Theorem xaddf 13118

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 11154	. . . . . 6 ⊢ 0 ∈ ℝ*
2		pnfxr 11161	. . . . . 6 ⊢ +∞ ∈ ℝ*
3	1, 2	ifcli 4518	. . . . 5 ⊢ if(𝑦 = -∞, 0, +∞) ∈ ℝ*
4	3	a1i 11	. . . 4 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → if(𝑦 = -∞, 0, +∞) ∈ ℝ*)
5		mnfxr 11164	. . . . . . 7 ⊢ -∞ ∈ ℝ*
6	1, 5	ifcli 4518	. . . . . 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 985	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑥 = +∞ ∨ 𝑥 = -∞) ↔ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞))
11		elxr 13010	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
12		3orass 1089	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞) ↔ (𝑥 ∈ ℝ ∨ (𝑥 = +∞ ∨ 𝑥 = -∞)))
13	11, 12	sylbb 219	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ* → (𝑥 ∈ ℝ ∨ (𝑥 = +∞ ∨ 𝑥 = -∞)))
14	13	ord 864	. . . . . . . . . . . . . . . 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 985	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑦 = +∞ ∨ 𝑦 = -∞) ↔ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞))
19		elxr 13010	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
20		3orass 1089	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) ↔ (𝑦 ∈ ℝ ∨ (𝑦 = +∞ ∨ 𝑦 = -∞)))
21	19, 20	sylbb 219	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ* → (𝑦 ∈ ℝ ∨ (𝑦 = +∞ ∨ 𝑦 = -∞)))
22	21	ord 864	. . . . . . . . . . . . . . . 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 11084	. . . . . . . . . . . . 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 11157	. . . . . . . . . . 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 4506	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑦 = +∞) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) ∈ ℝ*)
32	8, 31	ifclda 4506	. . . . . . 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 4506	. . . 4 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) ∈ ℝ*)
36	4, 35	ifclda 4506	. . 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 3172	. 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 13007	. . 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 7995	. 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 847   ∨ w3o 1085   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ifcif 4470   × cxp 5609  ⟶wf 6472  (class class class)co 7341  ℝcr 11000  0cc0 11001   + caddc 11004  +∞cpnf 11138  -∞cmnf 11139  ℝ^*cxr 11140   +_𝑒 cxad 13004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-1cn 11059  ax-addrcl 11062  ax-rnegex 11072  ax-cnre 11074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-pnf 11143  df-mnf 11144  df-xr 11145  df-xadd 13007

This theorem is referenced by:  xaddcl  13133  xrsadd  21317  xrofsup  32742  xrge0pluscn  33945  xrge0tmdALT  33951