Theorem rexadd 13159

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

Step	Hyp	Ref	Expression
1		rexr 11190	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
2		rexr 11190	. . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
3		xaddval 13150	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))
4	1, 2, 3	syl2an 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 11192	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
6		ifnefalse 4493	. . . . 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)))))
7	5, 6	syl 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 11193	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
9		ifnefalse 4493	. . . . 5 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
10	8, 9	syl 17	. . . 4 ⊢ (𝐴 ∈ ℝ → if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
11	7, 10	eqtrd 2772	. . 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 11192	. . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
13		ifnefalse 4493	. . . . 5 ⊢ (𝐵 ≠ +∞ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
14	12, 13	syl 17	. . . 4 ⊢ (𝐵 ∈ ℝ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
15		renemnf 11193	. . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
16		ifnefalse 4493	. . . . 5 ⊢ (𝐵 ≠ -∞ → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) = (𝐴 + 𝐵))
17	15, 16	syl 17	. . . 4 ⊢ (𝐵 ∈ ℝ → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) = (𝐴 + 𝐵))
18	14, 17	eqtrd 2772	. . 3 ⊢ (𝐵 ∈ ℝ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = (𝐴 + 𝐵))
19	11, 18	sylan9eq 2792	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) = (𝐴 + 𝐵))
20	4, 19	eqtrd 2772	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ifcif 4481  (class class class)co 7368  ℝcr 11037  0cc0 11038   + caddc 11041  +∞cpnf 11175  -∞cmnf 11176  ℝ^*cxr 11177   +_𝑒 cxad 13036

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-mulcl 11100  ax-i2m1 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-xadd 13039

This theorem is referenced by:  rexsub  13160  rexaddd  13161  xnn0xaddcl  13162  xaddnemnf  13163  xaddnepnf  13164  xnegid  13165  xaddcom  13167  xaddrid  13168  xnn0xadd0  13174  xnegdi  13175  xaddass  13176  xadddilem  13221  x2times  13226  hashunx  14321  hashunsnggt  14329  isxmet2d  24283  xmeter  24389  vtxdgfival  29555  1loopgrvd2  29589  vdegp1bi  29623  xlt2addrd  32849  xrsmulgzz  33101  xrge0slmod  33440  xrge0iifhom  34114  esumfsupre  34248  esumpfinvallem  34251  omssubadd  34477  probun  34596  heicant  37900  cntotbnd  38041  heiborlem6  38061  supxrgelem  45690  supxrge  45691  infrpge  45704  xrlexaddrp  45705  ovolsplit  46340  sge0tsms  46732  sge0pr  46746  sge0resplit  46758  sge0split  46761  sge0iunmptlemfi  46765  sge0iunmptlemre  46767  sge0xaddlem1  46785  sge0xaddlem2  46786  carageniuncllem1  46873  carageniuncllem2  46874  hoidmv1lelem2  46944  hoidmvlelem2  46948  hspmbllem3  46980