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 13285

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 13218	. . 3 ⊢ -_𝑒+∞ = -∞
2	1	oveq2i 7411	. 2 ⊢ (𝐴 ·_e -_𝑒+∞) = (𝐴 ·_e -∞)
3		simpl 482	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → 𝐴 ∈ ℝ^*)
4		pnfxr 11282	. . . 4 ⊢ +∞ ∈ ℝ^*
5		xmulneg2 13279	. . . 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 13283	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e +∞) = +∞)
8		xnegeq 13216	. . . . 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 2785	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → -_𝑒(𝐴 ·_e +∞) = -∞)
11	6, 10	eqtrd 2769	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e -_𝑒+∞) = -∞)
12	2, 11	eqtr3id 2783	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e -∞) = -∞)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 class class class wbr 5117 (class class class)co 7400 0cc0 11122 +∞cpnf 11259 -∞cmnf 11260 ℝ^*cxr 11261 < clt 11262 -_𝑒cxne 13118 ·_e cxmu 13120

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 2706 ax-sep 5264 ax-nul 5274 ax-pow 5333 ax-pr 5400 ax-un 7724 ax-cnex 11178 ax-resscn 11179 ax-1cn 11180 ax-icn 11181 ax-addcl 11182 ax-addrcl 11183 ax-mulcl 11184 ax-mulrcl 11185 ax-mulcom 11186 ax-addass 11187 ax-mulass 11188 ax-distr 11189 ax-i2m1 11190 ax-1ne0 11191 ax-1rid 11192 ax-rnegex 11193 ax-rrecex 11194 ax-cnre 11195 ax-pre-lttri 11196 ax-pre-lttrn 11197 ax-pre-ltadd 11198

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 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4882 df-br 5118 df-opab 5180 df-mpt 5200 df-id 5546 df-po 5559 df-so 5560 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-iota 6481 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-er 8714 df-en 8955 df-dom 8956 df-sdom 8957 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11461 df-neg 11462 df-xneg 13121 df-xmul 13123

This theorem is referenced by: xmulmnf2 13286 xadddilem 13303

W3C validator