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Theorem xmulgt0 13289

Description: Extended real version of mulgt0 11316. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xmulgt0	⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → 0 < (𝐴 ·_e 𝐵))

**Proof of Theorem xmulgt0**
Step	Hyp	Ref	Expression
1		simpr 484	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → 0 < 𝐴)
2		simpr 484	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → 0 < 𝐵)
3	1, 2	anim12i 612	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → (0 < 𝐴 ∧ 0 < 𝐵))
4		mulgt0 11316	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵))
5	4	an4s 659	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵))
6	5	ancoms 458	. . . . . 6 ⊢ (((0 < 𝐴 ∧ 0 < 𝐵) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → 0 < (𝐴 · 𝐵))
7		rexmul 13277	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))
8	7	adantl 481	. . . . . 6 ⊢ (((0 < 𝐴 ∧ 0 < 𝐵) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))
9	6, 8	breqtrrd 5171	. . . . 5 ⊢ (((0 < 𝐴 ∧ 0 < 𝐵) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → 0 < (𝐴 ·_e 𝐵))
10	3, 9	sylan 579	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → 0 < (𝐴 ·_e 𝐵))
11	10	anassrs 467	. . 3 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → 0 < (𝐴 ·_e 𝐵))
12		0ltpnf 13129	. . . . 5 ⊢ 0 < +∞
13		oveq2 7423	. . . . . 6 ⊢ (𝐵 = +∞ → (𝐴 ·_e 𝐵) = (𝐴 ·_e +∞))
14		xmulpnf1 13280	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e +∞) = +∞)
15	14	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → (𝐴 ·_e +∞) = +∞)
16	13, 15	sylan9eqr 2790	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐵 = +∞) → (𝐴 ·_e 𝐵) = +∞)
17	12, 16	breqtrrid 5181	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐵 = +∞) → 0 < (𝐴 ·_e 𝐵))
18	17	adantlr 714	. . 3 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) ∧ 𝐵 = +∞) → 0 < (𝐴 ·_e 𝐵))
19		simplrr 777	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) → 0 < 𝐵)
20		xmulasslem2 13288	. . . 4 ⊢ ((0 < 𝐵 ∧ 𝐵 = -∞) → 0 < (𝐴 ·_e 𝐵))
21	19, 20	sylan 579	. . 3 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) ∧ 𝐵 = -∞) → 0 < (𝐴 ·_e 𝐵))
22		simprl 770	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → 𝐵 ∈ ℝ^*)
23		elxr 13123	. . . . 5 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
24	22, 23	sylib 217	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
25	24	adantr 480	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
26	11, 18, 21, 25	mpjao3dan 1429	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) → 0 < (𝐴 ·_e 𝐵))
27		oveq1 7422	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 ·_e 𝐵) = (+∞ ·_e 𝐵))
28		xmulpnf2 13281	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → (+∞ ·_e 𝐵) = +∞)
29	28	adantl 481	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → (+∞ ·_e 𝐵) = +∞)
30	27, 29	sylan9eqr 2790	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 = +∞) → (𝐴 ·_e 𝐵) = +∞)
31	12, 30	breqtrrid 5181	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 = +∞) → 0 < (𝐴 ·_e 𝐵))
32		xmulasslem2 13288	. . 3 ⊢ ((0 < 𝐴 ∧ 𝐴 = -∞) → 0 < (𝐴 ·_e 𝐵))
33	32	ad4ant24 753	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 = -∞) → 0 < (𝐴 ·_e 𝐵))
34		simpll 766	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → 𝐴 ∈ ℝ^*)
35		elxr 13123	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
36	34, 35	sylib 217	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
37	26, 31, 33, 36	mpjao3dan 1429	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → 0 < (𝐴 ·_e 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1084 = wceq 1534 ∈ wcel 2099 class class class wbr 5143 (class class class)co 7415 ℝcr 11132 0cc0 11133 · cmul 11138 +∞cpnf 11270 -∞cmnf 11271 ℝ^*cxr 11272 < clt 11273 ·_e cxmu 13118

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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-ov 7418 df-oprab 7419 df-mpo 7420 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-xmul 13121

This theorem is referenced by: xmulge0 13290 xmulasslem3 13292

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