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Theorem xadddi2 12852

Description: The assumption that the multiplier be real in xadddi 12850 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 488	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
2		simp2l 1201	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → 𝐵 ∈ ℝ^*)
3	2	ad2antrr 726	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 ∈ ℝ) → 𝐵 ∈ ℝ^*)
4		simp3l 1203	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ^*)
5	4	ad2antrr 726	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 ∈ ℝ) → 𝐶 ∈ ℝ^*)
6		xadddi 12850	. . . 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 1373	. . 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 10852	. . . . . 6 ⊢ +∞ ∈ ℝ^*
9	4	adantr 484	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 𝐶 ∈ ℝ^*)
10		xmulcl 12828	. . . . . 6 ⊢ ((+∞ ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (+∞ ·_e 𝐶) ∈ ℝ^)
11	8, 9, 10	sylancr 590	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (+∞ ·_e 𝐶) ∈ ℝ^*)
12		simpl3r 1231	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 0 ≤ 𝐶)
13		0lepnf 12689	. . . . . . . . 9 ⊢ 0 ≤ +∞
14		xmulge0 12839	. . . . . . . . 9 ⊢ (((+∞ ∈ ℝ^* ∧ 0 ≤ +∞) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → 0 ≤ (+∞ ·_e 𝐶))
15	8, 13, 14	mpanl12 702	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶) → 0 ≤ (+∞ ·_e 𝐶))
16	4, 12, 15	syl2an2r 685	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 0 ≤ (+∞ ·_e 𝐶))
17		ge0nemnf 12728	. . . . . . 7 ⊢ (((+∞ ·_e 𝐶) ∈ ℝ^* ∧ 0 ≤ (+∞ ·_e 𝐶)) → (+∞ ·_e 𝐶) ≠ -∞)
18	11, 16, 17	syl2anc 587	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (+∞ ·_e 𝐶) ≠ -∞)
19	18	adantr 484	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = +∞) → (+∞ ·_e 𝐶) ≠ -∞)
20		xaddpnf2 12782	. . . . 5 ⊢ (((+∞ ·_e 𝐶) ∈ ℝ^* ∧ (+∞ ·_e 𝐶) ≠ -∞) → (+∞ +_𝑒 (+∞ ·_e 𝐶)) = +∞)
21	11, 19, 20	syl2an2r 685	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = +∞) → (+∞ +_𝑒 (+∞ ·_e 𝐶)) = +∞)
22		oveq1 7198	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ·_e 𝐵) = (+∞ ·_e 𝐵))
23		oveq1 7198	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ·_e 𝐶) = (+∞ ·_e 𝐶))
24	22, 23	oveq12d 7209	. . . . 5 ⊢ (𝐴 = +∞ → ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) = ((+∞ ·_e 𝐵) +_𝑒 (+∞ ·_e 𝐶)))
25		xmulpnf2 12830	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → (+∞ ·_e 𝐵) = +∞)
26	2, 25	sylan 583	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (+∞ ·_e 𝐵) = +∞)
27	26	oveq1d 7206	. . . . 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 2793	. . . 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 7198	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = (+∞ ·_e (𝐵 +_𝑒 𝐶)))
30		xaddcl 12794	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
31	2, 4, 30	syl2anc 587	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → (𝐵 +_𝑒 𝐶) ∈ ℝ^*)
32		0xr 10845	. . . . . . . 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 484	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 𝐵 ∈ ℝ^*)
35	31	adantr 484	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (𝐵 +_𝑒 𝐶) ∈ ℝ^*)
36		simpr 488	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 0 < 𝐵)
37	34	xaddid1d 12798	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (𝐵 +_𝑒 0) = 𝐵)
38		xleadd2a 12809	. . . . . . . . 9 ⊢ (((0 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 0 ≤ 𝐶) → (𝐵 +_𝑒 0) ≤ (𝐵 +_𝑒 𝐶))
39	33, 9, 34, 12, 38	syl31anc 1375	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (𝐵 +_𝑒 0) ≤ (𝐵 +_𝑒 𝐶))
40	37, 39	eqbrtrrd 5063	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 𝐵 ≤ (𝐵 +_𝑒 𝐶))
41	33, 34, 35, 36, 40	xrltletrd 12716	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 0 < (𝐵 +_𝑒 𝐶))
42		xmulpnf2 12830	. . . . . 6 ⊢ (((𝐵 +_𝑒 𝐶) ∈ ℝ^* ∧ 0 < (𝐵 +_𝑒 𝐶)) → (+∞ ·_e (𝐵 +_𝑒 𝐶)) = +∞)
43	31, 41, 42	syl2an2r 685	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (+∞ ·_e (𝐵 +_𝑒 𝐶)) = +∞)
44	29, 43	sylan9eqr 2793	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = +∞) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = +∞)
45	21, 28, 44	3eqtr4rd 2782	. . 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 10855	. . . . . . 7 ⊢ -∞ ∈ ℝ^*
47		xmulcl 12828	. . . . . . 7 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (-∞ ·_e 𝐶) ∈ ℝ^)
48	46, 9, 47	sylancr 590	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-∞ ·_e 𝐶) ∈ ℝ^*)
49		xmulneg1 12824	. . . . . . . . . . . 12 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (-_𝑒-∞ ·_e 𝐶) = -_𝑒(-∞ ·_e 𝐶))
50	46, 9, 49	sylancr 590	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-_𝑒-∞ ·_e 𝐶) = -_𝑒(-∞ ·_e 𝐶))
51		xnegmnf 12765	. . . . . . . . . . . 12 ⊢ -_𝑒-∞ = +∞
52	51	oveq1i 7201	. . . . . . . . . . 11 ⊢ (-_𝑒-∞ ·_e 𝐶) = (+∞ ·_e 𝐶)
53	50, 52	eqtr3di 2786	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → -_𝑒(-∞ ·_e 𝐶) = (+∞ ·_e 𝐶))
54		xnegpnf 12764	. . . . . . . . . . 11 ⊢ -_𝑒+∞ = -∞
55	54	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → -_𝑒+∞ = -∞)
56	53, 55	eqeq12d 2752	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-_𝑒(-∞ ·_e 𝐶) = -_𝑒+∞ ↔ (+∞ ·_e 𝐶) = -∞))
57		xneg11 12770	. . . . . . . . . 10 ⊢ (((-∞ ·_e 𝐶) ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (-_𝑒(-∞ ·_e 𝐶) = -_𝑒+∞ ↔ (-∞ ·_e 𝐶) = +∞))
58	48, 8, 57	sylancl 589	. . . . . . . . 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 2976	. . . . . . 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 12784	. . . . . 6 ⊢ (((-∞ ·_e 𝐶) ∈ ℝ^* ∧ (-∞ ·_e 𝐶) ≠ +∞) → (-∞ +_𝑒 (-∞ ·_e 𝐶)) = -∞)
63	48, 61, 62	syl2anc 587	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-∞ +_𝑒 (-∞ ·_e 𝐶)) = -∞)
64	63	adantr 484	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = -∞) → (-∞ +_𝑒 (-∞ ·_e 𝐶)) = -∞)
65		oveq1 7198	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 ·_e 𝐵) = (-∞ ·_e 𝐵))
66		oveq1 7198	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 ·_e 𝐶) = (-∞ ·_e 𝐶))
67	65, 66	oveq12d 7209	. . . . 5 ⊢ (𝐴 = -∞ → ((𝐴 ·_e 𝐵) +_𝑒 (𝐴 ·_e 𝐶)) = ((-∞ ·_e 𝐵) +_𝑒 (-∞ ·_e 𝐶)))
68		xmulmnf2 12832	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → (-∞ ·_e 𝐵) = -∞)
69	2, 68	sylan 583	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-∞ ·_e 𝐵) = -∞)
70	69	oveq1d 7206	. . . . 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 2793	. . . 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 7198	. . . . 5 ⊢ (𝐴 = -∞ → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = (-∞ ·_e (𝐵 +_𝑒 𝐶)))
73		xmulmnf2 12832	. . . . . 6 ⊢ (((𝐵 +_𝑒 𝐶) ∈ ℝ^* ∧ 0 < (𝐵 +_𝑒 𝐶)) → (-∞ ·_e (𝐵 +_𝑒 𝐶)) = -∞)
74	31, 41, 73	syl2an2r 685	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (-∞ ·_e (𝐵 +_𝑒 𝐶)) = -∞)
75	72, 74	sylan9eqr 2793	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = -∞) → (𝐴 ·_e (𝐵 +_𝑒 𝐶)) = -∞)
76	64, 71, 75	3eqtr4rd 2782	. . 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 1193	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 𝐴 ∈ ℝ^*)
78		elxr 12673	. . . 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 1433	. 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 1138	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → 𝐴 ∈ ℝ^*)
82		xmulcl 12828	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 ·_e 𝐶) ∈ ℝ^)
83	81, 4, 82	syl2anc 587	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → (𝐴 ·_e 𝐶) ∈ ℝ^*)
84	83	adantr 484	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 = 𝐵) → (𝐴 ·_e 𝐶) ∈ ℝ^*)
85		xaddid2 12797	. . . 4 ⊢ ((𝐴 ·_e 𝐶) ∈ ℝ^* → (0 +_𝑒 (𝐴 ·_e 𝐶)) = (𝐴 ·_e 𝐶))
86	84, 85	syl 17	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 = 𝐵) → (0 +_𝑒 (𝐴 ·_e 𝐶)) = (𝐴 ·_e 𝐶))
87		oveq2 7199	. . . . . 6 ⊢ (0 = 𝐵 → (𝐴 ·_e 0) = (𝐴 ·_e 𝐵))
88	87	eqcomd 2742	. . . . 5 ⊢ (0 = 𝐵 → (𝐴 ·_e 𝐵) = (𝐴 ·_e 0))
89		xmul01 12822	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ·_e 0) = 0)
90	89	3ad2ant1 1135	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → (𝐴 ·_e 0) = 0)
91	88, 90	sylan9eqr 2793	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 = 𝐵) → (𝐴 ·_e 𝐵) = 0)
92	91	oveq1d 7206	. . 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 7198	. . . . . 6 ⊢ (0 = 𝐵 → (0 +_𝑒 𝐶) = (𝐵 +_𝑒 𝐶))
94	93	eqcomd 2742	. . . . 5 ⊢ (0 = 𝐵 → (𝐵 +_𝑒 𝐶) = (0 +_𝑒 𝐶))
95		xaddid2 12797	. . . . . 6 ⊢ (𝐶 ∈ ℝ^* → (0 +_𝑒 𝐶) = 𝐶)
96	4, 95	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → (0 +_𝑒 𝐶) = 𝐶)
97	94, 96	sylan9eqr 2793	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 = 𝐵) → (𝐵 +_𝑒 𝐶) = 𝐶)
98	97	oveq2d 7207	. . 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 2782	. 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 1202	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) → 0 ≤ 𝐵)
101		xrleloe 12699	. . . 4 ⊢ ((0 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ 𝐵 ↔ (0 < 𝐵 ∨ 0 = 𝐵)))
102	32, 2, 101	sylancr 590	. . 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 959	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

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 ∨ w3o 1088 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 class class class wbr 5039 (class class class)co 7191 ℝcr 10693 0cc0 10694 +∞cpnf 10829 -∞cmnf 10830 ℝ^*cxr 10831 < clt 10832 ≤ cle 10833 -_𝑒cxne 12666 +_𝑒 cxad 12667 ·_e cxmu 12668

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-xneg 12669 df-xadd 12670 df-xmul 12671

This theorem is referenced by: xadddi2r 12853

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