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Theorem xmulgt0 13316

Description: Extended real version of mulgt0 11341. (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 483	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → 0 < 𝐴)
2		simpr 483	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → 0 < 𝐵)
3	1, 2	anim12i 611	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → (0 < 𝐴 ∧ 0 < 𝐵))
4		mulgt0 11341	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵))
5	4	an4s 658	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵))
6	5	ancoms 457	. . . . . 6 ⊢ (((0 < 𝐴 ∧ 0 < 𝐵) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → 0 < (𝐴 · 𝐵))
7		rexmul 13304	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))
8	7	adantl 480	. . . . . 6 ⊢ (((0 < 𝐴 ∧ 0 < 𝐵) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))
9	6, 8	breqtrrd 5181	. . . . 5 ⊢ (((0 < 𝐴 ∧ 0 < 𝐵) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → 0 < (𝐴 ·_e 𝐵))
10	3, 9	sylan 578	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → 0 < (𝐴 ·_e 𝐵))
11	10	anassrs 466	. . 3 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → 0 < (𝐴 ·_e 𝐵))
12		0ltpnf 13156	. . . . 5 ⊢ 0 < +∞
13		oveq2 7432	. . . . . 6 ⊢ (𝐵 = +∞ → (𝐴 ·_e 𝐵) = (𝐴 ·_e +∞))
14		xmulpnf1 13307	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e +∞) = +∞)
15	14	adantr 479	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → (𝐴 ·_e +∞) = +∞)
16	13, 15	sylan9eqr 2788	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐵 = +∞) → (𝐴 ·_e 𝐵) = +∞)
17	12, 16	breqtrrid 5191	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐵 = +∞) → 0 < (𝐴 ·_e 𝐵))
18	17	adantlr 713	. . 3 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) ∧ 𝐵 = +∞) → 0 < (𝐴 ·_e 𝐵))
19		simplrr 776	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) → 0 < 𝐵)
20		xmulasslem2 13315	. . . 4 ⊢ ((0 < 𝐵 ∧ 𝐵 = -∞) → 0 < (𝐴 ·_e 𝐵))
21	19, 20	sylan 578	. . 3 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) ∧ 𝐵 = -∞) → 0 < (𝐴 ·_e 𝐵))
22		simprl 769	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → 𝐵 ∈ ℝ^*)
23		elxr 13150	. . . . 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 479	. . 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 7431	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 ·_e 𝐵) = (+∞ ·_e 𝐵))
28		xmulpnf2 13308	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → (+∞ ·_e 𝐵) = +∞)
29	28	adantl 480	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → (+∞ ·_e 𝐵) = +∞)
30	27, 29	sylan9eqr 2788	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 = +∞) → (𝐴 ·_e 𝐵) = +∞)
31	12, 30	breqtrrid 5191	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 = +∞) → 0 < (𝐴 ·_e 𝐵))
32		xmulasslem2 13315	. . 3 ⊢ ((0 < 𝐴 ∧ 𝐴 = -∞) → 0 < (𝐴 ·_e 𝐵))
33	32	ad4ant24 752	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 = -∞) → 0 < (𝐴 ·_e 𝐵))
34		simpll 765	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → 𝐴 ∈ ℝ^*)
35		elxr 13150	. . 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 394 ∨ w3o 1083 = wceq 1534 ∈ wcel 2099 class class class wbr 5153 (class class class)co 7424 ℝcr 11157 0cc0 11158 · cmul 11163 +∞cpnf 11295 -∞cmnf 11296 ℝ^*cxr 11297 < clt 11298 ·_e cxmu 13145

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 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-xmul 13148

This theorem is referenced by: xmulge0 13317 xmulasslem3 13319

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