Theorem xaddmnf1 13261

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 11321	. . 3 ⊢ -∞ ∈ ℝ*
2		xaddval 13256	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
3	1, 2	mpan2 689	. 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
4		ifnefalse 4545	. . 3 ⊢ (𝐴 ≠ +∞ → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))))
5		mnfnepnf 11320	. . . . . 6 ⊢ -∞ ≠ +∞
6		ifnefalse 4545	. . . . . 6 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, 0, -∞) = -∞)
7	5, 6	ax-mp 5	. . . . 5 ⊢ if(-∞ = +∞, 0, -∞) = -∞
8		ifnefalse 4545	. . . . . . 7 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞)))
9	5, 8	ax-mp 5	. . . . . 6 ⊢ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞))
10		eqid 2726	. . . . . . 7 ⊢ -∞ = -∞
11	10	iftruei 4540	. . . . . 6 ⊢ if(-∞ = -∞, -∞, (𝐴 + -∞)) = -∞
12	9, 11	eqtri 2754	. . . . 5 ⊢ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞
13		ifeq12 4551	. . . . 5 ⊢ ((if(-∞ = +∞, 0, -∞) = -∞ ∧ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞) → if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞))
14	7, 12, 13	mp2an 690	. . . 4 ⊢ if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞)
15		ifid 4573	. . . 4 ⊢ if(𝐴 = -∞, -∞, -∞) = -∞
16	14, 15	eqtri 2754	. . 3 ⊢ if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = -∞
17	4, 16	eqtrdi 2782	. 2 ⊢ (𝐴 ≠ +∞ → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = -∞)
18	3, 17	sylan9eq 2786	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ifcif 4533  (class class class)co 7424  0cc0 11158   + caddc 11161  +∞cpnf 11295  -∞cmnf 11296  ℝ^*cxr 11297   +_𝑒 cxad 13144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-mulcl 11220  ax-i2m1 11226

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6506  df-fun 6556  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-pnf 11300  df-mnf 11301  df-xr 11302  df-xadd 13147

This theorem is referenced by:  xaddnepnf  13270  xaddcom  13273  xnegdi  13281  xleadd1a  13286  xsubge0  13294  xlesubadd  13296  xadddilem  13327  xblss2ps  24398  xblss2  24399