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Theorem xmulgt0 13202

Description: Extended real version of mulgt0 11232. (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 485	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → 0 < 𝐴)
2		simpr 485	. . . . . 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 11232	. . . . . . . 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 459	. . . . . 6 ⊢ (((0 < 𝐴 ∧ 0 < 𝐵) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → 0 < (𝐴 · 𝐵))
7		rexmul 13190	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))
8	7	adantl 482	. . . . . 6 ⊢ (((0 < 𝐴 ∧ 0 < 𝐵) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))
9	6, 8	breqtrrd 5133	. . . . 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 468	. . 3 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → 0 < (𝐴 ·_e 𝐵))
12		0ltpnf 13043	. . . . 5 ⊢ 0 < +∞
13		oveq2 7365	. . . . . 6 ⊢ (𝐵 = +∞ → (𝐴 ·_e 𝐵) = (𝐴 ·_e +∞))
14		xmulpnf1 13193	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e +∞) = +∞)
15	14	adantr 481	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → (𝐴 ·_e +∞) = +∞)
16	13, 15	sylan9eqr 2798	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐵 = +∞) → (𝐴 ·_e 𝐵) = +∞)
17	12, 16	breqtrrid 5143	. . . 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 13201	. . . 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 769	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → 𝐵 ∈ ℝ^*)
23		elxr 13037	. . . . 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 481	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
26	11, 18, 21, 25	mpjao3dan 1431	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 ∈ ℝ) → 0 < (𝐴 ·_e 𝐵))
27		oveq1 7364	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 ·_e 𝐵) = (+∞ ·_e 𝐵))
28		xmulpnf2 13194	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → (+∞ ·_e 𝐵) = +∞)
29	28	adantl 482	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → (+∞ ·_e 𝐵) = +∞)
30	27, 29	sylan9eqr 2798	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 = +∞) → (𝐴 ·_e 𝐵) = +∞)
31	12, 30	breqtrrid 5143	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) ∧ 𝐴 = +∞) → 0 < (𝐴 ·_e 𝐵))
32		xmulasslem2 13201	. . 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 13037	. . 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 1431	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → 0 < (𝐴 ·_e 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 (class class class)co 7357 ℝcr 11050 0cc0 11051 · cmul 11056 +∞cpnf 11186 -∞cmnf 11187 ℝ^*cxr 11188 < clt 11189 ·_e cxmu 13032

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-xmul 13035

This theorem is referenced by: xmulge0 13203 xmulasslem3 13205

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