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Theorem mulgt1 9010
Description: The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)
Assertion
Ref Expression
mulgt1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 1 < ( A x. B ) )
Proof of Theorem mulgt1
StepHypRef Expression
1 simpl 109 . . . . 5 |- ( ( 1 < A /\ 1 < B ) -> 1 < A )
21a1i 9 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> 1 < A ) )
3 0lt1 8273 . . . . . . . . 9 |- 0 < 1
4 0re 8146 . . . . . . . . . 10 |- 0 e. RR
5 1re 8145 . . . . . . . . . 10 |- 1 e. RR
6 lttr 8220 . . . . . . . . . 10 |- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
74, 5, 6mp3an12 1361 . . . . . . . . 9 |- ( A e. RR -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
83, 7mpani 430 . . . . . . . 8 |- ( A e. RR -> ( 1 < A -> 0 < A ) )
98adantr 276 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < A -> 0 < A ) )
10 ltmul2 9003 . . . . . . . . . . 11 |- ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B <-> ( A x. 1 ) < ( A x. B ) ) )
1110biimpd 144 . . . . . . . . . 10 |- ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) )
125, 11mp3an1 1358 . . . . . . . . 9 |- ( ( B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) )
1312exp32 365 . . . . . . . 8 |- ( B e. RR -> ( A e. RR -> ( 0 < A -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) ) ) )
1413impcom 125 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( 0 < A -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) ) )
159, 14syld 45 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < A -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) ) )
1615impd 254 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> ( A x. 1 ) < ( A x. B ) ) )
17 ax-1rid 8106 . . . . . . 7 |- ( A e. RR -> ( A x. 1 ) = A )
1817adantr 276 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A x. 1 ) = A )
1918breq1d 4093 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) < ( A x. B ) <-> A < ( A x. B ) ) )
2016, 19sylibd 149 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> A < ( A x. B ) ) )
212, 20jcad 307 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> ( 1 < A /\ A < ( A x. B ) ) ) )
22 remulcl 8127 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
23 lttr 8220 . . . . 5 |- ( ( 1 e. RR /\ A e. RR /\ ( A x. B ) e. RR ) -> ( ( 1 < A /\ A < ( A x. B ) ) -> 1 < ( A x. B ) ) )
245, 23mp3an1 1358 . . . 4 |- ( ( A e. RR /\ ( A x. B ) e. RR ) -> ( ( 1 < A /\ A < ( A x. B ) ) -> 1 < ( A x. B ) ) )
2522, 24syldan 282 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ A < ( A x. B ) ) -> 1 < ( A x. B ) ) )
2621, 25syld 45 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> 1 < ( A x. B ) ) )
2726imp 124 1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 1 < ( A x. B ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   RRcr 7998   0cc0 7999   1c1 8000   x. cmul 8004   < clt 8181
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-lttrn 8113  ax-pre-ltadd 8115  ax-pre-mulgt0 8116
This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-sub 8319  df-neg 8320
This theorem is referenced by:  mulgt1d  9083  addltmul  9348  uz2mulcl  9803
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