) ∈ ℝ^)
50	38, 31, 49	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → (𝐴 ·_e 𝐶) ∈ ℝ^)
51		xnegdi 13264	. . . . . 6 ⊢ (((𝐴 ·_e 𝐵) ∈ ℝ^* ∧ (𝐴 ·_e 𝐶) ∈ ℝ^*) → -_𝑒((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) = (-_𝑒(𝐴 ·_e 𝐵) +_𝑒 -_𝑒(𝐴 ·_e 𝐶)))
52	48, 50, 51	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → -_𝑒((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) = (-_𝑒(𝐴 ·_e 𝐵) +_𝑒 -_𝑒(𝐴 ·_e 𝐶)))
53	46, 52	eqtr4d 2773	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → ((-_𝑒𝐴 ·_e 𝐵) +_𝑒 (-_𝑒𝐴 ·_e 𝐶)) = -_𝑒((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
54	37, 41, 53	3eqtr3d 2778	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → -_𝑒(𝐴 ·_e (𝐵 +_𝑒 𝐶)) = -_𝑒((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
55		xmulcl 13289	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 +_𝑒 𝐶) ∈ ℝ^) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) ∈ ℝ^)
56	38, 39, 55	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) ∈ ℝ^)
57		xaddcl 13255	. . . . 5 ⊢ (((𝐴 ·_e 𝐵) ∈ ℝ^* ∧ (𝐴 ·_e 𝐶) ∈ ℝ^) → ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) ∈ ℝ^)
58	48, 50, 57	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) ∈ ℝ^)
59		xneg11 13231	. . . 4 ⊢ (((𝐴 ·_e (𝐵 +_𝑒 𝐶)) ∈ ℝ^* ∧ ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) ∈ ℝ^*) → (-_𝑒(𝐴 ·_e (𝐵 +_𝑒 𝐶)) = -_𝑒((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) ↔ (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶))))
60	56, 58, 59	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (-_𝑒(𝐴 ·_e (𝐵 +_𝑒 𝐶)) = -_𝑒((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) ↔ (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶))))
61	54, 60	mpbid 232	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
62		0re 11237	. . 3 ⊢ 0 ∈ ℝ
63		lttri4 11319	. . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 ∨ 0 = 𝐴 ∨ 𝐴 < 0))
64	62, 24, 63	sylancr 587	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (0 < 𝐴 ∨ 0 = 𝐴 ∨ 𝐴 < 0))
65	1, 23, 61, 64	mpjao3dan 1434	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1085 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 class class class wbr 5119 (class class class)co 7405 ℝcr 11128 0cc0 11129 ℝ^*cxr 11268 < clt 11269 -cneg 11467 -_𝑒cxne 13125 +_𝑒 cxad 13126 ·_e cxmu 13127

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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7988 df-2nd 7989 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-xneg 13128 df-xadd 13129 df-xmul 13130

This theorem is referenced by: xadddir 13312 xadddi2 13313

W3C validator