xadddi - Metamath Proof Explorer

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Theorem xadddi 13357

Description: Distributive property for extended real addition and multiplication. Like xaddass 13311, 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 13356	. 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 13301	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
5	2, 3, 4	syl2anc 583	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 0 = 𝐴) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
6		xmul02 13330	. . . . 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 11337	. . . . 5 ⊢ 0 ∈ ℝ^*
9		xaddrid 13303	. . . . 5 ⊢ (0 ∈ ℝ^* → (0 +_𝑒 0) = 0)
10	8, 9	ax-mp 5	. . . 4 ⊢ (0 +_𝑒 0) = 0
11	7, 10	eqtr4di 2798	. . 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 7463	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e (𝐵 +_𝑒 𝐶)) = (𝐴 ·_e (𝐵 +_𝑒 𝐶)))
14		xmul02 13330	. . . . . 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 7463	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e 𝐵) = (𝐴 ·_e 𝐵))
17	15, 16	eqtr3d 2782	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → 0 = (𝐴 ·_e 𝐵))
18		xmul02 13330	. . . . . 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 7463	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e 𝐶) = (𝐴 ·_e 𝐶))
21	19, 20	eqtr3d 2782	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → 0 = (𝐴 ·_e 𝐶))
22	17, 21	oveq12d 7466	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 +_𝑒 0) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
23	11, 13, 22	3eqtr3d 2788	. 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 13273	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
27		renegcl 11599	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
28	26, 27	eqeltrd 2844	. . . . . 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 11340	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
33		xlt0neg1 13281	. . . . . . 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 13356	. . . . 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 480	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → 𝐴 ∈ ℝ^)
39	30, 31, 4	syl2anc 583	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
40		xmulneg1 13331	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 +_𝑒 𝐶) ∈ ℝ^*) → (-_𝑒𝐴 ·_e (𝐵 +_𝑒 𝐶)) = -_𝑒(𝐴 ·_e (𝐵 +_𝑒 𝐶)))
41	38, 39, 40	syl2anc 583	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e (𝐵 +_𝑒 𝐶)) = -_𝑒(𝐴 ·_e (𝐵 +_𝑒 𝐶)))
42		xmulneg1 13331	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 ·_e 𝐵) = -_𝑒(𝐴 ·_e 𝐵))
43	38, 30, 42	syl2anc 583	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e 𝐵) = -_𝑒(𝐴 ·_e 𝐵))
44		xmulneg1 13331	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (-_𝑒𝐴 ·_e 𝐶) = -_𝑒(𝐴 ·_e 𝐶))
45	38, 31, 44	syl2anc 583	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e 𝐶) = -_𝑒(𝐴 ·_e 𝐶))
46	43, 45	oveq12d 7466	. . . . 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 13335	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ·_e 𝐵) ∈ ℝ^)
48	38, 30, 47	syl2anc 583	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → (𝐴 ·_e 𝐵) ∈ ℝ^)
49		xmulcl 13335	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 ·_e 𝐶) ∈ ℝ^)
50	38, 31, 49	syl2anc 583	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → (𝐴 ·_e 𝐶) ∈ ℝ^)
51		xnegdi 13310	. . . . . 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 583	. . . . 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 2783	. . . 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 2788	. . 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 13335	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 +_𝑒 𝐶) ∈ ℝ^) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) ∈ ℝ^)
56	38, 39, 55	syl2anc 583	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) ∈ ℝ^)
57		xaddcl 13301	. . . . 5 ⊢ (((𝐴 ·_e 𝐵) ∈ ℝ^* ∧ (𝐴 ·_e 𝐶) ∈ ℝ^) → ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) ∈ ℝ^)
58	48, 50, 57	syl2anc 583	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) ∈ ℝ^)
59		xneg11 13277	. . . 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 583	. . 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 11292	. . 3 ⊢ 0 ∈ ℝ
63		lttri4 11374	. . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 ∨ 0 = 𝐴 ∨ 𝐴 < 0))
64	62, 24, 63	sylancr 586	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (0 < 𝐴 ∨ 0 = 𝐴 ∨ 𝐴 < 0))
65	1, 23, 61, 64	mpjao3dan 1432	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 1086 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 class class class wbr 5166 (class class class)co 7448 ℝcr 11183 0cc0 11184 ℝ^*cxr 11323 < clt 11324 -cneg 11521 -_𝑒cxne 13172 +_𝑒 cxad 13173 ·_e cxmu 13174

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-xneg 13175 df-xadd 13176 df-xmul 13177

This theorem is referenced by: xadddir 13358 xadddi2 13359

W3C validator