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

Theorem xaddf 9876
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.
StepHypRef Expression
1 0xr 8035 . . . . . . 7 0 ∈ ℝ*
21a1i 9 . . . . . 6 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → 0 ∈ ℝ*)
3 pnfxr 8041 . . . . . . 7 +∞ ∈ ℝ*
43a1i 9 . . . . . 6 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → +∞ ∈ ℝ*)
5 xrmnfdc 9875 . . . . . . 7 (𝑦 ∈ ℝ*DECID 𝑦 = -∞)
65adantl 277 . . . . . 6 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → DECID 𝑦 = -∞)
72, 4, 6ifcldcd 3585 . . . . 5 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → if(𝑦 = -∞, 0, +∞) ∈ ℝ*)
87adantr 276 . . . 4 (((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ 𝑥 = +∞) → if(𝑦 = -∞, 0, +∞) ∈ ℝ*)
91a1i 9 . . . . . 6 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → 0 ∈ ℝ*)
10 mnfxr 8045 . . . . . . 7 -∞ ∈ ℝ*
1110a1i 9 . . . . . 6 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → -∞ ∈ ℝ*)
12 xrpnfdc 9874 . . . . . . 7 (𝑦 ∈ ℝ*DECID 𝑦 = +∞)
1312ad3antlr 493 . . . . . 6 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → DECID 𝑦 = +∞)
149, 11, 13ifcldcd 3585 . . . . 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 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 = -∞)
2019neqned 2367 . . . . . . . . . . 11 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑥 ≠ -∞)
21 xrnemnf 9809 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ*𝑥 ≠ -∞) ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞))
2221biimpi 120 . . . . . . . . . . 11 ((𝑥 ∈ ℝ*𝑥 ≠ -∞) → (𝑥 ∈ ℝ ∨ 𝑥 = +∞))
2318, 20, 22syl2anc 411 . . . . . . . . . 10 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 ∈ ℝ ∨ 𝑥 = +∞))
2417, 23ecased 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 2368 . . . . . . . . . . . 12 𝑦 = -∞ → 𝑦 ≠ -∞)
2827adantl 277 . . . . . . . . . . 11 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑦 ≠ -∞)
29 xrnemnf 9809 . . . . . . . . . . . 12 ((𝑦 ∈ ℝ*𝑦 ≠ -∞) ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞))
3029biimpi 120 . . . . . . . . . . 11 ((𝑦 ∈ ℝ*𝑦 ≠ -∞) → (𝑦 ∈ ℝ ∨ 𝑦 = +∞))
3126, 28, 30syl2anc 411 . . . . . . . . . 10 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑦 ∈ ℝ ∨ 𝑦 = +∞))
3225, 31ecased 1360 . . . . . . . . 9 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑦 ∈ ℝ)
3324, 32readdcld 8018 . . . . . . . 8 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 + 𝑦) ∈ ℝ)
3433rexrd 8038 . . . . . . 7 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 + 𝑦) ∈ ℝ*)
356ad3antrrr 492 . . . . . . 7 (((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) → DECID 𝑦 = -∞)
3616, 34, 35ifcldadc 3578 . . . . . 6 (((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) ∈ ℝ*)
3712ad3antlr 493 . . . . . 6 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) → DECID 𝑦 = +∞)
3815, 36, 37ifcldadc 3578 . . . . 5 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
39 xrmnfdc 9875 . . . . . 6 (𝑥 ∈ ℝ*DECID 𝑥 = -∞)
4039ad2antrr 488 . . . . 5 (((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → DECID 𝑥 = -∞)
4114, 38, 40ifcldadc 3578 . . . 4 (((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) ∈ ℝ*)
42 xrpnfdc 9874 . . . . 5 (𝑥 ∈ ℝ*DECID 𝑥 = +∞)
4342adantr 276 . . . 4 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → DECID 𝑥 = +∞)
448, 41, 43ifcldadc 3578 . . 3 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))) ∈ ℝ*)
4544rgen2a 2544 . 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 9805 . . 3 +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
4746fmpo 6227 . 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 145 1 +𝑒 :(ℝ* × ℝ*)⟶ℝ*
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wo 709  DECID wdc 835   = wceq 1364  wcel 2160  wne 2360  wral 2468  ifcif 3549   × cxp 4642  wf 5231  (class class class)co 5897  cr 7841  0cc0 7842   + caddc 7845  +∞cpnf 8020  -∞cmnf 8021  *cxr 8022   +𝑒 cxad 9802
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1re 7936  ax-addrcl 7939  ax-rnegex 7951
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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-pnf 8025  df-mnf 8026  df-xr 8027  df-xadd 9805
This theorem is referenced by:  xaddcl  9892
  Copyright terms: Public domain W3C validator