Theorem xaddf 13266

Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
xaddf	⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ*

Proof of Theorem xaddf

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

Step	Hyp	Ref	Expression
1		0xr 11308	. . . . . 6 ⊢ 0 ∈ ℝ*
2		pnfxr 11315	. . . . . 6 ⊢ +∞ ∈ ℝ*
3	1, 2	ifcli 4573	. . . . 5 ⊢ if(𝑦 = -∞, 0, +∞) ∈ ℝ*
4	3	a1i 11	. . . 4 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → if(𝑦 = -∞, 0, +∞) ∈ ℝ*)
5		mnfxr 11318	. . . . . . 7 ⊢ -∞ ∈ ℝ*
6	1, 5	ifcli 4573	. . . . . 6 ⊢ if(𝑦 = +∞, 0, -∞) ∈ ℝ*
7	6	a1i 11	. . . . 5 ⊢ ((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → if(𝑦 = +∞, 0, -∞) ∈ ℝ*)
8	2	a1i 11	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ 𝑦 = +∞) → +∞ ∈ ℝ*)
9	5	a1i 11	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑦 = +∞) ∧ 𝑦 = -∞) → -∞ ∈ ℝ*)
10		ioran 986	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑥 = +∞ ∨ 𝑥 = -∞) ↔ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞))
11		elxr 13158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
12		3orass 1090	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞) ↔ (𝑥 ∈ ℝ ∨ (𝑥 = +∞ ∨ 𝑥 = -∞)))
13	11, 12	sylbb 219	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ* → (𝑥 ∈ ℝ ∨ (𝑥 = +∞ ∨ 𝑥 = -∞)))
14	13	ord 865	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ* → (¬ 𝑥 ∈ ℝ → (𝑥 = +∞ ∨ 𝑥 = -∞)))
15	14	con1d 145	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ* → (¬ (𝑥 = +∞ ∨ 𝑥 = -∞) → 𝑥 ∈ ℝ))
16	15	imp 406	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ* ∧ ¬ (𝑥 = +∞ ∨ 𝑥 = -∞)) → 𝑥 ∈ ℝ)
17	10, 16	sylan2br 595	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) → 𝑥 ∈ ℝ)
18		ioran 986	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑦 = +∞ ∨ 𝑦 = -∞) ↔ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞))
19		elxr 13158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
20		3orass 1090	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) ↔ (𝑦 ∈ ℝ ∨ (𝑦 = +∞ ∨ 𝑦 = -∞)))
21	19, 20	sylbb 219	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ* → (𝑦 ∈ ℝ ∨ (𝑦 = +∞ ∨ 𝑦 = -∞)))
22	21	ord 865	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ* → (¬ 𝑦 ∈ ℝ → (𝑦 = +∞ ∨ 𝑦 = -∞)))
23	22	con1d 145	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ* → (¬ (𝑦 = +∞ ∨ 𝑦 = -∞) → 𝑦 ∈ ℝ))
24	23	imp 406	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ* ∧ ¬ (𝑦 = +∞ ∨ 𝑦 = -∞)) → 𝑦 ∈ ℝ)
25	18, 24	sylan2br 595	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ* ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞)) → 𝑦 ∈ ℝ)
26		readdcl 11238	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
27	17, 25, 26	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ (𝑦 ∈ ℝ* ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞))) → (𝑥 + 𝑦) ∈ ℝ)
28	27	rexrd 11311	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ (𝑦 ∈ ℝ* ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞))) → (𝑥 + 𝑦) ∈ ℝ*)
29	28	anassrs 467	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞)) → (𝑥 + 𝑦) ∈ ℝ*)
30	29	anassrs 467	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 + 𝑦) ∈ ℝ*)
31	9, 30	ifclda 4561	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑦 = +∞) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) ∈ ℝ*)
32	8, 31	ifclda 4561	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
33	32	an32s 652	. . . . . 6 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
34	33	anassrs 467	. . . . 5 ⊢ ((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
35	7, 34	ifclda 4561	. . . 4 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) ∈ ℝ*)
36	4, 35	ifclda 4561	. . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))) ∈ ℝ*)
37	36	rgen2 3199	. 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))) ∈ ℝ*
38		df-xadd 13155	. . 3 ⊢ +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
39	38	fmpo 8093	. 2 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))) ∈ ℝ* ↔ +𝑒 :(ℝ* × ℝ*)⟶ℝ*)
40	37, 39	mpbi 230	1 ⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ*

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 848   ∨ w3o 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ifcif 4525   × cxp 5683  ⟶wf 6557  (class class class)co 7431  ℝcr 11154  0cc0 11155   + caddc 11158  +∞cpnf 11292  -∞cmnf 11293  ℝ^*cxr 11294   +_𝑒 cxad 13152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addrcl 11216  ax-rnegex 11226  ax-cnre 11228

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-pnf 11297  df-mnf 11298  df-xr 11299  df-xadd 13155

This theorem is referenced by:  xaddcl  13281  xrsadd  21397  xrofsup  32771  xrge0pluscn  33939  xrge0tmdALT  33945