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Theorem xaddval 12005
Description: Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddval ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))

Proof of Theorem xaddval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 473 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
21eqeq1d 2623 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = +∞ ↔ 𝐴 = +∞))
3 simpr 477 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
43eqeq1d 2623 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = -∞ ↔ 𝐵 = -∞))
54ifbid 4085 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = -∞, 0, +∞) = if(𝐵 = -∞, 0, +∞))
61eqeq1d 2623 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = -∞ ↔ 𝐴 = -∞))
73eqeq1d 2623 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = +∞ ↔ 𝐵 = +∞))
87ifbid 4085 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = +∞, 0, -∞) = if(𝐵 = +∞, 0, -∞))
9 oveq12 6619 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 + 𝑦) = (𝐴 + 𝐵))
104, 9ifbieq2d 4088 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
117, 10ifbieq2d 4088 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
126, 8, 11ifbieq12d 4090 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) = if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))))
132, 5, 12ifbieq12d 4090 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))
14 df-xadd 11899 . 2 +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
15 c0ex 9986 . . . 4 0 ∈ V
16 pnfex 10045 . . . 4 +∞ ∈ V
1715, 16ifex 4133 . . 3 if(𝐵 = -∞, 0, +∞) ∈ V
18 mnfxr 10048 . . . . . 6 -∞ ∈ ℝ*
1918elexi 3202 . . . . 5 -∞ ∈ V
2015, 19ifex 4133 . . . 4 if(𝐵 = +∞, 0, -∞) ∈ V
21 ovex 6638 . . . . . 6 (𝐴 + 𝐵) ∈ V
2219, 21ifex 4133 . . . . 5 if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) ∈ V
2316, 22ifex 4133 . . . 4 if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) ∈ V
2420, 23ifex 4133 . . 3 if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) ∈ V
2517, 24ifex 4133 . 2 if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) ∈ V
2613, 14, 25ovmpt2a 6751 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  ifcif 4063  (class class class)co 6610  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-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-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  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-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-pnf 10028  df-mnf 10029  df-xr 10030  df-xadd 11899
This theorem is referenced by:  xaddpnf1  12008  xaddpnf2  12009  xaddmnf1  12010  xaddmnf2  12011  pnfaddmnf  12012  mnfaddpnf  12013  rexadd  12014
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