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Theorem xmulpnf1n 12851

Description: Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xmulpnf1n	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (𝐴 ·_e +∞) = -∞)

**Proof of Theorem xmulpnf1n**
Step	Hyp	Ref	Expression
1		simpl 486	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → 𝐴 ∈ ℝ^*)
2		pnfxr 10870	. . . . 5 ⊢ +∞ ∈ ℝ^*
3		xmulneg1 12842	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (-_𝑒𝐴 ·_e +∞) = -_𝑒(𝐴 ·_e +∞))
4	1, 2, 3	sylancl 589	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e +∞) = -_𝑒(𝐴 ·_e +∞))
5		xnegcl 12786	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒𝐴 ∈ ℝ^*)
6		xlt0neg1 12792	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 < 0 ↔ 0 < -_𝑒𝐴))
7	6	biimpa 480	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → 0 < -_𝑒𝐴)
8		xmulpnf1 12847	. . . . 5 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ 0 < -_𝑒𝐴) → (-_𝑒𝐴 ·_e +∞) = +∞)
9	5, 7, 8	syl2an2r 685	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e +∞) = +∞)
10	4, 9	eqtr3d 2776	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → -_𝑒(𝐴 ·_e +∞) = +∞)
11		xnegmnf 12783	. . 3 ⊢ -_𝑒-∞ = +∞
12	10, 11	eqtr4di 2792	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → -_𝑒(𝐴 ·_e +∞) = -_𝑒-∞)
13		xmulcl 12846	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝐴 ·_e +∞) ∈ ℝ^)
14	1, 2, 13	sylancl 589	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (𝐴 ·_e +∞) ∈ ℝ^*)
15		mnfxr 10873	. . 3 ⊢ -∞ ∈ ℝ^*
16		xneg11 12788	. . 3 ⊢ (((𝐴 ·_e +∞) ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → (-_𝑒(𝐴 ·_e +∞) = -_𝑒-∞ ↔ (𝐴 ·_e +∞) = -∞))
17	14, 15, 16	sylancl 589	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (-_𝑒(𝐴 ·_e +∞) = -_𝑒-∞ ↔ (𝐴 ·_e +∞) = -∞))
18	12, 17	mpbid 235	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (𝐴 ·_e +∞) = -∞)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 class class class wbr 5043 (class class class)co 7202 0cc0 10712 +∞cpnf 10847 -∞cmnf 10848 ℝ^*cxr 10849 < clt 10850 -_𝑒cxne 12684 ·_e cxmu 12686

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-po 5457 df-so 5458 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-xneg 12687 df-xmul 12689

This theorem is referenced by: xlemul1a 12861

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