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Theorem xmulpnf1n 13264

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 482	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → 𝐴 ∈ ℝ^*)
2		pnfxr 11275	. . . . 5 ⊢ +∞ ∈ ℝ^*
3		xmulneg1 13255	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (-_𝑒𝐴 ·_e +∞) = -_𝑒(𝐴 ·_e +∞))
4	1, 2, 3	sylancl 585	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e +∞) = -_𝑒(𝐴 ·_e +∞))
5		xnegcl 13199	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒𝐴 ∈ ℝ^*)
6		xlt0neg1 13205	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 < 0 ↔ 0 < -_𝑒𝐴))
7	6	biimpa 476	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → 0 < -_𝑒𝐴)
8		xmulpnf1 13260	. . . . 5 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ 0 < -_𝑒𝐴) → (-_𝑒𝐴 ·_e +∞) = +∞)
9	5, 7, 8	syl2an2r 682	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (-_𝑒𝐴 ·_e +∞) = +∞)
10	4, 9	eqtr3d 2773	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → -_𝑒(𝐴 ·_e +∞) = +∞)
11		xnegmnf 13196	. . 3 ⊢ -_𝑒-∞ = +∞
12	10, 11	eqtr4di 2789	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → -_𝑒(𝐴 ·_e +∞) = -_𝑒-∞)
13		xmulcl 13259	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝐴 ·_e +∞) ∈ ℝ^)
14	1, 2, 13	sylancl 585	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (𝐴 ·_e +∞) ∈ ℝ^*)
15		mnfxr 11278	. . 3 ⊢ -∞ ∈ ℝ^*
16		xneg11 13201	. . 3 ⊢ (((𝐴 ·_e +∞) ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → (-_𝑒(𝐴 ·_e +∞) = -_𝑒-∞ ↔ (𝐴 ·_e +∞) = -∞))
17	14, 15, 16	sylancl 585	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (-_𝑒(𝐴 ·_e +∞) = -_𝑒-∞ ↔ (𝐴 ·_e +∞) = -∞))
18	12, 17	mpbid 231	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (𝐴 ·_e +∞) = -∞)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 (class class class)co 7412 0cc0 11116 +∞cpnf 11252 -∞cmnf 11253 ℝ^*cxr 11254 < clt 11255 -_𝑒cxne 13096 ·_e cxmu 13098

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-xneg 13099 df-xmul 13101

This theorem is referenced by: xlemul1a 13274

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