Theorem xaddval 9980

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 8132	. . . . . 6 ⊢ 0 ∈ ℝ*
2	1	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 0 ∈ ℝ*)
3		pnfxr 8138	. . . . . 6 ⊢ +∞ ∈ ℝ*
4	3	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → +∞ ∈ ℝ*)
5		xrmnfdc 9978	. . . . . 6 ⊢ (𝐵 ∈ ℝ* → DECID 𝐵 = -∞)
6	5	adantl 277	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → DECID 𝐵 = -∞)
7	2, 4, 6	ifcldcd 3610	. . . 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 8142	. . . . . . 7 ⊢ -∞ ∈ ℝ*
10	9	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -∞ ∈ ℝ*)
11		xrpnfdc 9977	. . . . . . 7 ⊢ (𝐵 ∈ ℝ* → DECID 𝐵 = +∞)
12	11	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → DECID 𝐵 = +∞)
13	2, 10, 12	ifcldcd 3610	. . . . 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 2384	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ≠ -∞)
22		xrnemnf 9912	. . . . . . . . . . 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 1362	. . . . . . . 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 2384	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐵 ≠ -∞)
31		xrnemnf 9912	. . . . . . . . . . 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 1362	. . . . . . . 8 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐵 ∈ ℝ)
35	25, 34	readdcld 8115	. . . . . . 7 ⊢ ((((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 + 𝐵) ∈ ℝ)
36	35	rexrd 8135	. . . . . 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 3602	. . . . 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 3602	. . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) ∈ ℝ*)
41		xrmnfdc 9978	. . . . 5 ⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = -∞)
42	41	ad2antrr 488	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) → DECID 𝐴 = -∞)
43	14, 40, 42	ifcldadc 3602	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) → if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) ∈ ℝ*)
44		xrpnfdc 9977	. . . 4 ⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = +∞)
45	44	adantr 276	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → DECID 𝐴 = +∞)
46	8, 43, 45	ifcldadc 3602	. 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 2215	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = +∞ ↔ 𝐴 = +∞))
49		simpr 110	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
50	49	eqeq1d 2215	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = -∞ ↔ 𝐵 = -∞))
51	50	ifbid 3594	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = -∞, 0, +∞) = if(𝐵 = -∞, 0, +∞))
52	47	eqeq1d 2215	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = -∞ ↔ 𝐴 = -∞))
53	49	eqeq1d 2215	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = +∞ ↔ 𝐵 = +∞))
54	53	ifbid 3594	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = +∞, 0, -∞) = if(𝐵 = +∞, 0, -∞))
55		oveq12 5963	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 + 𝑦) = (𝐴 + 𝐵))
56	50, 55	ifbieq2d 3597	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
57	53, 56	ifbieq2d 3597	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
58	52, 54, 57	ifbieq12d 3599	. . . 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 3599	. . 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 9908	. . 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 6085	. 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 1351	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 710  DECID wdc 836   = wceq 1373   ∈ wcel 2177   ≠ wne 2377  ifcif 3573  (class class class)co 5954  ℝcr 7937  0cc0 7938   + caddc 7941  +∞cpnf 8117  -∞cmnf 8118  ℝ^*cxr 8119   +_𝑒 cxad 9905

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-1re 8032  ax-addrcl 8035  ax-rnegex 8047

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3001  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-opab 4111  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-iota 5238  df-fun 5279  df-fv 5285  df-ov 5957  df-oprab 5958  df-mpo 5959  df-pnf 8122  df-mnf 8123  df-xr 8124  df-xadd 9908

This theorem is referenced by:  xaddpnf1  9981  xaddpnf2  9982  xaddmnf1  9983  xaddmnf2  9984  pnfaddmnf  9985  mnfaddpnf  9986  rexadd  9987