Theorem xaddpnf1 13240

Description: Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xaddpnf1	⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)

Proof of Theorem xaddpnf1

Step	Hyp	Ref	Expression
1		pnfxr 11251	. . 3 ⊢ +∞ ∈ ℝ*
2		xaddval 13237	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))))
3	1, 2	mpan2 703	. 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))))
4		pnfnemnf 11252	. . . . 5 ⊢ +∞ ≠ -∞
5		ifnefalse 4495	. . . . 5 ⊢ (+∞ ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
6	4, 5	mp1i 14	. . . 4 ⊢ (𝐴 ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
7		ifnefalse 4495	. . . . 5 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))
8		eqid 2765	. . . . . 6 ⊢ +∞ = +∞
9	8	iftruei 4490	. . . . 5 ⊢ if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))) = +∞
10	7, 9	eqtrdi 2816	. . . 4 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = +∞)
11	6, 10	ifeq12d 4505	. . 3 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = if(𝐴 = +∞, +∞, +∞))
12		ifid 4524	. . 3 ⊢ if(𝐴 = +∞, +∞, +∞) = +∞
13	11, 12	eqtrdi 2816	. 2 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = +∞)
14	3, 13	sylan9eq 2820	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ifcif 4483  (class class class)co 7400  0cc0 11088   + caddc 11091  +∞cpnf 11228  -∞cmnf 11229  ℝ^*cxr 11230   +_𝑒 cxad 13123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-cnex 11144  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-mulcl 11150  ax-i2m1 11156

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-pnf 11233  df-mnf 11234  df-xr 11235  df-xadd 13126

This theorem is referenced by:  xnn0xaddcl  13249  xaddnemnf  13250  xaddcom  13254  xnn0xadd0  13261  xnegdi  13262  xaddass  13263  xleadd1a  13267  xlt2add  13274  xsubge0  13275  xlesubadd  13277  xadddilem  13308  xrsdsreclblem  21520  isxmet2d  24441  xrge0iifhom  34239  esumpr2  34369  hasheuni  34387  carsgclctunlem2  34621  ovolsplit  46561  sge0pr  46967  sge0split  46982  sge0xadd  47008