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Theorem xaddf 9831
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 7994 . . . . . . 7 0 ∈ ℝ*
21a1i 9 . . . . . 6 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → 0 ∈ ℝ*)
3 pnfxr 8000 . . . . . . 7 +∞ ∈ ℝ*
43a1i 9 . . . . . 6 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → +∞ ∈ ℝ*)
5 xrmnfdc 9830 . . . . . . 7 (𝑦 ∈ ℝ*DECID 𝑦 = -∞)
65adantl 277 . . . . . 6 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → DECID 𝑦 = -∞)
72, 4, 6ifcldcd 3569 . . . . 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 8004 . . . . . . 7 -∞ ∈ ℝ*
1110a1i 9 . . . . . 6 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → -∞ ∈ ℝ*)
12 xrpnfdc 9829 . . . . . . 7 (𝑦 ∈ ℝ*DECID 𝑦 = +∞)
1312ad3antlr 493 . . . . . 6 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ 𝑥 = -∞) → DECID 𝑦 = +∞)
149, 11, 13ifcldcd 3569 . . . . 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 2354 . . . . . . . . . . 11 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑥 ≠ -∞)
21 xrnemnf 9764 . . . . . . . . . . . 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 1349 . . . . . . . . 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 2355 . . . . . . . . . . . 12 𝑦 = -∞ → 𝑦 ≠ -∞)
2827adantl 277 . . . . . . . . . . 11 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑦 ≠ -∞)
29 xrnemnf 9764 . . . . . . . . . . . 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 1349 . . . . . . . . 9 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → 𝑦 ∈ ℝ)
3324, 32readdcld 7977 . . . . . . . 8 ((((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) ∧ ¬ 𝑦 = +∞) ∧ ¬ 𝑦 = -∞) → (𝑥 + 𝑦) ∈ ℝ)
3433rexrd 7997 . . . . . . 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 3563 . . . . . 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 3563 . . . . 5 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) ∧ ¬ 𝑥 = -∞) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) ∈ ℝ*)
39 xrmnfdc 9830 . . . . . 6 (𝑥 ∈ ℝ*DECID 𝑥 = -∞)
4039ad2antrr 488 . . . . 5 (((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → DECID 𝑥 = -∞)
4114, 38, 40ifcldadc 3563 . . . 4 (((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ 𝑥 = +∞) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) ∈ ℝ*)
42 xrpnfdc 9829 . . . . 5 (𝑥 ∈ ℝ*DECID 𝑥 = +∞)
4342adantr 276 . . . 4 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → DECID 𝑥 = +∞)
448, 41, 43ifcldadc 3563 . . 3 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))) ∈ ℝ*)
4544rgen2a 2531 . 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 9760 . . 3 +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
4746fmpo 6196 . 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 708  DECID wdc 834   = wceq 1353  wcel 2148  wne 2347  wral 2455  ifcif 3534   × cxp 4621  wf 5208  (class class class)co 5869  cr 7801  0cc0 7802   + caddc 7805  +∞cpnf 7979  -∞cmnf 7980  *cxr 7981   +𝑒 cxad 9757
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899  ax-rnegex 7911
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-xadd 9760
This theorem is referenced by:  xaddcl  9847
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