Theorem xaddval 10230

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 8366	. . . . . 6 ⊢ 0 ∈ ℝ*
2	1	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 0 ∈ ℝ*)
3		pnfxr 8372	. . . . . 6 ⊢ +∞ ∈ ℝ*
4	3	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → +∞ ∈ ℝ*)
5		xrmnfdc 10228	. . . . . 6 ⊢ (𝐵 ∈ ℝ* → DECID 𝐵 = -∞)
6	5	adantl 277	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → DECID 𝐵 = -∞)
7	2, 4, 6	ifcldcd 3678	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐵 = -∞, 0, +∞) ∈ ℝ*)
8	7	adantr 276	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐴 = +∞) → if(𝐵 = -∞, 0, +∞) ∈ ℝ*)
9		mnfxr 8376	. . . . . . 7 ⊢ -∞ ∈ ℝ*
10	9	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -∞ ∈ ℝ*)
11		xrpnfdc 10227	. . . . . . 7 ⊢ (𝐵 ∈ ℝ* → DECID 𝐵 = +∞)
12	11	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → DECID 𝐵 = +∞)
13	2, 10, 12	ifcldcd 3678	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐵 = +∞, 0, -∞) ∈ ℝ*)
14	13	ad2antrr 492	. . . 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 548	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐴 = +∞)
18		simpl 109	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ∈ ℝ*)
19	18	ad4antr 498	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ∈ ℝ*)
20		simpllr 540	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐴 = -∞)
21	20	neqned 2427	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ≠ -∞)
22		xrnemnf 10162	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
23	22	biimpi 120	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
24	19, 21, 23	syl2anc 415	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
25	17, 24	ecased 1390	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ∈ ℝ)
26		simplr 533	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐵 = +∞)
27		simpr 110	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ℝ*)
28	27	ad4antr 498	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐵 ∈ ℝ*)
29		simpr 110	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐵 = -∞)
30	29	neqned 2427	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐵 ≠ -∞)
31		xrnemnf 10162	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞) ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
32	31	biimpi 120	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
33	28, 30, 32	syl2anc 415	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
34	26, 33	ecased 1390	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐵 ∈ ℝ)
35	25, 34	readdcld 8349	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 + 𝐵) ∈ ℝ)
36	35	rexrd 8369	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 + 𝐵) ∈ ℝ*)
37	6	ad3antrrr 496	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) → DECID 𝐵 = -∞)
38	16, 36, 37	ifcldadc 3670	. . . . 5 ⊢ (((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) ∈ ℝ*)
39	12	ad2antrr 492	. . . . 5 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → DECID 𝐵 = +∞)
40	15, 38, 39	ifcldadc 3670	. . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) ∈ ℝ*)
41		xrmnfdc 10228	. . . . 5 ⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = -∞)
42	41	ad2antrr 492	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) → DECID 𝐴 = -∞)
43	14, 40, 42	ifcldadc 3670	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) → if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) ∈ ℝ*)
44		xrpnfdc 10227	. . . 4 ⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = +∞)
45	44	adantr 276	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → DECID 𝐴 = +∞)
46	8, 43, 45	ifcldadc 3670	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) ∈ ℝ*)
47		simpl 109	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
48	47	eqeq1d 2247	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = +∞ ↔ 𝐴 = +∞))
49		simpr 110	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
50	49	eqeq1d 2247	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = -∞ ↔ 𝐵 = -∞))
51	50	ifbid 3662	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = -∞, 0, +∞) = if(𝐵 = -∞, 0, +∞))
52	47	eqeq1d 2247	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = -∞ ↔ 𝐴 = -∞))
53	49	eqeq1d 2247	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = +∞ ↔ 𝐵 = +∞))
54	53	ifbid 3662	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = +∞, 0, -∞) = if(𝐵 = +∞, 0, -∞))
55		oveq12 6088	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 + 𝑦) = (𝐴 + 𝐵))
56	50, 55	ifbieq2d 3665	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
57	53, 56	ifbieq2d 3665	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
58	52, 54, 57	ifbieq12d 3667	. . . 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 3667	. . 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 10158	. . 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 6212	. 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 1379	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 104   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ifcif 3638  (class class class)co 6079  ℝcr 8172  0cc0 8173   + caddc 8176  +∞cpnf 8351  -∞cmnf 8352  ℝ^*cxr 8353   +_𝑒 cxad 10155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270  ax-rnegex 8282

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-pnf 8356  df-mnf 8357  df-xr 8358  df-xadd 10158

This theorem is referenced by:  xaddpnf1  10231  xaddpnf2  10232  xaddmnf1  10233  xaddmnf2  10234  pnfaddmnf  10235  mnfaddpnf  10236  rexadd  10237