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Theorem xmulgt0 13243

Description: Extended real version of mulgt0 11251. (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 613	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → (0 < 𝐴 ∧ 0 < 𝐵))
4		mulgt0 11251	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵))
5	4	an4s 660	. . . . . . 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 13231	. . . . . . 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 5135	. . . . 5 ⊢ (((0 < 𝐴 ∧ 0 < 𝐵) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → 0 < (𝐴 ·_e 𝐵))
10	3, 9	sylan 580	. . . 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 13082	. . . . 5 ⊢ 0 < +∞
13		oveq2 7395	. . . . . 6 ⊢ (𝐵 = +∞ → (𝐴 ·_e 𝐵) = (𝐴 ·_e +∞))
14		xmulpnf1 13234	. . . . . . 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 2786	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐵 = +∞) → (𝐴 ·_e 𝐵) = +∞)
17	12, 16	breqtrrid 5145	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐵 = +∞) → 0 < (𝐴 ·_e 𝐵))
18	17	adantlr 715	. . 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 13242	. . . 4 ⊢ ((0 < 𝐵 ∧ 𝐵 = -∞) → 0 < (𝐴 ·_e 𝐵))
21	19, 20	sylan 580	. . 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 13076	. . . . 5 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
24	22, 23	sylib 218	. . . 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 1434	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) → 0 < (𝐴 ·_e 𝐵))
27		oveq1 7394	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 ·_e 𝐵) = (+∞ ·_e 𝐵))
28		xmulpnf2 13235	. . . . 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 2786	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 = +∞) → (𝐴 ·_e 𝐵) = +∞)
31	12, 30	breqtrrid 5145	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 = +∞) → 0 < (𝐴 ·_e 𝐵))
32		xmulasslem2 13242	. . 3 ⊢ ((0 < 𝐴 ∧ 𝐴 = -∞) → 0 < (𝐴 ·_e 𝐵))
33	32	ad4ant24 754	. 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 13076	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
36	34, 35	sylib 218	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
37	26, 31, 33, 36	mpjao3dan 1434	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 1085 = wceq 1540 ∈ wcel 2109 class class class wbr 5107 (class class class)co 7387 ℝcr 11067 0cc0 11068 · cmul 11073 +∞cpnf 11205 -∞cmnf 11206 ℝ^*cxr 11207 < clt 11208 ·_e cxmu 13071

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-xmul 13074

This theorem is referenced by: xmulge0 13244 xmulasslem3 13246

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