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Theorem rexadd 13160
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 11209 . . 3 (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
2 rexr 11209 . . 3 (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
3 xaddval 13151 . . 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 597 . 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 11211 . . . . 5 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
6 ifnefalse 4502 . . . . 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 11212 . . . . 5 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
9 ifnefalse 4502 . . . . 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 2773 . . 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 11211 . . . . 5 (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
13 ifnefalse 4502 . . . . 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 11212 . . . . 5 (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
16 ifnefalse 4502 . . . . 5 (𝐵 ≠ -∞ → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) = (𝐴 + 𝐵))
1715, 16syl 17 . . . 4 (𝐵 ∈ ℝ → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) = (𝐴 + 𝐵))
1814, 17eqtrd 2773 . . 3 (𝐵 ∈ ℝ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = (𝐴 + 𝐵))
1911, 18sylan9eq 2793 . 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 2773 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2940  ifcif 4490  (class class class)co 7361  cr 11058  0cc0 11059   + caddc 11062  +∞cpnf 11194  -∞cmnf 11195  *cxr 11196   +𝑒 cxad 13039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-mulcl 11121  ax-i2m1 11127
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-xadd 13042
This theorem is referenced by:  rexsub  13161  rexaddd  13162  xnn0xaddcl  13163  xaddnemnf  13164  xaddnepnf  13165  xnegid  13166  xaddcom  13168  xaddid1  13169  xnn0xadd0  13175  xnegdi  13176  xaddass  13177  xadddilem  13222  x2times  13227  hashunx  14295  hashunsnggt  14303  isxmet2d  23703  xmeter  23809  vtxdgfival  28466  1loopgrvd2  28500  vdegp1bi  28534  xlt2addrd  31717  xrsmulgzz  31925  xrge0slmod  32194  xrge0iifhom  32582  esumfsupre  32734  esumpfinvallem  32737  omssubadd  32964  probun  33083  heicant  36163  cntotbnd  36305  heiborlem6  36325  supxrgelem  43662  supxrge  43663  infrpge  43676  xrlexaddrp  43677  ovolsplit  44319  sge0tsms  44711  sge0pr  44725  sge0resplit  44737  sge0split  44740  sge0iunmptlemfi  44744  sge0iunmptlemre  44746  sge0xaddlem1  44764  sge0xaddlem2  44765  carageniuncllem1  44852  carageniuncllem2  44853  hoidmv1lelem2  44923  hoidmvlelem2  44927  hspmbllem3  44959
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