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Theorem fprodge1 31837
Description: If all of the terms of a finite product are larger 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 1re 9584 . . . 4  |-  1  e.  RR
21rexri 9635 . . 3  |-  1  e.  RR*
32a1i 11 . 2  |-  ( ph  ->  1  e.  RR* )
4 pnfxr 11324 . . 3  |- +oo  e.  RR*
54a1i 11 . 2  |-  ( ph  -> +oo  e.  RR* )
6 fprodge1.ph . . 3  |-  F/ k
ph
7 icossre 11608 . . . . . 6  |-  ( ( 1  e.  RR  /\ +oo  e.  RR* )  ->  (
1 [,) +oo )  C_  RR )
81, 4, 7mp2an 670 . . . . 5  |-  ( 1 [,) +oo )  C_  RR
9 ax-resscn 9538 . . . . 5  |-  RR  C_  CC
108, 9sstri 3498 . . . 4  |-  ( 1 [,) +oo )  C_  CC
1110a1i 11 . . 3  |-  ( ph  ->  ( 1 [,) +oo )  C_  CC )
122a1i 11 . . . . 5  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  1  e.  RR* )
134a1i 11 . . . . 5  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  -> +oo  e.  RR* )
148sseli 3485 . . . . . . . 8  |-  ( x  e.  ( 1 [,) +oo )  ->  x  e.  RR )
1514adantr 463 . . . . . . 7  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  x  e.  RR )
168sseli 3485 . . . . . . . 8  |-  ( y  e.  ( 1 [,) +oo )  ->  y  e.  RR )
1716adantl 464 . . . . . . 7  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  y  e.  RR )
1815, 17remulcld 9613 . . . . . 6  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  ( x  x.  y )  e.  RR )
1918rexrd 9632 . . . . 5  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  ( x  x.  y )  e.  RR* )
20 1t1e1 10679 . . . . . . . 8  |-  ( 1  x.  1 )  =  1
2120eqcomi 2467 . . . . . . 7  |-  1  =  ( 1  x.  1 )
2221a1i 11 . . . . . 6  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  1  =  ( 1  x.  1 ) )
231a1i 11 . . . . . . 7  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  1  e.  RR )
24 0le1 10072 . . . . . . . 8  |-  0  <_  1
2524a1i 11 . . . . . . 7  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  0  <_  1
)
262a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 1 [,) +oo )  ->  1  e. 
RR* )
274a1i 11 . . . . . . . . 9  |-  ( x  e.  ( 1 [,) +oo )  -> +oo  e.  RR* )
28 id 22 . . . . . . . . 9  |-  ( x  e.  ( 1 [,) +oo )  ->  x  e.  ( 1 [,) +oo ) )
29 icogelb 31781 . . . . . . . . 9  |-  ( ( 1  e.  RR*  /\ +oo  e.  RR*  /\  x  e.  ( 1 [,) +oo ) )  ->  1  <_  x )
3026, 27, 28, 29syl3anc 1226 . . . . . . . 8  |-  ( x  e.  ( 1 [,) +oo )  ->  1  <_  x )
3130adantr 463 . . . . . . 7  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  1  <_  x
)
322a1i 11 . . . . . . . . 9  |-  ( y  e.  ( 1 [,) +oo )  ->  1  e. 
RR* )
334a1i 11 . . . . . . . . 9  |-  ( y  e.  ( 1 [,) +oo )  -> +oo  e.  RR* )
34 id 22 . . . . . . . . 9  |-  ( y  e.  ( 1 [,) +oo )  ->  y  e.  ( 1 [,) +oo ) )
35 icogelb 31781 . . . . . . . . 9  |-  ( ( 1  e.  RR*  /\ +oo  e.  RR*  /\  y  e.  ( 1 [,) +oo ) )  ->  1  <_  y )
3632, 33, 34, 35syl3anc 1226 . . . . . . . 8  |-  ( y  e.  ( 1 [,) +oo )  ->  1  <_ 
y )
3736adantl 464 . . . . . . 7  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  1  <_  y
)
3823, 15, 23, 17, 25, 25, 31, 37lemul12ad 10483 . . . . . 6  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  ( 1  x.  1 )  <_  (
x  x.  y ) )
3922, 38eqbrtrd 4459 . . . . 5  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  1  <_  (
x  x.  y ) )
4018ltpnfd 31720 . . . . 5  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  ( x  x.  y )  < +oo )
4112, 13, 19, 39, 40elicod 31790 . . . 4  |-  ( ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo ) )  ->  ( x  x.  y )  e.  ( 1 [,) +oo )
)
4241adantl 464 . . 3  |-  ( (
ph  /\  ( x  e.  ( 1 [,) +oo )  /\  y  e.  ( 1 [,) +oo )
) )  ->  (
x  x.  y )  e.  ( 1 [,) +oo ) )
43 fprodge1.a . . 3  |-  ( ph  ->  A  e.  Fin )
442a1i 11 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  1  e.  RR* )
454a1i 11 . . . 4  |-  ( (
ph  /\  k  e.  A )  -> +oo  e.  RR* )
46 fprodge1.b . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
4746rexrd 9632 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR* )
48 fprodge1.ge . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  1  <_  B )
4946ltpnfd 31720 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  B  < +oo )
5044, 45, 47, 48, 49elicod 31790 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 1 [,) +oo ) )
51 1le1 10173 . . . . . 6  |-  1  <_  1
52 ltpnf 11334 . . . . . . 7  |-  ( 1  e.  RR  ->  1  < +oo )
531, 52ax-mp 5 . . . . . 6  |-  1  < +oo
541, 51, 533pm3.2i 1172 . . . . 5  |-  ( 1  e.  RR  /\  1  <_  1  /\  1  < +oo )
55 elico2 11591 . . . . . 6  |-  ( ( 1  e.  RR  /\ +oo  e.  RR* )  ->  (
1  e.  ( 1 [,) +oo )  <->  ( 1  e.  RR  /\  1  <_  1  /\  1  < +oo ) ) )
561, 4, 55mp2an 670 . . . . 5  |-  ( 1  e.  ( 1 [,) +oo )  <->  ( 1  e.  RR  /\  1  <_ 
1  /\  1  < +oo ) )
5754, 56mpbir 209 . . . 4  |-  1  e.  ( 1 [,) +oo )
5857a1i 11 . . 3  |-  ( ph  ->  1  e.  ( 1 [,) +oo ) )
596, 11, 42, 43, 50, 58fprodcllemf 31830 . 2  |-  ( ph  ->  prod_ k  e.  A  B  e.  ( 1 [,) +oo ) )
60 icogelb 31781 . 2  |-  ( ( 1  e.  RR*  /\ +oo  e.  RR*  /\  prod_ k  e.  A  B  e.  ( 1 [,) +oo ) )  ->  1  <_  prod_ k  e.  A  B )
613, 5, 59, 60syl3anc 1226 1  |-  ( ph  ->  1  <_  prod_ k  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398   F/wnf 1621    e. wcel 1823    C_ wss 3461   class class class wbr 4439  (class class class)co 6270   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    x. cmul 9486   +oocpnf 9614   RR*cxr 9616    < clt 9617    <_ cle 9618   [,)cico 11534   prod_cprod 13794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-ico 11538  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-prod 13795
This theorem is referenced by:  fprodle  31843
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