Theorem xaddf 9936

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 8090	. . . . . . 7 ⊢ 0 ∈ ℝ*
2	1	a1i 9	. . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 0 ∈ ℝ*)
3		pnfxr 8096	. . . . . . 7 ⊢ +∞ ∈ ℝ*
4	3	a1i 9	. . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → +∞ ∈ ℝ*)
5		xrmnfdc 9935	. . . . . . 7 ⊢ (𝑦 ∈ ℝ* → DECID 𝑦 = -∞)
6	5	adantl 277	. . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → DECID 𝑦 = -∞)
7	2, 4, 6	ifcldcd 3598	. . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → if(𝑦 = -∞, 0, +∞) ∈ ℝ*)
8	7	adantr 276	. . . 4 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → if(𝑦 = -∞, 0, +∞) ∈ ℝ*)
9	1	a1i 9	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → 0 ∈ ℝ*)
10		mnfxr 8100	. . . . . . 7 ⊢ -∞ ∈ ℝ*
11	10	a1i 9	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → -∞ ∈ ℝ*)
12		xrpnfdc 9934	. . . . . . 7 ⊢ (𝑦 ∈ ℝ* → DECID 𝑦 = +∞)
13	12	ad3antlr 493	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → DECID 𝑦 = +∞)
14	9, 11, 13	ifcldcd 3598	. . . . 5 ⊢ ((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → if(𝑦 = +∞, 0, -∞) ∈ ℝ*)
15	3	a1i 9	. . . . . 6 ⊢ (((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ 𝑦 = +∞) → +∞ ∈ ℝ*)
16	10	a1i 9	. . . . . . 7 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ 𝑦 = -∞) → -∞ ∈ ℝ*)
17		simp-4r 542	. . . . . . . . . 10 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → ¬ 𝑥 = +∞)
18		simp-5l 543	. . . . . . . . . . 11 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑥 ∈ ℝ*)
19		simpllr 534	. . . . . . . . . . . 12 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → ¬ 𝑥 = -∞)
20	19	neqned 2374	. . . . . . . . . . 11 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑥 ≠ -∞)
21		xrnemnf 9869	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞) ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞))
22	21	biimpi 120	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞) → (𝑥 ∈ ℝ ∨ 𝑥 = +∞))
23	18, 20, 22	syl2anc 411	. . . . . . . . . 10 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 ∈ ℝ ∨ 𝑥 = +∞))
24	17, 23	ecased 1360	. . . . . . . . 9 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑥 ∈ ℝ)
25		simplr 528	. . . . . . . . . 10 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → ¬ 𝑦 = +∞)
26		simp-5r 544	. . . . . . . . . . 11 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑦 ∈ ℝ*)
27		neqne 2375	. . . . . . . . . . . 12 ⊢ (¬ 𝑦 = -∞ → 𝑦 ≠ -∞)
28	27	adantl 277	. . . . . . . . . . 11 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑦 ≠ -∞)
29		xrnemnf 9869	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞) ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞))
30	29	biimpi 120	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞) → (𝑦 ∈ ℝ ∨ 𝑦 = +∞))
31	26, 28, 30	syl2anc 411	. . . . . . . . . 10 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑦 ∈ ℝ ∨ 𝑦 = +∞))
32	25, 31	ecased 1360	. . . . . . . . 9 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑦 ∈ ℝ)
33	24, 32	readdcld 8073	. . . . . . . 8 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 + 𝑦) ∈ ℝ)
34	33	rexrd 8093	. . . . . . 7 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 + 𝑦) ∈ ℝ*)
35	6	ad3antrrr 492	. . . . . . 7 ⊢ (((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) → DECID 𝑦 = -∞)
36	16, 34, 35	ifcldadc 3591	. . . . . 6 ⊢ (((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) ∈ ℝ*)
37	12	ad3antlr 493	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) → DECID 𝑦 = +∞)
38	15, 36, 37	ifcldadc 3591	. . . . 5 ⊢ ((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
39		xrmnfdc 9935	. . . . . 6 ⊢ (𝑥 ∈ ℝ* → DECID 𝑥 = -∞)
40	39	ad2antrr 488	. . . . 5 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → DECID 𝑥 = -∞)
41	14, 38, 40	ifcldadc 3591	. . . 4 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) ∈ ℝ*)
42		xrpnfdc 9934	. . . . 5 ⊢ (𝑥 ∈ ℝ* → DECID 𝑥 = +∞)
43	42	adantr 276	. . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → DECID 𝑥 = +∞)
44	8, 41, 43	ifcldadc 3591	. . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))) ∈ ℝ*)
45	44	rgen2a 2551	. 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))) ∈ ℝ*
46		df-xadd 9865	. . 3 ⊢ +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
47	46	fmpo 6268	. 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*)
48	45, 47	mpbi 145	1 ⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ*

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  ∀wral 2475  ifcif 3562   × cxp 4662  ⟶wf 5255  (class class class)co 5925  ℝcr 7895  0cc0 7896   + caddc 7899  +∞cpnf 8075  -∞cmnf 8076  ℝ^*cxr 8077   +_𝑒 cxad 9862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993  ax-rnegex 8005

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-xr 8082  df-xadd 9865

This theorem is referenced by:  xaddcl  9952