xmulmnf1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xmulmnf1		Structured version Visualization version GIF version

Theorem xmulmnf1 13010

Description: Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xmulmnf1	⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e -∞) = -∞)

**Proof of Theorem xmulmnf1**
Step	Hyp	Ref	Expression
1		xnegpnf 12943	. . 3 ⊢ -_𝑒+∞ = -∞
2	1	oveq2i 7286	. 2 ⊢ (𝐴 ·_e -_𝑒+∞) = (𝐴 ·_e -∞)
3		simpl 483	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → 𝐴 ∈ ℝ^*)
4		pnfxr 11029	. . . 4 ⊢ +∞ ∈ ℝ^*
5		xmulneg2 13004	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐴 ·_e -_𝑒+∞) = -_𝑒(𝐴 ·_e +∞))
6	3, 4, 5	sylancl 586	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e -_𝑒+∞) = -_𝑒(𝐴 ·_e +∞))
7		xmulpnf1 13008	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e +∞) = +∞)
8		xnegeq 12941	. . . . 5 ⊢ ((𝐴 ·_e +∞) = +∞ → -_𝑒(𝐴 ·_e +∞) = -_𝑒+∞)
9	7, 8	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → -_𝑒(𝐴 ·_e +∞) = -_𝑒+∞)
10	9, 1	eqtrdi 2794	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → -_𝑒(𝐴 ·_e +∞) = -∞)
11	6, 10	eqtrd 2778	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e -_𝑒+∞) = -∞)
12	2, 11	eqtr3id 2792	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e -∞) = -∞)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 0cc0 10871 +∞cpnf 11006 -∞cmnf 11007 ℝ^*cxr 11008 < clt 11009 -_𝑒cxne 12845 ·_e cxmu 12847

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-xneg 12848 df-xmul 12850

This theorem is referenced by: xmulmnf2 13011 xadddilem 13028

W3C validator