Theorem xaddval 9469

Description: Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xaddval	⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))

Proof of Theorem xaddval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0xr 7684	. . . . . 6 ⊢ 0 ∈ ℝ*
2	1	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 0 ∈ ℝ*)
3		pnfxr 7690	. . . . . 6 ⊢ +∞ ∈ ℝ*
4	3	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → +∞ ∈ ℝ*)
5		xrmnfdc 9467	. . . . . 6 ⊢ (𝐵 ∈ ℝ* → DECID 𝐵 = -∞)
6	5	adantl 273	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → DECID 𝐵 = -∞)
7	2, 4, 6	ifcldcd 3454	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐵 = -∞, 0, +∞) ∈ ℝ*)
8	7	adantr 272	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐴 = +∞) → if(𝐵 = -∞, 0, +∞) ∈ ℝ*)
9		mnfxr 7694	. . . . . . 7 ⊢ -∞ ∈ ℝ*
10	9	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -∞ ∈ ℝ*)
11		xrpnfdc 9466	. . . . . . 7 ⊢ (𝐵 ∈ ℝ* → DECID 𝐵 = +∞)
12	11	adantl 273	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → DECID 𝐵 = +∞)
13	2, 10, 12	ifcldcd 3454	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐵 = +∞, 0, -∞) ∈ ℝ*)
14	13	ad2antrr 475	. . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → if(𝐵 = +∞, 0, -∞) ∈ ℝ*)
15	3	a1i 9	. . . . 5 ⊢ (((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ 𝐵 = +∞) → +∞ ∈ ℝ*)
16	9	a1i 9	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → -∞ ∈ ℝ*)
17		simp-4r 512	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐴 = +∞)
18		simpl 108	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ∈ ℝ*)
19	18	ad4antr 481	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ∈ ℝ*)
20		simpllr 504	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐴 = -∞)
21	20	neqned 2274	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ≠ -∞)
22		xrnemnf 9405	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
23	22	biimpi 119	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
24	19, 21, 23	syl2anc 406	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
25	17, 24	ecased 1295	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ∈ ℝ)
26		simplr 500	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐵 = +∞)
27		simpr 109	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ℝ*)
28	27	ad4antr 481	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐵 ∈ ℝ*)
29		simpr 109	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐵 = -∞)
30	29	neqned 2274	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐵 ≠ -∞)
31		xrnemnf 9405	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞) ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
32	31	biimpi 119	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
33	28, 30, 32	syl2anc 406	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
34	26, 33	ecased 1295	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐵 ∈ ℝ)
35	25, 34	readdcld 7667	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 + 𝐵) ∈ ℝ)
36	35	rexrd 7687	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 + 𝐵) ∈ ℝ*)
37	6	ad3antrrr 479	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) → DECID 𝐵 = -∞)
38	16, 36, 37	ifcldadc 3448	. . . . 5 ⊢ (((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) ∈ ℝ*)
39	12	ad2antrr 475	. . . . 5 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → DECID 𝐵 = +∞)
40	15, 38, 39	ifcldadc 3448	. . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) ∈ ℝ*)
41		xrmnfdc 9467	. . . . 5 ⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = -∞)
42	41	ad2antrr 475	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) → DECID 𝐴 = -∞)
43	14, 40, 42	ifcldadc 3448	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) → if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) ∈ ℝ*)
44		xrpnfdc 9466	. . . 4 ⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = +∞)
45	44	adantr 272	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → DECID 𝐴 = +∞)
46	8, 43, 45	ifcldadc 3448	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) ∈ ℝ*)
47		simpl 108	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
48	47	eqeq1d 2108	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = +∞ ↔ 𝐴 = +∞))
49		simpr 109	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
50	49	eqeq1d 2108	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = -∞ ↔ 𝐵 = -∞))
51	50	ifbid 3440	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = -∞, 0, +∞) = if(𝐵 = -∞, 0, +∞))
52	47	eqeq1d 2108	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = -∞ ↔ 𝐴 = -∞))
53	49	eqeq1d 2108	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = +∞ ↔ 𝐵 = +∞))
54	53	ifbid 3440	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = +∞, 0, -∞) = if(𝐵 = +∞, 0, -∞))
55		oveq12 5715	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 + 𝑦) = (𝐴 + 𝐵))
56	50, 55	ifbieq2d 3443	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
57	53, 56	ifbieq2d 3443	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
58	52, 54, 57	ifbieq12d 3445	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) = if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))))
59	48, 51, 58	ifbieq12d 3445	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))
60		df-xadd 9401	. . 3 ⊢ +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
61	59, 60	ovmpoga 5832	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))
62	46, 61	mpd3an3 1284	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 670  DECID wdc 786   = wceq 1299   ∈ wcel 1448   ≠ wne 2267  ifcif 3421  (class class class)co 5706  ℝcr 7499  0cc0 7500   + caddc 7503  +∞cpnf 7669  -∞cmnf 7670  ℝ^*cxr 7671   +_𝑒 cxad 9398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1re 7589  ax-addrcl 7592  ax-rnegex 7604

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-xadd 9401

This theorem is referenced by:  xaddpnf1  9470  xaddpnf2  9471  xaddmnf1  9472  xaddmnf2  9473  pnfaddmnf  9474  mnfaddpnf  9475  rexadd  9476