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Theorem xadddi2 13457

Description: The assumption that the multiplier be real in xadddi 13455 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 13455	. . . 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 1399	. . 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 11393	. . . . . 6 ⊢ +∞ ∈ ℝ^*
9	4	adantr 486	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 𝐶 ∈ ℝ^*)
10		xmulcl 13433	. . . . . 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 13292	. . . . . . . . 9 ⊢ 0 ≤ +∞
14		xmulge0 13444	. . . . . . . . 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 13333	. . . . . . 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 13387	. . . . 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 7453	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ·_e 𝐵) = (+∞ ·_e 𝐵))
23		oveq1 7453	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ·_e 𝐶) = (+∞ ·_e 𝐶))
24	22, 23	oveq12d 7464	. . . . 5 ⊢ (𝐴 = +∞ → ((𝐴 ·_e 𝐵) +_e (𝐴 ·_e 𝐶)) = ((+∞ ·_e 𝐵) +_e (+∞ ·_e 𝐶)))
25		xmulpnf2 13435	. . . . . . 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 7461	. . . . 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 2820	. . . 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 7453	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴 ·_e (𝐵 +_e 𝐶)) = (+∞ ·_e (𝐵 +_e 𝐶)))
30		xaddcl 13399	. . . . . . 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 11386	. . . . . . . 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 13403	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (𝐵 +_e 0) = 𝐵)
38		xleadd2a 13414	. . . . . . . . 9 ⊢ (((0 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 0 ≤ 𝐶) → (𝐵 +_e 0) ≤ (𝐵 +_e 𝐶))
39	33, 9, 34, 12, 38	syl31anc 1401	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → (𝐵 +_e 0) ≤ (𝐵 +_e 𝐶))
40	37, 39	eqbrtrrd 5133	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 𝐵 ≤ (𝐵 +_e 𝐶))
41	33, 34, 35, 36, 40	xrltletrd 13320	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → 0 < (𝐵 +_e 𝐶))
42		xmulpnf2 13435	. . . . . 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 2820	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = +∞) → (𝐴 ·_e (𝐵 +_e 𝐶)) = +∞)
45	21, 28, 44	3eqtr4rd 2809	. . 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 11396	. . . . . . 7 ⊢ -∞ ∈ ℝ^*
47		xmulcl 13433	. . . . . . 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 13429	. . . . . . . . . . . 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 13370	. . . . . . . . . . . 12 ⊢ -_e-∞ = +∞
52	51	oveq1i 7456	. . . . . . . . . . 11 ⊢ ( -_e-∞ ·_e 𝐶) = (+∞ ·_e 𝐶)
53	50, 52	eqtr3di 2813	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → -_e(-∞ ·_e 𝐶) = (+∞ ·_e 𝐶))
54		xnegpnf 13369	. . . . . . . . . . 11 ⊢ -_e+∞ = -∞
55	54	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → -_e+∞ = -∞)
56	53, 55	eqeq12d 2779	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) → ( -_e(-∞ ·_e 𝐶) = -_e+∞ ↔ (+∞ ·_e 𝐶) = -∞))
57		xneg11 13375	. . . . . . . . . 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 3003	. . . . . . 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 13389	. . . . . 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 7453	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 ·_e 𝐵) = (-∞ ·_e 𝐵))
66		oveq1 7453	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 ·_e 𝐶) = (-∞ ·_e 𝐶))
67	65, 66	oveq12d 7464	. . . . 5 ⊢ (𝐴 = -∞ → ((𝐴 ·_e 𝐵) +_e (𝐴 ·_e 𝐶)) = ((-∞ ·_e 𝐵) +_e (-∞ ·_e 𝐶)))
68		xmulmnf2 13437	. . . . . . 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 7461	. . . . 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 2820	. . . 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 7453	. . . . 5 ⊢ (𝐴 = -∞ → (𝐴 ·_e (𝐵 +_e 𝐶)) = (-∞ ·_e (𝐵 +_e 𝐶)))
73		xmulmnf2 13437	. . . . . 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 2820	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 < 𝐵) ∧ 𝐴 = -∞) → (𝐴 ·_e (𝐵 +_e 𝐶)) = -∞)
76	64, 71, 75	3eqtr4rd 2809	. . 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 13275	. . . 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 1460	. 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 13433	. . . . . 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 13402	. . . 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 7454	. . . . . 6 ⊢ (0 = 𝐵 → (𝐴 ·_e 0) = (𝐴 ·_e 𝐵))
88	87	eqcomd 2769	. . . . 5 ⊢ (0 = 𝐵 → (𝐴 ·_e 𝐵) = (𝐴 ·_e 0))
89		xmul01 13427	. . . . . 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 2820	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 = 𝐵) → (𝐴 ·_e 𝐵) = 0)
92	91	oveq1d 7461	. . 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 7453	. . . . . 6 ⊢ (0 = 𝐵 → (0 +_e 𝐶) = (𝐵 +_e 𝐶))
94	93	eqcomd 2769	. . . . 5 ⊢ (0 = 𝐵 → (𝐵 +_e 𝐶) = (0 +_e 𝐶))
95		xaddlid 13402	. . . . . 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 2820	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 ≤ 𝐶)) ∧ 0 = 𝐵) → (𝐵 +_e 𝐶) = 𝐶)
98	97	oveq2d 7462	. . 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 2809	. 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 13303	. . . 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 2147 ≠ wne 2959 class class class wbr 5106 (class class class)co 7446 ℝcr 11229 0cc0 11230 +∞cpnf 11370 -∞cmnf 11371 ℝ^*cxr 11372 < clt 11373 ≤ cle 11374 -_ecxne 13268 +_e cxad 13269 ·_e cxmu 13270

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-po 5564 df-so 5565 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-1st 8028 df-2nd 8029 df-er 8748 df-en 9005 df-dom 9006 df-sdom 9007 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-xneg 13271 df-xadd 13272 df-xmul 13273

This theorem is used by: xadddi2r 13458

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