MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexadd Structured version   Visualization version   GIF version

Theorem rexadd 12014
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 10037 . . 3 (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
2 rexr 10037 . . 3 (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
3 xaddval 12005 . . 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 494 . 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 10039 . . . . 5 (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
6 ifnefalse 4075 . . . . 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 10040 . . . . 5 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
9 ifnefalse 4075 . . . . 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 2655 . . 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 10039 . . . . 5 (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
13 ifnefalse 4075 . . . . 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 10040 . . . . 5 (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
16 ifnefalse 4075 . . . . 5 (𝐵 ≠ -∞ → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) = (𝐴 + 𝐵))
1715, 16syl 17 . . . 4 (𝐵 ∈ ℝ → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) = (𝐴 + 𝐵))
1814, 17eqtrd 2655 . . 3 (𝐵 ∈ ℝ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = (𝐴 + 𝐵))
1911, 18sylan9eq 2675 . 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 2655 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  ifcif 4063  (class class class)co 6610  cr 9887  0cc0 9888   + caddc 9891  +∞cpnf 10023  -∞cmnf 10024  *cxr 10025   +𝑒 cxad 11896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-mulcl 9950  ax-i2m1 9956
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-xadd 11899
This theorem is referenced by:  rexsub  12015  rexaddd  12016  xnn0xaddcl  12017  xaddnemnf  12018  xaddnepnf  12019  xnegid  12020  xaddcom  12022  xaddid1  12023  xnn0xadd0  12028  xnegdi  12029  xaddass  12030  xpncan  12032  xleadd1a  12034  xadddilem  12075  x2times  12080  hashunx  13123  isxmet2d  22055  ismet2  22061  mettri2  22069  prdsxmetlem  22096  bl2in  22128  xblss2ps  22129  xmeter  22161  methaus  22248  metustexhalf  22284  metdcnlem  22562  metnrmlem3  22587  iscau3  22999  vtxdgfival  26269  1loopgrvd2  26302  vdegp1bi  26336  xlt2addrd  29390  xrsmulgzz  29487  xrge0slmod  29653  xrge0iifhom  29789  esumfsupre  29938  esumpfinvallem  29941  omssubadd  30167  probun  30286  heicant  33111  cntotbnd  33262  heiborlem6  33282  supxrgelem  39048  supxrge  39049  infrpge  39062  xrlexaddrp  39063  ovolsplit  39538  sge0tsms  39930  sge0pr  39944  sge0resplit  39956  sge0split  39959  sge0iunmptlemfi  39963  sge0iunmptlemre  39965  sge0xaddlem1  39983  sge0xaddlem2  39984  carageniuncllem1  40068  carageniuncllem2  40069  hoidmv1lelem2  40139  hoidmvlelem2  40143  hspmbllem3  40175
  Copyright terms: Public domain W3C validator