Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  xaddf GIF version

 Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0xr 7832 . . . . . . 7 0 ∈ ℝ*
21a1i 9 . . . . . 6 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → 0 ∈ ℝ*)
3 pnfxr 7838 . . . . . . 7 +∞ ∈ ℝ*
43a1i 9 . . . . . 6 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → +∞ ∈ ℝ*)
5 xrmnfdc 9652 . . . . . . 7 (𝑦 ∈ ℝ*DECID 𝑦 = -∞)
65adantl 275 . . . . . 6 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → DECID 𝑦 = -∞)
72, 4, 6ifcldcd 3508 . . . . 5 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → if(𝑦 = -∞, 0, +∞) ∈ ℝ*)
87adantr 274 . . . 4 (((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → if(𝑦 = -∞, 0, +∞) ∈ ℝ*)
91a1i 9 . . . . . 6 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → 0 ∈ ℝ*)
10 mnfxr 7842 . . . . . . 7 -∞ ∈ ℝ*
1110a1i 9 . . . . . 6 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → -∞ ∈ ℝ*)
12 xrpnfdc 9651 . . . . . . 7 (𝑦 ∈ ℝ*DECID 𝑦 = +∞)
1312ad3antlr 485 . . . . . 6 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → DECID 𝑦 = +∞)
149, 11, 13ifcldcd 3508 . . . . 5 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → if(𝑦 = +∞, 0, -∞) ∈ ℝ*)
153a1i 9 . . . . . 6 (((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ 𝑦 = +∞) → +∞ ∈ ℝ*)
1610a1i 9 . . . . . . 7 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ 𝑦 = -∞) → -∞ ∈ ℝ*)
17 simp-4r 532 . . . . . . . . . 10 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → ¬ 𝑥 = +∞)
18 simp-5l 533 . . . . . . . . . . 11 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑥 ∈ ℝ*)
19 simpllr 524 . . . . . . . . . . . 12 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → ¬ 𝑥 = -∞)
2019neqned 2316 . . . . . . . . . . 11 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑥 ≠ -∞)
21 xrnemnf 9590 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ*𝑥 ≠ -∞) ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞))
2221biimpi 119 . . . . . . . . . . 11 ((𝑥 ∈ ℝ*𝑥 ≠ -∞) → (𝑥 ∈ ℝ ∨ 𝑥 = +∞))
2318, 20, 22syl2anc 409 . . . . . . . . . 10 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 ∈ ℝ ∨ 𝑥 = +∞))
2417, 23ecased 1328 . . . . . . . . 9 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑥 ∈ ℝ)
25 simplr 520 . . . . . . . . . 10 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → ¬ 𝑦 = +∞)
26 simp-5r 534 . . . . . . . . . . 11 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑦 ∈ ℝ*)
27 neqne 2317 . . . . . . . . . . . 12 𝑦 = -∞ → 𝑦 ≠ -∞)
2827adantl 275 . . . . . . . . . . 11 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑦 ≠ -∞)
29 xrnemnf 9590 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ*𝑦 ≠ -∞) ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞))
3029biimpi 119 . . . . . . . . . . 11 ((𝑦 ∈ ℝ*𝑦 ≠ -∞) → (𝑦 ∈ ℝ ∨ 𝑦 = +∞))
3126, 28, 30syl2anc 409 . . . . . . . . . 10 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑦 ∈ ℝ ∨ 𝑦 = +∞))
3225, 31ecased 1328 . . . . . . . . 9 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑦 ∈ ℝ)
3324, 32readdcld 7815 . . . . . . . 8 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 + 𝑦) ∈ ℝ)
3433rexrd 7835 . . . . . . 7 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 + 𝑦) ∈ ℝ*)
356ad3antrrr 484 . . . . . . 7 (((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) → DECID 𝑦 = -∞)
3616, 34, 35ifcldadc 3502 . . . . . 6 (((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) ∈ ℝ*)
3712ad3antlr 485 . . . . . 6 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) → DECID 𝑦 = +∞)
3815, 36, 37ifcldadc 3502 . . . . 5 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
39 xrmnfdc 9652 . . . . . 6 (𝑥 ∈ ℝ*DECID 𝑥 = -∞)
4039ad2antrr 480 . . . . 5 (((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → DECID 𝑥 = -∞)
4114, 38, 40ifcldadc 3502 . . . 4 (((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) ∈ ℝ*)
42 xrpnfdc 9651 . . . . 5 (𝑥 ∈ ℝ*DECID 𝑥 = +∞)
4342adantr 274 . . . 4 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → DECID 𝑥 = +∞)
448, 41, 43ifcldadc 3502 . . 3 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))) ∈ ℝ*)
4544rgen2a 2487 . 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 9586 . . 3 +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
4746fmpo 6103 . 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*)
4845, 47mpbi 144 1 +𝑒 :(ℝ* × ℝ*)⟶ℝ*
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 698  DECID wdc 820   = wceq 1332   ∈ wcel 1481   ≠ wne 2309  ∀wral 2417  ifcif 3475   × cxp 4541  ⟶wf 5123  (class class class)co 5778  ℝcr 7639  0cc0 7640   + caddc 7643  +∞cpnf 7817  -∞cmnf 7818  ℝ*cxr 7819   +𝑒 cxad 9583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-cnex 7731  ax-resscn 7732  ax-1re 7734  ax-addrcl 7737  ax-rnegex 7749 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-if 3476  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-fv 5135  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-pnf 7822  df-mnf 7823  df-xr 7824  df-xadd 9586 This theorem is referenced by:  xaddcl  9669
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