Theorem xmulgt0 12113

Description: Extended real version of mulgt0 10115. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xmulgt0	|- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) -> 0 < ( A xe B ) )

Proof of Theorem xmulgt0

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . 6 |- ( ( A e. RR* /\ 0 < A ) -> 0 < A )
2		simpr 477	. . . . . 6 |- ( ( B e. RR* /\ 0 < B ) -> 0 < B )
3	1, 2	anim12i 590	. . . . 5 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) -> ( 0 < A /\ 0 < B ) )
4		mulgt0 10115	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A x. B ) )
5	4	an4s 869	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < B ) ) -> 0 < ( A x. B ) )
6	5	ancoms 469	. . . . . 6 |- ( ( ( 0 < A /\ 0 < B ) /\ ( A e. RR /\ B e. RR ) ) -> 0 < ( A x. B ) )
7		rexmul 12101	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A xe B ) = ( A x. B ) )
8	7	adantl 482	. . . . . 6 |- ( ( ( 0 < A /\ 0 < B ) /\ ( A e. RR /\ B e. RR ) ) -> ( A xe B ) = ( A x. B ) )
9	6, 8	breqtrrd 4681	. . . . 5 |- ( ( ( 0 < A /\ 0 < B ) /\ ( A e. RR /\ B e. RR ) ) -> 0 < ( A xe B ) )
10	3, 9	sylan 488	. . . 4 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) /\ ( A e. RR /\ B e. RR ) ) -> 0 < ( A xe B ) )
11	10	anassrs 680	. . 3 |- ( ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) /\ A e. RR ) /\ B e. RR ) -> 0 < ( A xe B ) )
12		0ltpnf 11956	. . . . 5 |- 0 < +oo
13		oveq2 6658	. . . . . 6 |- ( B = +oo -> ( A xe B ) = ( A xe +oo ) )
14		xmulpnf1 12104	. . . . . . 7 |- ( ( A e. RR* /\ 0 < A ) -> ( A xe +oo ) = +oo )
15	14	adantr 481	. . . . . 6 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) -> ( A xe +oo ) = +oo )
16	13, 15	sylan9eqr 2678	. . . . 5 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) /\ B = +oo ) -> ( A xe B ) = +oo )
17	12, 16	syl5breqr 4691	. . . 4 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) /\ B = +oo ) -> 0 < ( A xe B ) )
18	17	adantlr 751	. . 3 |- ( ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) /\ A e. RR ) /\ B = +oo ) -> 0 < ( A xe B ) )
19		simplrr 801	. . . 4 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) /\ A e. RR ) -> 0 < B )
20		xmulasslem2 12112	. . . 4 |- ( ( 0 < B /\ B = -oo ) -> 0 < ( A xe B ) )
21	19, 20	sylan 488	. . 3 |- ( ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) /\ A e. RR ) /\ B = -oo ) -> 0 < ( A xe B ) )
22		simprl 794	. . . . 5 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) -> B e. RR* )
23		elxr 11950	. . . . 5 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
24	22, 23	sylib 208	. . . 4 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
25	24	adantr 481	. . 3 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) /\ A e. RR ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
26	11, 18, 21, 25	mpjao3dan 1395	. 2 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) /\ A e. RR ) -> 0 < ( A xe B ) )
27		oveq1 6657	. . . 4 |- ( A = +oo -> ( A xe B ) = ( +oo xe B ) )
28		xmulpnf2 12105	. . . . 5 |- ( ( B e. RR* /\ 0 < B ) -> ( +oo xe B ) = +oo )
29	28	adantl 482	. . . 4 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) -> ( +oo xe B ) = +oo )
30	27, 29	sylan9eqr 2678	. . 3 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) /\ A = +oo ) -> ( A xe B ) = +oo )
31	12, 30	syl5breqr 4691	. 2 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) /\ A = +oo ) -> 0 < ( A xe B ) )
32		simplr 792	. . 3 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) -> 0 < A )
33		xmulasslem2 12112	. . 3 |- ( ( 0 < A /\ A = -oo ) -> 0 < ( A xe B ) )
34	32, 33	sylan 488	. 2 |- ( ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) /\ A = -oo ) -> 0 < ( A xe B ) )
35		simpll 790	. . 3 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) -> A e. RR* )
36		elxr 11950	. . 3 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
37	35, 36	sylib 208	. 2 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
38	26, 31, 34, 37	mpjao3dan 1395	1 |- ( ( ( A e. RR* /\ 0 < A ) /\ ( B e. RR* /\ 0 < B ) ) -> 0 < ( A xe B ) )

Syntax hints:   -> wi 4   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   RRcr 9935   0cc0 9936   x. cmul 9941   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   xecxmu 11945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-xmul 11948

This theorem is referenced by:  xmulge0  12114  xmulasslem3  12116