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Theorem rexadd 12617
Description: The extended real addition operation when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
rexadd ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))

Proof of Theorem rexadd
StepHypRef Expression
1 rexr 10680 . . 3 (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
2 rexr 10680 . . 3 (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
3 xaddval 12608 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))
41, 2, 3syl2an 598 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))
5 renepnf 10682 . . . . 5 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
6 ifnefalse 4440 . . . . 5 (𝐴 ≠ +∞ → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) = if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))))
75, 6syl 17 . . . 4 (𝐴 ∈ ℝ → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) = if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))))
8 renemnf 10683 . . . . 5 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
9 ifnefalse 4440 . . . . 5 (𝐴 ≠ -∞ → if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
108, 9syl 17 . . . 4 (𝐴 ∈ ℝ → if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
117, 10eqtrd 2836 . . 3 (𝐴 ∈ ℝ → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
12 renepnf 10682 . . . . 5 (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
13 ifnefalse 4440 . . . . 5 (𝐵 ≠ +∞ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
1412, 13syl 17 . . . 4 (𝐵 ∈ ℝ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
15 renemnf 10683 . . . . 5 (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
16 ifnefalse 4440 . . . . 5 (𝐵 ≠ -∞ → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) = (𝐴 + 𝐵))
1715, 16syl 17 . . . 4 (𝐵 ∈ ℝ → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) = (𝐴 + 𝐵))
1814, 17eqtrd 2836 . . 3 (𝐵 ∈ ℝ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = (𝐴 + 𝐵))
1911, 18sylan9eq 2856 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) = (𝐴 + 𝐵))
204, 19eqtrd 2836 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wne 2990  ifcif 4428  (class class class)co 7139  cr 10529  0cc0 10530   + caddc 10533  +∞cpnf 10665  -∞cmnf 10666  *cxr 10667   +𝑒 cxad 12497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-mulcl 10592  ax-i2m1 10598
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-xadd 12500
This theorem is referenced by:  rexsub  12618  rexaddd  12619  xnn0xaddcl  12620  xaddnemnf  12621  xaddnepnf  12622  xnegid  12623  xaddcom  12625  xaddid1  12626  xnn0xadd0  12632  xnegdi  12633  xaddass  12634  xadddilem  12679  x2times  12684  hashunx  13747  hashunsnggt  13755  isxmet2d  22938  xmeter  23044  vtxdgfival  27263  1loopgrvd2  27297  vdegp1bi  27331  xlt2addrd  30512  xrsmulgzz  30716  xrge0slmod  30972  xrge0iifhom  31294  esumfsupre  31444  esumpfinvallem  31447  omssubadd  31672  probun  31791  heicant  35091  cntotbnd  35233  heiborlem6  35253  supxrgelem  41962  supxrge  41963  infrpge  41976  xrlexaddrp  41977  ovolsplit  42623  sge0tsms  43012  sge0pr  43026  sge0resplit  43038  sge0split  43041  sge0iunmptlemfi  43045  sge0iunmptlemre  43047  sge0xaddlem1  43065  sge0xaddlem2  43066  carageniuncllem1  43153  carageniuncllem2  43154  hoidmv1lelem2  43224  hoidmvlelem2  43228  hspmbllem3  43260
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