Theorem xaddval 9937

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 8090	. . . . . 6 ⊢ 0 ∈ ℝ*
2	1	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 0 ∈ ℝ*)
3		pnfxr 8096	. . . . . 6 ⊢ +∞ ∈ ℝ*
4	3	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → +∞ ∈ ℝ*)
5		xrmnfdc 9935	. . . . . 6 ⊢ (𝐵 ∈ ℝ* → DECID 𝐵 = -∞)
6	5	adantl 277	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → DECID 𝐵 = -∞)
7	2, 4, 6	ifcldcd 3598	. . . 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 8100	. . . . . . 7 ⊢ -∞ ∈ ℝ*
10	9	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -∞ ∈ ℝ*)
11		xrpnfdc 9934	. . . . . . 7 ⊢ (𝐵 ∈ ℝ* → DECID 𝐵 = +∞)
12	11	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → DECID 𝐵 = +∞)
13	2, 10, 12	ifcldcd 3598	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐵 = +∞, 0, -∞) ∈ ℝ*)
14	13	ad2antrr 488	. . . 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 542	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐴 = +∞)
18		simpl 109	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ∈ ℝ*)
19	18	ad4antr 494	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ∈ ℝ*)
20		simpllr 534	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐴 = -∞)
21	20	neqned 2374	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ≠ -∞)
22		xrnemnf 9869	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
23	22	biimpi 120	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
24	19, 21, 23	syl2anc 411	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
25	17, 24	ecased 1360	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ∈ ℝ)
26		simplr 528	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐵 = +∞)
27		simpr 110	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ℝ*)
28	27	ad4antr 494	. . . . . . . . . 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 2374	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐵 ≠ -∞)
31		xrnemnf 9869	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞) ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
32	31	biimpi 120	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
33	28, 30, 32	syl2anc 411	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
34	26, 33	ecased 1360	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐵 ∈ ℝ)
35	25, 34	readdcld 8073	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 + 𝐵) ∈ ℝ)
36	35	rexrd 8093	. . . . . 6 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 + 𝐵) ∈ ℝ*)
37	6	ad3antrrr 492	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) → DECID 𝐵 = -∞)
38	16, 36, 37	ifcldadc 3591	. . . . 5 ⊢ (((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) ∈ ℝ*)
39	12	ad2antrr 488	. . . . 5 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → DECID 𝐵 = +∞)
40	15, 38, 39	ifcldadc 3591	. . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) ∈ ℝ*)
41		xrmnfdc 9935	. . . . 5 ⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = -∞)
42	41	ad2antrr 488	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) → DECID 𝐴 = -∞)
43	14, 40, 42	ifcldadc 3591	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) → if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) ∈ ℝ*)
44		xrpnfdc 9934	. . . 4 ⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = +∞)
45	44	adantr 276	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → DECID 𝐴 = +∞)
46	8, 43, 45	ifcldadc 3591	. 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 2205	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = +∞ ↔ 𝐴 = +∞))
49		simpr 110	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
50	49	eqeq1d 2205	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = -∞ ↔ 𝐵 = -∞))
51	50	ifbid 3583	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = -∞, 0, +∞) = if(𝐵 = -∞, 0, +∞))
52	47	eqeq1d 2205	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = -∞ ↔ 𝐴 = -∞))
53	49	eqeq1d 2205	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = +∞ ↔ 𝐵 = +∞))
54	53	ifbid 3583	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = +∞, 0, -∞) = if(𝐵 = +∞, 0, -∞))
55		oveq12 5934	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 + 𝑦) = (𝐴 + 𝐵))
56	50, 55	ifbieq2d 3586	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
57	53, 56	ifbieq2d 3586	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
58	52, 54, 57	ifbieq12d 3588	. . . 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 3588	. . 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 9865	. . 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 6056	. 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 1349	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 709  DECID wdc 835   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  ifcif 3562  (class class class)co 5925  ℝcr 7895  0cc0 7896   + caddc 7899  +∞cpnf 8075  -∞cmnf 8076  ℝ^*cxr 8077   +_𝑒 cxad 9862

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993  ax-rnegex 8005

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-xadd 9865

This theorem is referenced by:  xaddpnf1  9938  xaddpnf2  9939  xaddmnf1  9940  xaddmnf2  9941  pnfaddmnf  9942  mnfaddpnf  9943  rexadd  9944