Theorem xaddf 12055

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 10086	. . . . . 6 ⊢ 0 ∈ ℝ*
2		pnfxr 10092	. . . . . 6 ⊢ +∞ ∈ ℝ*
3	1, 2	keepel 4155	. . . . 5 ⊢ if(𝑦 = -∞, 0, +∞) ∈ ℝ*
4	3	a1i 11	. . . 4 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → if(𝑦 = -∞, 0, +∞) ∈ ℝ*)
5		mnfxr 10096	. . . . . . 7 ⊢ -∞ ∈ ℝ*
6	1, 5	keepel 4155	. . . . . 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 511	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑥 = +∞ ∨ 𝑥 = -∞) ↔ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞))
11		elxr 11950	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
12		3orass 1040	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞) ↔ (𝑥 ∈ ℝ ∨ (𝑥 = +∞ ∨ 𝑥 = -∞)))
13	11, 12	sylbb 209	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ* → (𝑥 ∈ ℝ ∨ (𝑥 = +∞ ∨ 𝑥 = -∞)))
14	13	ord 392	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ* → (¬ 𝑥 ∈ ℝ → (𝑥 = +∞ ∨ 𝑥 = -∞)))
15	14	con1d 139	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ* → (¬ (𝑥 = +∞ ∨ 𝑥 = -∞) → 𝑥 ∈ ℝ))
16	15	imp 445	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ* ∧ ¬ (𝑥 = +∞ ∨ 𝑥 = -∞)) → 𝑥 ∈ ℝ)
17	10, 16	sylan2br 493	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) → 𝑥 ∈ ℝ)
18		ioran 511	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑦 = +∞ ∨ 𝑦 = -∞) ↔ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞))
19		elxr 11950	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
20		3orass 1040	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) ↔ (𝑦 ∈ ℝ ∨ (𝑦 = +∞ ∨ 𝑦 = -∞)))
21	19, 20	sylbb 209	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ* → (𝑦 ∈ ℝ ∨ (𝑦 = +∞ ∨ 𝑦 = -∞)))
22	21	ord 392	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ* → (¬ 𝑦 ∈ ℝ → (𝑦 = +∞ ∨ 𝑦 = -∞)))
23	22	con1d 139	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ* → (¬ (𝑦 = +∞ ∨ 𝑦 = -∞) → 𝑦 ∈ ℝ))
24	23	imp 445	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ* ∧ ¬ (𝑦 = +∞ ∨ 𝑦 = -∞)) → 𝑦 ∈ ℝ)
25	18, 24	sylan2br 493	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ* ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞)) → 𝑦 ∈ ℝ)
26		readdcl 10019	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
27	17, 25, 26	syl2an 494	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ (𝑦 ∈ ℝ* ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞))) → (𝑥 + 𝑦) ∈ ℝ)
28	27	rexrd 10089	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ (𝑦 ∈ ℝ* ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞))) → (𝑥 + 𝑦) ∈ ℝ*)
29	28	anassrs 680	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞)) → (𝑥 + 𝑦) ∈ ℝ*)
30	29	anassrs 680	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 + 𝑦) ∈ ℝ*)
31	9, 30	ifclda 4120	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑦 = +∞) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) ∈ ℝ*)
32	8, 31	ifclda 4120	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ*) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
33	32	an32s 846	. . . . . 6 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
34	33	anassrs 680	. . . . 5 ⊢ ((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
35	7, 34	ifclda 4120	. . . 4 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) ∈ ℝ*)
36	4, 35	ifclda 4120	. . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))) ∈ ℝ*)
37	36	rgen2a 2977	. 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 11947	. . 3 ⊢ +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
39	38	fmpt2 7237	. 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 220	1 ⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ*

Syntax hints:  ¬ wn 3   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ifcif 4086   × cxp 5112  ⟶wf 5884  (class class class)co 6650  ℝcr 9935  0cc0 9936   + caddc 9939  +∞cpnf 10071  -∞cmnf 10072  ℝ^*cxr 10073   +_𝑒 cxad 11944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-pnf 10076  df-mnf 10077  df-xr 10078  df-xadd 11947

This theorem is referenced by:  xaddcl  12070  xrsadd  19763  xrofsup  29533  xrge0pluscn  29986  xrge0tmdOLD  29991