Theorem xaddf 13167

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 11183	. . . . . 6 ⊢ 0 ∈ ℝ*
2		pnfxr 11190	. . . . . 6 ⊢ +∞ ∈ ℝ*
3	1, 2	ifcli 4502	. . . . 5 ⊢ if(𝑦 = -∞, 0, +∞) ∈ ℝ*
4	3	a1i 11	. . . 4 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → if(𝑦 = -∞, 0, +∞) ∈ ℝ*)
5		mnfxr 11193	. . . . . . 7 ⊢ -∞ ∈ ℝ*
6	1, 5	ifcli 4502	. . . . . 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 991	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑥 = +∞ ∨ 𝑥 = -∞) ↔ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞))
11		elxr 13058	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
12		3orass 1095	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞) ↔ (𝑥 ∈ ℝ ∨ (𝑥 = +∞ ∨ 𝑥 = -∞)))
13	11, 12	sylbb 220	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ* → (𝑥 ∈ ℝ ∨ (𝑥 = +∞ ∨ 𝑥 = -∞)))
14	13	ord 870	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ* → (¬ 𝑥 ∈ ℝ → (𝑥 = +∞ ∨ 𝑥 = -∞)))
15	14	con1d 145	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ* → (¬ (𝑥 = +∞ ∨ 𝑥 = -∞) → 𝑥 ∈ ℝ))
16	15	imp 407	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ* ∧ ¬ (𝑥 = +∞ ∨ 𝑥 = -∞)) → 𝑥 ∈ ℝ)
17	10, 16	sylan2br 601	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) → 𝑥 ∈ ℝ)
18		ioran 991	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑦 = +∞ ∨ 𝑦 = -∞) ↔ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞))
19		elxr 13058	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
20		3orass 1095	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) ↔ (𝑦 ∈ ℝ ∨ (𝑦 = +∞ ∨ 𝑦 = -∞)))
21	19, 20	sylbb 220	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ* → (𝑦 ∈ ℝ ∨ (𝑦 = +∞ ∨ 𝑦 = -∞)))
22	21	ord 870	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ* → (¬ 𝑦 ∈ ℝ → (𝑦 = +∞ ∨ 𝑦 = -∞)))
23	22	con1d 145	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ* → (¬ (𝑦 = +∞ ∨ 𝑦 = -∞) → 𝑦 ∈ ℝ))
24	23	imp 407	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ* ∧ ¬ (𝑦 = +∞ ∨ 𝑦 = -∞)) → 𝑦 ∈ ℝ)
25	18, 24	sylan2br 601	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ* ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞)) → 𝑦 ∈ ℝ)
26		readdcl 11112	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
27	17, 25, 26	syl2an 602	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ (𝑦 ∈ ℝ* ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞))) → (𝑥 + 𝑦) ∈ ℝ)
28	27	rexrd 11186	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ (𝑦 ∈ ℝ* ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞))) → (𝑥 + 𝑦) ∈ ℝ*)
29	28	anassrs 468	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞)) → (𝑥 + 𝑦) ∈ ℝ*)
30	29	anassrs 468	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 + 𝑦) ∈ ℝ*)
31	9, 30	ifclda 4490	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑦 = +∞) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) ∈ ℝ*)
32	8, 31	ifclda 4490	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
33	32	an32s 658	. . . . . 6 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
34	33	anassrs 468	. . . . 5 ⊢ ((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
35	7, 34	ifclda 4490	. . . 4 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) ∈ ℝ*)
36	4, 35	ifclda 4490	. . 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 3179	. 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 13055	. . 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 8010	. 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 231	1 ⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ*

Syntax hints:  ¬ wn 3   ∧ wa 396   ∨ wo 853   ∨ w3o 1091   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ifcif 4454   × cxp 5616  ⟶wf 6481  (class class class)co 7356  ℝcr 11028  0cc0 11029   + caddc 11032  +∞cpnf 11167  -∞cmnf 11168  ℝ^*cxr 11169   +_𝑒 cxad 13052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-1cn 11087  ax-addrcl 11090  ax-rnegex 11100  ax-cnre 11102

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-pnf 11172  df-mnf 11173  df-xr 11174  df-xadd 13055

This theorem is referenced by:  xaddcl  13182  xrsadd  21365  xrofsup  32859  xrge0pluscn  34124  xrge0tmdALT  34130