Theorem xaddf 10177

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 8320	. . . . . . 7 ⊢ 0 ∈ ℝ*
2	1	a1i 9	. . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 0 ∈ ℝ*)
3		pnfxr 8326	. . . . . . 7 ⊢ +∞ ∈ ℝ*
4	3	a1i 9	. . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → +∞ ∈ ℝ*)
5		xrmnfdc 10176	. . . . . . 7 ⊢ (𝑦 ∈ ℝ* → DECID 𝑦 = -∞)
6	5	adantl 277	. . . . . 6 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → DECID 𝑦 = -∞)
7	2, 4, 6	ifcldcd 3660	. . . . 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 8330	. . . . . . 7 ⊢ -∞ ∈ ℝ*
11	10	a1i 9	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → -∞ ∈ ℝ*)
12		xrpnfdc 10175	. . . . . . 7 ⊢ (𝑦 ∈ ℝ* → DECID 𝑦 = +∞)
13	12	ad3antlr 493	. . . . . 6 ⊢ ((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → DECID 𝑦 = +∞)
14	9, 11, 13	ifcldcd 3660	. . . . 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 544	. . . . . . . . . 10 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → ¬ 𝑥 = +∞)
18		simp-5l 545	. . . . . . . . . . 11 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑥 ∈ ℝ*)
19		simpllr 536	. . . . . . . . . . . 12 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → ¬ 𝑥 = -∞)
20	19	neqned 2419	. . . . . . . . . . 11 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑥 ≠ -∞)
21		xrnemnf 10110	. . . . . . . . . . . 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 1386	. . . . . . . . 9 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑥 ∈ ℝ)
25		simplr 529	. . . . . . . . . 10 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → ¬ 𝑦 = +∞)
26		simp-5r 546	. . . . . . . . . . 11 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑦 ∈ ℝ*)
27		neqne 2420	. . . . . . . . . . . 12 ⊢ (¬ 𝑦 = -∞ → 𝑦 ≠ -∞)
28	27	adantl 277	. . . . . . . . . . 11 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑦 ≠ -∞)
29		xrnemnf 10110	. . . . . . . . . . . 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 1386	. . . . . . . . 9 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑦 ∈ ℝ)
33	24, 32	readdcld 8303	. . . . . . . 8 ⊢ ((((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 + 𝑦) ∈ ℝ)
34	33	rexrd 8323	. . . . . . 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 3652	. . . . . 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 3652	. . . . 5 ⊢ ((((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
39		xrmnfdc 10176	. . . . . 6 ⊢ (𝑥 ∈ ℝ* → DECID 𝑥 = -∞)
40	39	ad2antrr 488	. . . . 5 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → DECID 𝑥 = -∞)
41	14, 38, 40	ifcldadc 3652	. . . 4 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) ∈ ℝ*)
42		xrpnfdc 10175	. . . . 5 ⊢ (𝑥 ∈ ℝ* → DECID 𝑥 = +∞)
43	42	adantr 276	. . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → DECID 𝑥 = +∞)
44	8, 41, 43	ifcldadc 3652	. . 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 2596	. 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 10106	. . 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 6397	. 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 716  DECID wdc 842   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  ∀wral 2520  ifcif 3620   × cxp 4747  ⟶wf 5348  (class class class)co 6050  ℝcr 8126  0cc0 8127   + caddc 8130  +∞cpnf 8305  -∞cmnf 8306  ℝ^*cxr 8307   +_𝑒 cxad 10103

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224  ax-rnegex 8236

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-pnf 8310  df-mnf 8311  df-xr 8312  df-xadd 10106

This theorem is referenced by:  xaddcl  10193