Theorem xaddmnf1 9861

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 8027	. . . 4 ⊢ -∞ ∈ ℝ*
2		xaddval 9858	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
3	1, 2	mpan2 425	. . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
4	3	adantr 276	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
5		ifnefalse 3557	. . 3 ⊢ (𝐴 ≠ +∞ → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))))
6		mnfnepnf 8026	. . . . . 6 ⊢ -∞ ≠ +∞
7		ifnefalse 3557	. . . . . 6 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, 0, -∞) = -∞)
8	6, 7	ax-mp 5	. . . . 5 ⊢ if(-∞ = +∞, 0, -∞) = -∞
9		ifnefalse 3557	. . . . . . 7 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞)))
10	6, 9	ax-mp 5	. . . . . 6 ⊢ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞))
11		eqid 2187	. . . . . . 7 ⊢ -∞ = -∞
12	11	iftruei 3552	. . . . . 6 ⊢ if(-∞ = -∞, -∞, (𝐴 + -∞)) = -∞
13	10, 12	eqtri 2208	. . . . 5 ⊢ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞
14		ifeq12 3562	. . . . 5 ⊢ ((if(-∞ = +∞, 0, -∞) = -∞ ∧ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞) → if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞))
15	8, 13, 14	mp2an 426	. . . 4 ⊢ if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞)
16		xrmnfdc 9856	. . . . 5 ⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = -∞)
17		ifiddc 3580	. . . . 5 ⊢ (DECID 𝐴 = -∞ → if(𝐴 = -∞, -∞, -∞) = -∞)
18	16, 17	syl 14	. . . 4 ⊢ (𝐴 ∈ ℝ* → if(𝐴 = -∞, -∞, -∞) = -∞)
19	15, 18	eqtrid 2232	. . 3 ⊢ (𝐴 ∈ ℝ* → if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = -∞)
20	5, 19	sylan9eqr 2242	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = -∞)
21	4, 20	eqtrd 2220	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 835   = wceq 1363   ∈ wcel 2158   ≠ wne 2357  ifcif 3546  (class class class)co 5888  0cc0 7824   + caddc 7827  +∞cpnf 8002  -∞cmnf 8003  ℝ^*cxr 8004   +_𝑒 cxad 9783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921  ax-rnegex 7933

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-xadd 9786

This theorem is referenced by:  xaddnepnf  9871  xaddcom  9874  xnegdi  9881  xleadd1a  9886  xsubge0  9894  xposdif  9895  xlesubadd  9896  xleaddadd  9900  xblss2ps  14144  xblss2  14145