Theorem prodgt0gt0 8821

Description: Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 8822 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 8001	. . 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 8599	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> A # 0 )
6	1, 5	rerecclapd 8804	. . 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 8820	. . . 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 8093	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < ( ( A x. B ) x. ( 1 / A ) ) )
11	3	recnd 7999	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) e. CC )
12	1	recnd 7999	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> A e. CC )
13	11, 12, 5	divrecapd 8763	. . 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 7999	. . . . 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 8765	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( ( A x. B ) / A ) = B )
18	13, 17	eqtr3d 2222	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( ( A x. B ) x. ( 1 / A ) ) = B )
19	10, 18	breqtrd 4041	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < B )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2158   class class class wbr 4015  (class class class)co 5888   CCcc 7822   RRcr 7823   0cc0 7824   1c1 7825   x. cmul 7829   < clt 8005   / cdiv 8642

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643

This theorem is referenced by:  prodgt0  8822