Theorem xaddpnf1 13178

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 11199	. . 3 ⊢ +∞ ∈ ℝ*
2		xaddval 13175	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))))
3	1, 2	mpan2 692	. 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))))
4		pnfnemnf 11200	. . . . 5 ⊢ +∞ ≠ -∞
5		ifnefalse 4478	. . . . 5 ⊢ (+∞ ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
6	4, 5	mp1i 13	. . . 4 ⊢ (𝐴 ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
7		ifnefalse 4478	. . . . 5 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))
8		eqid 2736	. . . . . 6 ⊢ +∞ = +∞
9	8	iftruei 4473	. . . . 5 ⊢ if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))) = +∞
10	7, 9	eqtrdi 2787	. . . 4 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = +∞)
11	6, 10	ifeq12d 4488	. . 3 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = if(𝐴 = +∞, +∞, +∞))
12		ifid 4507	. . 3 ⊢ if(𝐴 = +∞, +∞, +∞) = +∞
13	11, 12	eqtrdi 2787	. 2 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = +∞)
14	3, 13	sylan9eq 2791	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ifcif 4466  (class class class)co 7367  0cc0 11038   + caddc 11041  +∞cpnf 11176  -∞cmnf 11177  ℝ^*cxr 11178   +_𝑒 cxad 13061

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 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-pnf 11181  df-mnf 11182  df-xr 11183  df-xadd 13064

This theorem is referenced by:  xnn0xaddcl  13187  xaddnemnf  13188  xaddcom  13192  xnn0xadd0  13199  xnegdi  13200  xaddass  13201  xleadd1a  13205  xlt2add  13212  xsubge0  13213  xlesubadd  13215  xadddilem  13246  xrsdsreclblem  21393  isxmet2d  24292  xrge0iifhom  34081  esumpr2  34211  hasheuni  34229  carsgclctunlem2  34463  ovolsplit  46416  sge0pr  46822  sge0split  46837  sge0xadd  46863