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Theorem xadddi2 13408

Description: The assumption that the multiplier be real in xadddi 13406 can be relaxed if the addends have the same sign. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xadddi2	⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))

**Proof of Theorem xadddi2**
Step	Hyp	Ref	Expression
1		simpr 490	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
2		simp2l 1218	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → 𝐵 ∈ ℝ^*)
3	2	ad2antrr 739	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 ∈ ℝ) → 𝐵 ∈ ℝ^*)
4		simp3l 1220	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ^*)
5	4	ad2antrr 739	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 ∈ ℝ) → 𝐶 ∈ ℝ^*)
6		xadddi 13406	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
7	1, 3, 5, 6	syl3anc 1398	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 ∈ ℝ) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
8		pnfxr 11344	. . . . . 6 ⊢ +∞ ∈ ℝ^*
9	4	adantr 486	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 𝐶 ∈ ℝ^*)
10		xmulcl 13384	. . . . . 6 ⊢ ((+∞ ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (+∞ ·_e 𝐶) ∈ ℝ^)
11	8, 9, 10	sylancr 599	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (+∞ ·_e 𝐶) ∈ ℝ^*)
12		simpl3r 1248	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 0 ≤ 𝐶)
13		0lepnf 13243	. . . . . . . . 9 ⊢ 0 ≤ +∞
14		xmulge0 13395	. . . . . . . . 9 ⊢ (((+∞ ∈ ℝ^* ∧ 0 ≤ +∞) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → 0 ≤ (+∞ ·_e 𝐶))
15	8, 13, 14	mpanl12 715	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶) → 0 ≤ (+∞ ·_e 𝐶))
16	4, 12, 15	syl2an2r 698	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 0 ≤ (+∞ ·_e 𝐶))
17		ge0nemnf 13284	. . . . . . 7 ⊢ (((+∞ ·_e 𝐶) ∈ ℝ^* ∧ 0 ≤ (+∞ ·_e 𝐶)) → (+∞ ·_e 𝐶) ≠ -∞)
18	11, 16, 17	syl2anc 596	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (+∞ ·_e 𝐶) ≠ -∞)
19	18	adantr 486	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = +∞) → (+∞ ·_e 𝐶) ≠ -∞)
20		xaddpnf2 13338	. . . . 5 ⊢ (((+∞ ·_e 𝐶) ∈ ℝ^* ∧ (+∞ ·_e 𝐶) ≠ -∞) → (+∞ +_𝑒 (+∞ ·_e 𝐶)) = +∞)
21	11, 19, 20	syl2an2r 698	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = +∞) → (+∞ +_𝑒 (+∞ ·_e 𝐶)) = +∞)
22		oveq1 7419	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ·_e 𝐵) = (+∞ ·_e 𝐵))
23		oveq1 7419	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ·_e 𝐶) = (+∞ ·_e 𝐶))
24	22, 23	oveq12d 7430	. . . . 5 ⊢ (𝐴 = +∞ → ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) = ((+∞ ·_e 𝐵) +_𝑒 (+∞ ·_e 𝐶)))
25		xmulpnf2 13386	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → (+∞ ·_e 𝐵) = +∞)
26	2, 25	sylan 592	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (+∞ ·_e 𝐵) = +∞)
27	26	oveq1d 7427	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → ((+∞ ·_e 𝐵) +_𝑒 (+∞ ·_e 𝐶)) = (+∞ +_𝑒 (+∞ ·_e 𝐶)))
28	24, 27	sylan9eqr 2818	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = +∞) → ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) = (+∞ +_𝑒 (+∞ ·_e 𝐶)))
29		oveq1 7419	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = (+∞ ·_e (𝐵 +_𝑒 𝐶)))
30		xaddcl 13350	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
31	2, 4, 30	syl2anc 596	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → (𝐵 +_𝑒 𝐶) ∈ ℝ^*)
32		0xr 11337	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
33	32	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 0 ∈ ℝ^*)
34	2	adantr 486	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 𝐵 ∈ ℝ^*)
35	31	adantr 486	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (𝐵 +_𝑒 𝐶) ∈ ℝ^*)
36		simpr 490	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 0 < 𝐵)
37	34	xaddridd 13354	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (𝐵 +_𝑒 0) = 𝐵)
38		xleadd2a 13365	. . . . . . . . 9 ⊢ (((0 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 0 ≤ 𝐶) → (𝐵 +_𝑒 0) ≤ (𝐵 +_𝑒 𝐶))
39	33, 9, 34, 12, 38	syl31anc 1400	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (𝐵 +_𝑒 0) ≤ (𝐵 +_𝑒 𝐶))
40	37, 39	eqbrtrrd 5129	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 𝐵 ≤ (𝐵 +_𝑒 𝐶))
41	33, 34, 35, 36, 40	xrltletrd 13271	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 0 < (𝐵 +_𝑒 𝐶))
42		xmulpnf2 13386	. . . . . 6 ⊢ (((𝐵 +_𝑒 𝐶) ∈ ℝ^* ∧ 0 < (𝐵 +_𝑒 𝐶)) → (+∞ ·_e (𝐵 +_𝑒 𝐶)) = +∞)
43	31, 41, 42	syl2an2r 698	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (+∞ ·_e (𝐵 +_𝑒 𝐶)) = +∞)
44	29, 43	sylan9eqr 2818	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = +∞) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = +∞)
45	21, 28, 44	3eqtr4rd 2807	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = +∞) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
46		mnfxr 11347	. . . . . . 7 ⊢ -∞ ∈ ℝ^*
47		xmulcl 13384	. . . . . . 7 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (-∞ ·_e 𝐶) ∈ ℝ^)
48	46, 9, 47	sylancr 599	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-∞ ·_e 𝐶) ∈ ℝ^*)
49		xmulneg1 13380	. . . . . . . . . . . 12 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (-_𝑒-∞ ·_e 𝐶) = -_𝑒(-∞ ·_e 𝐶))
50	46, 9, 49	sylancr 599	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-_𝑒-∞ ·_e 𝐶) = -_𝑒(-∞ ·_e 𝐶))
51		xnegmnf 13321	. . . . . . . . . . . 12 ⊢ -_𝑒-∞ = +∞
52	51	oveq1i 7422	. . . . . . . . . . 11 ⊢ (-_𝑒-∞ ·_e 𝐶) = (+∞ ·_e 𝐶)
53	50, 52	eqtr3di 2811	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → -_𝑒(-∞ ·_e 𝐶) = (+∞ ·_e 𝐶))
54		xnegpnf 13320	. . . . . . . . . . 11 ⊢ -_𝑒+∞ = -∞
55	54	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → -_𝑒+∞ = -∞)
56	53, 55	eqeq12d 2777	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-_𝑒(-∞ ·_e 𝐶) = -_𝑒+∞ ↔ (+∞ ·_e 𝐶) = -∞))
57		xneg11 13326	. . . . . . . . . 10 ⊢ (((-∞ ·_e 𝐶) ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (-_𝑒(-∞ ·_e 𝐶) = -_𝑒+∞ ↔ (-∞ ·_e 𝐶) = +∞))
58	48, 8, 57	sylancl 598	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-_𝑒(-∞ ·_e 𝐶) = -_𝑒+∞ ↔ (-∞ ·_e 𝐶) = +∞))
59	56, 58	bitr3d 284	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → ((+∞ ·_e 𝐶) = -∞ ↔ (-∞ ·_e 𝐶) = +∞))
60	59	necon3bid 3000	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → ((+∞ ·_e 𝐶) ≠ -∞ ↔ (-∞ ·_e 𝐶) ≠ +∞))
61	18, 60	mpbid 235	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-∞ ·_e 𝐶) ≠ +∞)
62		xaddmnf2 13340	. . . . . 6 ⊢ (((-∞ ·_e 𝐶) ∈ ℝ^* ∧ (-∞ ·_e 𝐶) ≠ +∞) → (-∞ +_𝑒 (-∞ ·_e 𝐶)) = -∞)
63	48, 61, 62	syl2anc 596	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-∞ +_𝑒 (-∞ ·_e 𝐶)) = -∞)
64	63	adantr 486	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = -∞) → (-∞ +_𝑒 (-∞ ·_e 𝐶)) = -∞)
65		oveq1 7419	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 ·_e 𝐵) = (-∞ ·_e 𝐵))
66		oveq1 7419	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 ·_e 𝐶) = (-∞ ·_e 𝐶))
67	65, 66	oveq12d 7430	. . . . 5 ⊢ (𝐴 = -∞ → ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) = ((-∞ ·_e 𝐵) +_𝑒 (-∞ ·_e 𝐶)))
68		xmulmnf2 13388	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → (-∞ ·_e 𝐵) = -∞)
69	2, 68	sylan 592	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-∞ ·_e 𝐵) = -∞)
70	69	oveq1d 7427	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → ((-∞ ·_e 𝐵) +_𝑒 (-∞ ·_e 𝐶)) = (-∞ +_𝑒 (-∞ ·_e 𝐶)))
71	67, 70	sylan9eqr 2818	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = -∞) → ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) = (-∞ +_𝑒 (-∞ ·_e 𝐶)))
72		oveq1 7419	. . . . 5 ⊢ (𝐴 = -∞ → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = (-∞ ·_e (𝐵 +_𝑒 𝐶)))
73		xmulmnf2 13388	. . . . . 6 ⊢ (((𝐵 +_𝑒 𝐶) ∈ ℝ^* ∧ 0 < (𝐵 +_𝑒 𝐶)) → (-∞ ·_e (𝐵 +_𝑒 𝐶)) = -∞)
74	31, 41, 73	syl2an2r 698	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-∞ ·_e (𝐵 +_𝑒 𝐶)) = -∞)
75	72, 74	sylan9eqr 2818	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = -∞) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = -∞)
76	64, 71, 75	3eqtr4rd 2807	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = -∞) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
77		simpl1 1210	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 𝐴 ∈ ℝ^*)
78		elxr 13226	. . . 4 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
79	77, 78	sylib 221	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
80	7, 45, 76, 79	mpjao3dan 1459	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
81		simp1 1154	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → 𝐴 ∈ ℝ^*)
82		xmulcl 13384	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 ·_e 𝐶) ∈ ℝ^)
83	81, 4, 82	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → (𝐴 ·_e 𝐶) ∈ ℝ^*)
84	83	adantr 486	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 = 𝐵) → (𝐴 ·_e 𝐶) ∈ ℝ^*)
85		xaddlid 13353	. . . 4 ⊢ ((𝐴 ·_e 𝐶) ∈ ℝ^* → (0 +_𝑒 (𝐴 ·_e 𝐶)) = (𝐴 ·_e 𝐶))
86	84, 85	syl 18	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 = 𝐵) → (0 +_𝑒 (𝐴 ·_e 𝐶)) = (𝐴 ·_e 𝐶))
87		oveq2 7420	. . . . . 6 ⊢ (0 = 𝐵 → (𝐴 ·_e 0) = (𝐴 ·_e 𝐵))
88	87	eqcomd 2767	. . . . 5 ⊢ (0 = 𝐵 → (𝐴 ·_e 𝐵) = (𝐴 ·_e 0))
89		xmul01 13378	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ·_e 0) = 0)
90	89	3ad2ant1 1151	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → (𝐴 ·_e 0) = 0)
91	88, 90	sylan9eqr 2818	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 = 𝐵) → (𝐴 ·_e 𝐵) = 0)
92	91	oveq1d 7427	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 = 𝐵) → ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) = (0 +_𝑒 (𝐴 ·_e 𝐶)))
93		oveq1 7419	. . . . . 6 ⊢ (0 = 𝐵 → (0 +_𝑒 𝐶) = (𝐵 +_𝑒 𝐶))
94	93	eqcomd 2767	. . . . 5 ⊢ (0 = 𝐵 → (𝐵 +_𝑒 𝐶) = (0 +_𝑒 𝐶))
95		xaddlid 13353	. . . . . 6 ⊢ (𝐶 ∈ ℝ^* → (0 +_𝑒 𝐶) = 𝐶)
96	4, 95	syl 18	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → (0 +_𝑒 𝐶) = 𝐶)
97	94, 96	sylan9eqr 2818	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 = 𝐵) → (𝐵 +_𝑒 𝐶) = 𝐶)
98	97	oveq2d 7428	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 = 𝐵) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = (𝐴 ·_e 𝐶))
99	86, 92, 98	3eqtr4rd 2807	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 = 𝐵) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))
100		simp2r 1219	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → 0 ≤ 𝐵)
101		xrleloe 13254	. . . 4 ⊢ ((0 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ 𝐵 ↔ (0 < 𝐵 ∨ 0 = 𝐵)))
102	32, 2, 101	sylancr 599	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → (0 ≤ 𝐵 ↔ (0 < 𝐵 ∨ 0 = 𝐵)))
103	100, 102	mpbid 235	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → (0 < 𝐵 ∨ 0 = 𝐵))
104	80, 99, 103	mpjaodan 973	1 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 class class class wbr 5103 (class class class)co 7412 ℝcr 11180 0cc0 11181 +∞cpnf 11321 -∞cmnf 11322 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 -_𝑒cxne 13219 +_𝑒 cxad 13220 ·_e cxmu 13221

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7990 df-2nd 7991 df-er 8701 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-xneg 13222 df-xadd 13223 df-xmul 13224

This theorem is used by: xadddi2r 13409

W3C validator