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Theorem mulge0 10546
Description: The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
mulge0 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> 0 <_ ( A x. B ) )
Proof of Theorem mulge0
StepHypRef Expression
1 0red 10041 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> 0 e. RR )
2 simpl 473 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
31, 2leloed 10180 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) )
4 simpr 477 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
51, 4leloed 10180 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ B <-> ( 0 < B \/ 0 = B ) ) )
63, 5anbi12d 747 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <_ A /\ 0 <_ B ) <-> ( ( 0 < A \/ 0 = A ) /\ ( 0 < B \/ 0 = B ) ) ) )
7 0red 10041 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < B ) ) -> 0 e. RR )
8 simpll 790 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < B ) ) -> A e. RR )
9 simplr 792 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < B ) ) -> B e. RR )
108, 9remulcld 10070 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < B ) ) -> ( A x. B ) e. RR )
11 mulgt0 10115 . . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A x. B ) )
1211an4s 869 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < B ) ) -> 0 < ( A x. B ) )
137, 10, 12ltled 10185 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < B ) ) -> 0 <_ ( A x. B ) )
1413ex 450 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) -> 0 <_ ( A x. B ) ) )
15 0re 10040 . . . . . . . . 9 |- 0 e. RR
16 leid 10133 . . . . . . . . 9 |- ( 0 e. RR -> 0 <_ 0 )
1715, 16ax-mp 5 . . . . . . . 8 |- 0 <_ 0
184recnd 10068 . . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
1918mul02d 10234 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( 0 x. B ) = 0 )
2017, 19syl5breqr 4691 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> 0 <_ ( 0 x. B ) )
21 oveq1 6657 . . . . . . . 8 |- ( 0 = A -> ( 0 x. B ) = ( A x. B ) )
2221breq2d 4665 . . . . . . 7 |- ( 0 = A -> ( 0 <_ ( 0 x. B ) <-> 0 <_ ( A x. B ) ) )
2320, 22syl5ibcom 235 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( 0 = A -> 0 <_ ( A x. B ) ) )
2423adantrd 484 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 = A /\ 0 < B ) -> 0 <_ ( A x. B ) ) )
252recnd 10068 . . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
2625mul01d 10235 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A x. 0 ) = 0 )
2717, 26syl5breqr 4691 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> 0 <_ ( A x. 0 ) )
28 oveq2 6658 . . . . . . . 8 |- ( 0 = B -> ( A x. 0 ) = ( A x. B ) )
2928breq2d 4665 . . . . . . 7 |- ( 0 = B -> ( 0 <_ ( A x. 0 ) <-> 0 <_ ( A x. B ) ) )
3027, 29syl5ibcom 235 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( 0 = B -> 0 <_ ( A x. B ) ) )
3130adantld 483 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 = B ) -> 0 <_ ( A x. B ) ) )
3230adantld 483 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 = A /\ 0 = B ) -> 0 <_ ( A x. B ) ) )
3314, 24, 31, 32ccased 988 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( 0 < A \/ 0 = A ) /\ ( 0 < B \/ 0 = B ) ) -> 0 <_ ( A x. B ) ) )
346, 33sylbid 230 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <_ A /\ 0 <_ B ) -> 0 <_ ( A x. B ) ) )
3534imp 445 . 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 <_ B ) ) -> 0 <_ ( A x. B ) )
3635an4s 869 1 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> 0 <_ ( A x. B ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   RRcr 9935   0cc0 9936   x. cmul 9941   < clt 10074   <_ cle 10075
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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080
This theorem is referenced by:  mulge0i  10575  mulge0d  10604  mulge0b  10893  ge0mulcl  12285  expge0  12896  bernneq  12990  sqrtmul  14000  sqreulem  14099  amgm2  14109  efcllem  14808  nmoco  22541  iihalf1  22730  iimulcl  22736  mbfi1fseqlem1  23482  mbfi1fseqlem3  23484  mbfi1fseqlem5  23486  2lgslem1a1  25114  dchrisumlem3  25180  dchrvmasumlem2  25187  chpdifbndlem2  25243  cnlnadjlem7  28932  leopmuli  28992  reofld  29840  stoweidlem24  40241  hoidmvlelem1  40809
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