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Theorem xadddi 12125

Description: Distributive property for extended real addition and multiplication. Like xaddass 12079, 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 12124	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 < 𝐴) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
2		simpl2 1065	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 0 = 𝐴) → 𝐵 ∈ ℝ^)
3		simpl3 1066	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 0 = 𝐴) → 𝐶 ∈ ℝ^)
4		xaddcl 12070	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
5	2, 3, 4	syl2anc 693	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 0 = 𝐴) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
6		xmul02 12098	. . . . 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 10086	. . . . 5 ⊢ 0 ∈ ℝ^*
9		xaddid1 12072	. . . . 5 ⊢ (0 ∈ ℝ^* → (0 +_𝑒 0) = 0)
10	8, 9	ax-mp 5	. . . 4 ⊢ (0 +_𝑒 0) = 0
11	7, 10	syl6eqr 2674	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e (𝐵 +_𝑒 𝐶)) = (0 +_𝑒 0))
12		simpr 477	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → 0 = 𝐴)
13	12	oveq1d 6665	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e (𝐵 +_𝑒 𝐶)) = (𝐴 ·_e (𝐵 +_𝑒 𝐶)))
14		xmul02 12098	. . . . . 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 6665	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e 𝐵) = (𝐴 ·_e 𝐵))
17	15, 16	eqtr3d 2658	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → 0 = (𝐴 ·_e 𝐵))
18		xmul02 12098	. . . . . 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 6665	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e 𝐶) = (𝐴 ·_e 𝐶))
21	19, 20	eqtr3d 2658	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → 0 = (𝐴 ·_e 𝐶))
22	17, 21	oveq12d 6668	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 +_𝑒 0) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
23	11, 13, 22	3eqtr3d 2664	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
24		simp1 1061	. . . . . . 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 12042	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
27		renegcl 10344	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
28	26, 27	eqeltrd 2701	. . . . . 6 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ)
29	25, 28	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → -_𝑒𝐴 ∈ ℝ)
30		simpl2 1065	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → 𝐵 ∈ ℝ^)
31		simpl3 1066	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → 𝐶 ∈ ℝ^)
32	24	rexrd 10089	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
33		xlt0neg1 12050	. . . . . . 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 501	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → 0 < -_𝑒𝐴)
36		xadddilem 12124	. . . . 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 1329	. . . 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 693	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
40		xmulneg1 12099	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 +_𝑒 𝐶) ∈ ℝ^*) → (-_𝑒𝐴 ·_e (𝐵 +_𝑒 𝐶)) = -_𝑒(𝐴 ·_e (𝐵 +_𝑒 𝐶)))
41	38, 39, 40	syl2anc 693	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e (𝐵 +_𝑒 𝐶)) = -_𝑒(𝐴 ·_e (𝐵 +_𝑒 𝐶)))
42		xmulneg1 12099	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 ·_e 𝐵) = -_𝑒(𝐴 ·_e 𝐵))
43	38, 30, 42	syl2anc 693	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e 𝐵) = -_𝑒(𝐴 ·_e 𝐵))
44		xmulneg1 12099	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (-_𝑒𝐴 ·_e 𝐶) = -_𝑒(𝐴 ·_e 𝐶))
45	38, 31, 44	syl2anc 693	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e 𝐶) = -_𝑒(𝐴 ·_e 𝐶))
46	43, 45	oveq12d 6668	. . . . 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 12103	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ·_e 𝐵) ∈ ℝ^)
48	38, 30, 47	syl2anc 693	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → (𝐴 ·_e 𝐵) ∈ ℝ^)
49		xmulcl 12103	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 ·_e 𝐶) ∈ ℝ^)
50	38, 31, 49	syl2anc 693	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → (𝐴 ·_e 𝐶) ∈ ℝ^)
51		xnegdi 12078	. . . . . 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 693	. . . . 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 2659	. . . 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 2664	. . 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 12103	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 +_𝑒 𝐶) ∈ ℝ^) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) ∈ ℝ^)
56	38, 39, 55	syl2anc 693	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) ∈ ℝ^)
57		xaddcl 12070	. . . . 5 ⊢ (((𝐴 ·_e 𝐵) ∈ ℝ^* ∧ (𝐴 ·_e 𝐶) ∈ ℝ^) → ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) ∈ ℝ^)
58	48, 50, 57	syl2anc 693	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 0) → ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) ∈ ℝ^)
59		xneg11 12046	. . . 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 693	. . 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 222	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 0) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
62		0re 10040	. . 3 ⊢ 0 ∈ ℝ
63		lttri4 10122	. . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 ∨ 0 = 𝐴 ∨ 𝐴 < 0))
64	62, 24, 63	sylancr 695	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (0 < 𝐴 ∨ 0 = 𝐴 ∨ 𝐴 < 0))
65	1, 23, 61, 64	mpjao3dan 1395	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 196 ∧ wa 384 ∨ w3o 1036 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 0cc0 9936 ℝ^*cxr 10073 < clt 10074 -cneg 10267 -_𝑒cxne 11943 +_𝑒 cxad 11944 ·_e cxmu 11945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-xneg 11946 df-xadd 11947 df-xmul 11948

This theorem is referenced by: xadddir 12126 xadddi2 12127

W3C validator