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Theorem xadddi 13256

Description: Distributive property for extended real addition and multiplication. Like xaddass 13210, 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 13255	. 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 13200	. . . . . 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 13229	. . . . 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 11243	. . . . 5 ⊢ 0 ∈ ℝ^*
9		xaddrid 13202	. . . . 5 ⊢ (0 ∈ ℝ^* → (0 +_𝑒 0) = 0)
10	8, 9	ax-mp 5	. . . 4 ⊢ (0 +_𝑒 0) = 0
11	7, 10	eqtr4di 2789	. . 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 7408	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e (𝐵 +_𝑒 𝐶)) = (𝐴 ·_e (𝐵 +_𝑒 𝐶)))
14		xmul02 13229	. . . . . 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 7408	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e 𝐵) = (𝐴 ·_e 𝐵))
17	15, 16	eqtr3d 2773	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → 0 = (𝐴 ·_e 𝐵))
18		xmul02 13229	. . . . . 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 7408	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 ·_e 𝐶) = (𝐴 ·_e 𝐶))
21	19, 20	eqtr3d 2773	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → 0 = (𝐴 ·_e 𝐶))
22	17, 21	oveq12d 7411	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 0 = 𝐴) → (0 +_𝑒 0) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
23	11, 13, 22	3eqtr3d 2779	. 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 13172	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
27		renegcl 11505	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
28	26, 27	eqeltrd 2832	. . . . . 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 11246	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
33		xlt0neg1 13180	. . . . . . 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 13255	. . . . 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 13230	. . . . 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 13230	. . . . . . 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 13230	. . . . . . 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 7411	. . . . 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 13234	. . . . . . 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 13234	. . . . . . 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 13209	. . . . . 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 2774	. . . 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 2779	. . 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 13234	. . . . 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 13200	. . . . 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 13176	. . . 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 11198	. . 3 ⊢ 0 ∈ ℝ
63		lttri4 11280	. . 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 5141 (class class class)co 7393 ℝcr 11091 0cc0 11092 ℝ^*cxr 11229 < clt 11230 -cneg 11427 -_𝑒cxne 13071 +_𝑒 cxad 13072 ·_e cxmu 13073

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 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168

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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-xneg 13074 df-xadd 13075 df-xmul 13076

This theorem is referenced by: xadddir 13257 xadddi2 13258

W3C validator