Theorem xaddmnf1 11933

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 9975	. . 3 ⊢ -∞ ∈ ℝ*
2		xaddval 11928	. . 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		ifnefalse 4048	. . 3 ⊢ (𝐴 ≠ +∞ → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))))
5		mnfnepnf 9974	. . . . . 6 ⊢ -∞ ≠ +∞
6		ifnefalse 4048	. . . . . 6 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, 0, -∞) = -∞)
7	5, 6	ax-mp 5	. . . . 5 ⊢ if(-∞ = +∞, 0, -∞) = -∞
8		ifnefalse 4048	. . . . . . 7 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞)))
9	5, 8	ax-mp 5	. . . . . 6 ⊢ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞))
10		eqid 2610	. . . . . . 7 ⊢ -∞ = -∞
11	10	iftruei 4043	. . . . . 6 ⊢ if(-∞ = -∞, -∞, (𝐴 + -∞)) = -∞
12	9, 11	eqtri 2632	. . . . 5 ⊢ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞
13		ifeq12 4053	. . . . 5 ⊢ ((if(-∞ = +∞, 0, -∞) = -∞ ∧ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞) → if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞))
14	7, 12, 13	mp2an 704	. . . 4 ⊢ if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞)
15		ifid 4075	. . . 4 ⊢ if(𝐴 = -∞, -∞, -∞) = -∞
16	14, 15	eqtri 2632	. . 3 ⊢ if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = -∞
17	4, 16	syl6eq 2660	. 2 ⊢ (𝐴 ≠ +∞ → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = -∞)
18	3, 17	sylan9eq 2664	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ifcif 4036  (class class class)co 6549  0cc0 9815   + caddc 9818  +∞cpnf 9950  -∞cmnf 9951  ℝ^*cxr 9952   +_𝑒 cxad 11820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-mulcl 9877  ax-i2m1 9883

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pnf 9955  df-mnf 9956  df-xr 9957  df-xadd 11823

This theorem is referenced by:  xaddnepnf  11942  xaddcom  11945  xnegdi  11950  xleadd1a  11955  xsubge0  11963  xlesubadd  11965  xadddilem  11996  xblss2ps  22016  xblss2  22017