Theorem prodgt0gt0 8839

Description: Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 8840 for the same theorem with 0 < A replaced by the weaker condition 0 <_ A. (Contributed by Jim Kingdon, 29-Feb-2020.)

Assertion

Ref	Expression
prodgt0gt0	|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < B )

Proof of Theorem prodgt0gt0

Step	Hyp	Ref	Expression
1		simpll 527	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> A e. RR )
2		simplr 528	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> B e. RR )
3	1, 2	remulcld 8019	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) e. RR )
4		simprl 529	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < A )
5	1, 4	gt0ap0d 8617	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> A # 0 )
6	1, 5	rerecclapd 8822	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( 1 / A ) e. RR )
7		simprr 531	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < ( A x. B ) )
8		recgt0 8838	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )
9	8	ad2ant2r 509	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < ( 1 / A ) )
10	3, 6, 7, 9	mulgt0d 8111	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < ( ( A x. B ) x. ( 1 / A ) ) )
11	3	recnd 8017	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) e. CC )
12	1	recnd 8017	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> A e. CC )
13	11, 12, 5	divrecapd 8781	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( ( A x. B ) / A ) = ( ( A x. B ) x. ( 1 / A ) ) )
14		simpr 110	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
15	14	recnd 8017	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
16	15	adantr 276	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> B e. CC )
17	16, 12, 5	divcanap3d 8783	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( ( A x. B ) / A ) = B )
18	13, 17	eqtr3d 2224	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( ( A x. B ) x. ( 1 / A ) ) = B )
19	10, 18	breqtrd 4044	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < B )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2160   class class class wbr 4018  (class class class)co 5897   CCcc 7840   RRcr 7841   0cc0 7842   1c1 7843   x. cmul 7847   < clt 8023   / cdiv 8660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661

This theorem is referenced by:  prodgt0  8840