Theorem xaddmnf1 13267

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

Assertion

Ref	Expression
xaddmnf1	⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)

Proof of Theorem xaddmnf1

Step	Hyp	Ref	Expression
1		mnfxr 11316	. . 3 ⊢ -∞ ∈ ℝ*
2		xaddval 13262	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
3	1, 2	mpan2 691	. 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
4		ifnefalse 4543	. . 3 ⊢ (𝐴 ≠ +∞ → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))))
5		mnfnepnf 11315	. . . . . 6 ⊢ -∞ ≠ +∞
6		ifnefalse 4543	. . . . . 6 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, 0, -∞) = -∞)
7	5, 6	ax-mp 5	. . . . 5 ⊢ if(-∞ = +∞, 0, -∞) = -∞
8		ifnefalse 4543	. . . . . . 7 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞)))
9	5, 8	ax-mp 5	. . . . . 6 ⊢ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞))
10		eqid 2735	. . . . . . 7 ⊢ -∞ = -∞
11	10	iftruei 4538	. . . . . 6 ⊢ if(-∞ = -∞, -∞, (𝐴 + -∞)) = -∞
12	9, 11	eqtri 2763	. . . . 5 ⊢ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞
13		ifeq12 4549	. . . . 5 ⊢ ((if(-∞ = +∞, 0, -∞) = -∞ ∧ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞) → if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞))
14	7, 12, 13	mp2an 692	. . . 4 ⊢ if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞)
15		ifid 4571	. . . 4 ⊢ if(𝐴 = -∞, -∞, -∞) = -∞
16	14, 15	eqtri 2763	. . 3 ⊢ if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = -∞
17	4, 16	eqtrdi 2791	. 2 ⊢ (𝐴 ≠ +∞ → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = -∞)
18	3, 17	sylan9eq 2795	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ifcif 4531  (class class class)co 7431  0cc0 11153   + caddc 11156  +∞cpnf 11290  -∞cmnf 11291  ℝ^*cxr 11292   +_𝑒 cxad 13150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-mulcl 11215  ax-i2m1 11221

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-pnf 11295  df-mnf 11296  df-xr 11297  df-xadd 13153

This theorem is referenced by:  xaddnepnf  13276  xaddcom  13279  xnegdi  13287  xleadd1a  13292  xsubge0  13300  xlesubadd  13302  xadddilem  13333  xblss2ps  24427  xblss2  24428