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| Mirrors > Home > MPE Home > Th. List > xmulpnf1n | Structured version Visualization version GIF version | ||
| Description: Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| xmulpnf1n | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (𝐴 ·e +∞) = -∞) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpl 482 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 𝐴 ∈ ℝ*) | |
| 2 | pnfxr 11316 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
| 3 | xmulneg1 13312 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-𝑒𝐴 ·e +∞) = -𝑒(𝐴 ·e +∞)) | |
| 4 | 1, 2, 3 | sylancl 586 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (-𝑒𝐴 ·e +∞) = -𝑒(𝐴 ·e +∞)) | 
| 5 | xnegcl 13256 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) | |
| 6 | xlt0neg1 13262 | . . . . . 6 ⊢ (𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴)) | |
| 7 | 6 | biimpa 476 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 0 < -𝑒𝐴) | 
| 8 | xmulpnf1 13317 | . . . . 5 ⊢ ((-𝑒𝐴 ∈ ℝ* ∧ 0 < -𝑒𝐴) → (-𝑒𝐴 ·e +∞) = +∞) | |
| 9 | 5, 7, 8 | syl2an2r 685 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (-𝑒𝐴 ·e +∞) = +∞) | 
| 10 | 4, 9 | eqtr3d 2778 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → -𝑒(𝐴 ·e +∞) = +∞) | 
| 11 | xnegmnf 13253 | . . 3 ⊢ -𝑒-∞ = +∞ | |
| 12 | 10, 11 | eqtr4di 2794 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → -𝑒(𝐴 ·e +∞) = -𝑒-∞) | 
| 13 | xmulcl 13316 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 ·e +∞) ∈ ℝ*) | |
| 14 | 1, 2, 13 | sylancl 586 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (𝐴 ·e +∞) ∈ ℝ*) | 
| 15 | mnfxr 11319 | . . 3 ⊢ -∞ ∈ ℝ* | |
| 16 | xneg11 13258 | . . 3 ⊢ (((𝐴 ·e +∞) ∈ ℝ* ∧ -∞ ∈ ℝ*) → (-𝑒(𝐴 ·e +∞) = -𝑒-∞ ↔ (𝐴 ·e +∞) = -∞)) | |
| 17 | 14, 15, 16 | sylancl 586 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (-𝑒(𝐴 ·e +∞) = -𝑒-∞ ↔ (𝐴 ·e +∞) = -∞)) | 
| 18 | 12, 17 | mpbid 232 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (𝐴 ·e +∞) = -∞) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 (class class class)co 7432 0cc0 11156 +∞cpnf 11293 -∞cmnf 11294 ℝ*cxr 11295 < clt 11296 -𝑒cxne 13152 ·e cxmu 13154 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-xneg 13155 df-xmul 13157 | 
| This theorem is referenced by: xlemul1a 13331 | 
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