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Theorem fprodge1 11646
Description: If all of the terms of a finite product are greater than or equal to  1, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
fprodge1.ph  |-  F/ k
ph
fprodge1.a  |-  ( ph  ->  A  e.  Fin )
fprodge1.b  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
fprodge1.ge  |-  ( (
ph  /\  k  e.  A )  ->  1  <_  B )
Assertion
Ref Expression
fprodge1  |-  ( ph  ->  1  <_  prod_ k  e.  A  B )
Distinct variable group:    A, k
Allowed substitution hints:    ph( k)    B( k)

Proof of Theorem fprodge1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1xr 8015 . 2  |-  1  e.  RR*
2 pnfxr 8009 . 2  |- +oo  e.  RR*
3 fprodge1.ph . . 3  |-  F/ k
ph
4 1re 7955 . . . . . 6  |-  1  e.  RR
5 icossre 9953 . . . . . 6  |-  ( ( 1  e.  RR  /\ +oo  e.  RR* )  ->  (
1 [,) +oo )  C_  RR )
64, 2, 5mp2an 426 . . . . 5  |-  ( 1 [,) +oo )  C_  RR
7 ax-resscn 7902 . . . . 5  |-  RR  C_  CC
86, 7sstri 3164 . . . 4  |-  ( 1 [,) +oo )  C_  CC
98a1i 9 . . 3  |-  ( ph  ->  ( 1 [,) +oo )  C_  CC )
101a1i 9 . . . . 5  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  1  e.  RR* )
112a1i 9 . . . . 5  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  -> +oo  e.  RR* )
126sseli 3151 . . . . . . . 8  |-  ( x  e.  ( 1 [,) +oo )  ->  x  e.  RR )
1312adantr 276 . . . . . . 7  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  x  e.  RR )
146sseli 3151 . . . . . . . 8  |-  ( y  e.  ( 1 [,) +oo )  ->  y  e.  RR )
1514adantl 277 . . . . . . 7  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  y  e.  RR )
1613, 15remulcld 7987 . . . . . 6  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  ( x  x.  y )  e.  RR )
1716rexrd 8006 . . . . 5  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  ( x  x.  y )  e.  RR* )
18 1t1e1 9070 . . . . . 6  |-  ( 1  x.  1 )  =  1
194a1i 9 . . . . . . 7  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  1  e.  RR )
20 0le1 8437 . . . . . . . 8  |-  0  <_  1
2120a1i 9 . . . . . . 7  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  0  <_  1
)
22 icogelb 10265 . . . . . . . . 9  |-  ( ( 1  e.  RR*  /\ +oo  e.  RR*  /\  x  e.  ( 1 [,) +oo ) )  ->  1  <_  x )
231, 2, 22mp3an12 1327 . . . . . . . 8  |-  ( x  e.  ( 1 [,) +oo )  ->  1  <_  x )
2423adantr 276 . . . . . . 7  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  1  <_  x
)
25 icogelb 10265 . . . . . . . . 9  |-  ( ( 1  e.  RR*  /\ +oo  e.  RR*  /\  y  e.  ( 1 [,) +oo ) )  ->  1  <_  y )
261, 2, 25mp3an12 1327 . . . . . . . 8  |-  ( y  e.  ( 1 [,) +oo )  ->  1  <_ 
y )
2726adantl 277 . . . . . . 7  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  1  <_  y
)
2819, 13, 19, 15, 21, 21, 24, 27lemul12ad 8898 . . . . . 6  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  ( 1  x.  1 )  <_  (
x  x.  y ) )
2918, 28eqbrtrrid 4039 . . . . 5  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  1  <_  (
x  x.  y ) )
3016ltpnfd 9780 . . . . 5  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  ( x  x.  y )  < +oo )
3110, 11, 17, 29, 30elicod 10264 . . . 4  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  ( x  x.  y )  e.  ( 1 [,) +oo )
)
3231adantl 277 . . 3  |-  ( (
ph  /\  ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo )
) )  ->  (
x  x.  y )  e.  ( 1 [,) +oo ) )
33 fprodge1.a . . 3  |-  ( ph  ->  A  e.  Fin )
341a1i 9 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  1  e.  RR* )
352a1i 9 . . . 4  |-  ( (
ph  /\  k  e.  A )  -> +oo  e.  RR* )
36 fprodge1.b . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
3736rexrd 8006 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR* )
38 fprodge1.ge . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  1  <_  B )
3936ltpnfd 9780 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  B  < +oo )
4034, 35, 37, 38, 39elicod 10264 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 1 [,) +oo ) )
41 1le1 8528 . . . . 5  |-  1  <_  1
42 ltpnf 9779 . . . . . 6  |-  ( 1  e.  RR  ->  1  < +oo )
434, 42ax-mp 5 . . . . 5  |-  1  < +oo
44 elico2 9936 . . . . . 6  |-  ( ( 1  e.  RR  /\ +oo  e.  RR* )  ->  (
1  e.  ( 1 [,) +oo )  <->  ( 1  e.  RR  /\  1  <_  1  /\  1  < +oo ) ) )
454, 2, 44mp2an 426 . . . . 5  |-  ( 1  e.  ( 1 [,) +oo )  <->  ( 1  e.  RR  /\  1  <_ 
1  /\  1  < +oo ) )
464, 41, 43, 45mpbir3an 1179 . . . 4  |-  1  e.  ( 1 [,) +oo )
4746a1i 9 . . 3  |-  ( ph  ->  1  e.  ( 1 [,) +oo ) )
483, 9, 32, 33, 40, 47fprodcllemf 11620 . 2  |-  ( ph  ->  prod_ k  e.  A  B  e.  ( 1 [,) +oo ) )
49 icogelb 10265 . 2  |-  ( ( 1  e.  RR*  /\ +oo  e.  RR*  /\  prod_ k  e.  A  B  e.  ( 1 [,) +oo ) )  ->  1  <_  prod_ k  e.  A  B )
501, 2, 48, 49mp3an12i 1341 1  |-  ( ph  ->  1  <_  prod_ k  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978   F/wnf 1460    e. wcel 2148    C_ wss 3129   class class class wbr 4003  (class class class)co 5874   Fincfn 6739   CCcc 7808   RRcr 7809   0cc0 7810   1c1 7811    x. cmul 7815   +oocpnf 7988   RR*cxr 7990    < clt 7991    <_ cle 7992   [,)cico 9889   prod_cprod 11557
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-ico 9893  df-fz 10008  df-fzo 10142  df-seqfrec 10445  df-exp 10519  df-ihash 10755  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-clim 11286  df-proddc 11558
This theorem is referenced by: (None)
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