xadddi - Metamath Proof Explorer

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Theorem xadddi 13214

Description: Distributive property for extended real addition and multiplication. Like xaddass 13168, 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 13213	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 < 𝐴) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
2		simpl2 1192	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 0 = 𝐴) → 𝐵 ∈ ℝ^)
3		simpl3 1193	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 0 = 𝐴) → 𝐶 ∈ ℝ^)
4		xaddcl 13158	. . . . . 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 13187	. . . . 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 11202	. . . . 5 ⊢ 0 ∈ ℝ^*
9		xaddid1 13160	. . . . 5 ⊢ (0 ∈ ℝ^* → (0 +_𝑒 0) = 0)
10	8, 9	ax-mp 5	. . . 4 ⊢ (0 +_𝑒 0) = 0
11	7, 10	eqtr4di 2794	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e (𝐵 +_𝑒 𝐶)) = (0 +_𝑒 0))
12		simpr 485	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → 0 = 𝐴)
13	12	oveq1d 7372	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e (𝐵 +_𝑒 𝐶)) = (𝐴 ·_e (𝐵 +_𝑒 𝐶)))
14		xmul02 13187	. . . . . 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 7372	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e 𝐵) = (𝐴 ·_e 𝐵))
17	15, 16	eqtr3d 2778	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → 0 = (𝐴 ·_e 𝐵))
18		xmul02 13187	. . . . . 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 7372	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e 𝐶) = (𝐴 ·_e 𝐶))
21	19, 20	eqtr3d 2778	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → 0 = (𝐴 ·_e 𝐶))
22	17, 21	oveq12d 7375	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 +_𝑒 0) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
23	11, 13, 22	3eqtr3d 2784	. 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 481	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
26		rexneg 13130	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
27		renegcl 11464	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
28	26, 27	eqeltrd 2838	. . . . . 6 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ)
29	25, 28	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → -_𝑒𝐴 ∈ ℝ)
30		simpl2 1192	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → 𝐵 ∈ ℝ^)
31		simpl3 1193	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → 𝐶 ∈ ℝ^)
32	24	rexrd 11205	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
33		xlt0neg1 13138	. . . . . . 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 477	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → 0 < -_𝑒𝐴)
36		xadddilem 13213	. . . . 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 1373	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((-_𝑒𝐴 ·_e 𝐵) +_𝑒 (-_𝑒𝐴 ·_e 𝐶)))
38	32	adantr 481	. . . . 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 13188	. . . . 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 13188	. . . . . . 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 13188	. . . . . . 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 7375	. . . . 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 13192	. . . . . . 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 13192	. . . . . . 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 13167	. . . . . 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 2779	. . . 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 2784	. . 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 13192	. . . . 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 13158	. . . . 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 13134	. . . 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 231	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
62		0re 11157	. . 3 ⊢ 0 ∈ ℝ
63		lttri4 11239	. . 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 1431	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 205 ∧ wa 396 ∨ w3o 1086 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 (class class class)co 7357 ℝcr 11050 0cc0 11051 ℝ^*cxr 11188 < clt 11189 -cneg 11386 -_𝑒cxne 13030 +_𝑒 cxad 13031 ·_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

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-xadd 13034 df-xmul 13035

This theorem is referenced by: xadddir 13215 xadddi2 13216

W3C validator