xaddf - Metamath Proof Explorer

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Theorem xaddf 13154

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 11212	. . . . . 6 ⊢ 0 ∈ ℝ^*
2		pnfxr 11219	. . . . . 6 ⊢ +∞ ∈ ℝ^*
3	1, 2	ifcli 4539	. . . . 5 ⊢ if(𝑦 = -∞, 0, +∞) ∈ ℝ^*
4	3	a1i 11	. . . 4 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) ∧ 𝑥 = +∞) → if(𝑦 = -∞, 0, +∞) ∈ ℝ^)
5		mnfxr 11222	. . . . . . 7 ⊢ -∞ ∈ ℝ^*
6	1, 5	ifcli 4539	. . . . . 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 983	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑥 = +∞ ∨ 𝑥 = -∞) ↔ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞))
11		elxr 13047	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ^* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞))
12		3orass 1091	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞) ↔ (𝑥 ∈ ℝ ∨ (𝑥 = +∞ ∨ 𝑥 = -∞)))
13	11, 12	sylbb 218	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ^* → (𝑥 ∈ ℝ ∨ (𝑥 = +∞ ∨ 𝑥 = -∞)))
14	13	ord 863	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ^* → (¬ 𝑥 ∈ ℝ → (𝑥 = +∞ ∨ 𝑥 = -∞)))
15	14	con1d 145	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ^* → (¬ (𝑥 = +∞ ∨ 𝑥 = -∞) → 𝑥 ∈ ℝ))
16	15	imp 408	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ^* ∧ ¬ (𝑥 = +∞ ∨ 𝑥 = -∞)) → 𝑥 ∈ ℝ)
17	10, 16	sylan2br 596	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ^* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) → 𝑥 ∈ ℝ)
18		ioran 983	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑦 = +∞ ∨ 𝑦 = -∞) ↔ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞))
19		elxr 13047	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ^* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
20		3orass 1091	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) ↔ (𝑦 ∈ ℝ ∨ (𝑦 = +∞ ∨ 𝑦 = -∞)))
21	19, 20	sylbb 218	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ^* → (𝑦 ∈ ℝ ∨ (𝑦 = +∞ ∨ 𝑦 = -∞)))
22	21	ord 863	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ^* → (¬ 𝑦 ∈ ℝ → (𝑦 = +∞ ∨ 𝑦 = -∞)))
23	22	con1d 145	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ^* → (¬ (𝑦 = +∞ ∨ 𝑦 = -∞) → 𝑦 ∈ ℝ))
24	23	imp 408	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ^* ∧ ¬ (𝑦 = +∞ ∨ 𝑦 = -∞)) → 𝑦 ∈ ℝ)
25	18, 24	sylan2br 596	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ^* ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞)) → 𝑦 ∈ ℝ)
26		readdcl 11144	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
27	17, 25, 26	syl2an 597	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ^* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ (𝑦 ∈ ℝ^* ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞))) → (𝑥 + 𝑦) ∈ ℝ)
28	27	rexrd 11215	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ^* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ (𝑦 ∈ ℝ^* ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞))) → (𝑥 + 𝑦) ∈ ℝ^*)
29	28	anassrs 469	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ^* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ^) ∧ (¬ 𝑦 = +∞ ∧ ¬ 𝑦 = -∞)) → (𝑥 + 𝑦) ∈ ℝ^)
30	29	anassrs 469	. . . . . . . . 9 ⊢ (((((𝑥 ∈ ℝ^* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ^) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 + 𝑦) ∈ ℝ^)
31	9, 30	ifclda 4527	. . . . . . . 8 ⊢ ((((𝑥 ∈ ℝ^* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ^) ∧ ¬ 𝑦 = +∞) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) ∈ ℝ^)
32	8, 31	ifclda 4527	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ^* ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) ∧ 𝑦 ∈ ℝ^) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ^)
33	32	an32s 651	. . . . . 6 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) ∧ (¬ 𝑥 = +∞ ∧ ¬ 𝑥 = -∞)) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ^)
34	33	anassrs 469	. . . . 5 ⊢ ((((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ^)
35	7, 34	ifclda 4527	. . . 4 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^) ∧ ¬ 𝑥 = +∞) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) ∈ ℝ^)
36	4, 35	ifclda 4527	. . 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 3191	. 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 13044	. . 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 8006	. 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 229	1 ⊢ +_𝑒 :(ℝ^* × ℝ^)⟶ℝ^

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 397 ∨ wo 846 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ifcif 4492 × cxp 5637 ⟶wf 6498 (class class class)co 7363 ℝcr 11060 0cc0 11061 + caddc 11064 +∞cpnf 11196 -∞cmnf 11197 ℝ^*cxr 11198 +_𝑒 cxad 13041

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2703 ax-sep 5262 ax-nul 5269 ax-pow 5326 ax-pr 5390 ax-un 7678 ax-cnex 11117 ax-1cn 11119 ax-addrcl 11122 ax-rnegex 11132 ax-cnre 11134

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4289 df-if 4493 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4872 df-iun 4962 df-br 5112 df-opab 5174 df-mpt 5195 df-id 5537 df-xp 5645 df-rel 5646 df-cnv 5647 df-co 5648 df-dm 5649 df-rn 5650 df-res 5651 df-ima 5652 df-iota 6454 df-fun 6504 df-fn 6505 df-f 6506 df-fv 6510 df-oprab 7367 df-mpo 7368 df-1st 7927 df-2nd 7928 df-pnf 11201 df-mnf 11202 df-xr 11203 df-xadd 13044

This theorem is referenced by: xaddcl 13169 xrsadd 20852 xrofsup 31741 xrge0pluscn 32611 xrge0tmdALT 32617

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