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Theorem lgslem3 20537
Description: The set  Z of all integers with absolute value at most  1 is closed under multiplication. (Contributed by Mario Carneiro, 4-Feb-2015.)
Hypothesis
Ref Expression
lgslem2.z  |-  Z  =  { x  e.  ZZ  |  ( abs `  x
)  <_  1 }
Assertion
Ref Expression
lgslem3  |-  ( ( A  e.  Z  /\  B  e.  Z )  ->  ( A  x.  B
)  e.  Z )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    Z( x)

Proof of Theorem lgslem3
StepHypRef Expression
1 zmulcl 10066 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  x.  B
)  e.  ZZ )
21ad2ant2r 727 . . 3  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  ( A  x.  B )  e.  ZZ )
3 zcn 10029 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  CC )
4 zcn 10029 . . . . . 6  |-  ( B  e.  ZZ  ->  B  e.  CC )
5 absmul 11779 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  x.  B )
)  =  ( ( abs `  A )  x.  ( abs `  B
) ) )
63, 4, 5syl2an 463 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( abs `  ( A  x.  B )
)  =  ( ( abs `  A )  x.  ( abs `  B
) ) )
76ad2ant2r 727 . . . 4  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  ( abs `  ( A  x.  B ) )  =  ( ( abs `  A
)  x.  ( abs `  B ) ) )
8 abscl 11763 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
9 absge0 11772 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  0  <_  ( abs `  A
) )
108, 9jca 518 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
113, 10syl 15 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  (
( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
1211adantr 451 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
13 1re 8837 . . . . . . . . 9  |-  1  e.  RR
1413a1i 10 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  1  e.  RR )
15 abscl 11763 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  ( abs `  B )  e.  RR )
16 absge0 11772 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  0  <_  ( abs `  B
) )
1715, 16jca 518 . . . . . . . . . 10  |-  ( B  e.  CC  ->  (
( abs `  B
)  e.  RR  /\  0  <_  ( abs `  B
) ) )
184, 17syl 15 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  (
( abs `  B
)  e.  RR  /\  0  <_  ( abs `  B
) ) )
1918adantl 452 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( abs `  B
)  e.  RR  /\  0  <_  ( abs `  B
) ) )
20 lemul12a 9614 . . . . . . . 8  |-  ( ( ( ( ( abs `  A )  e.  RR  /\  0  <_  ( abs `  A ) )  /\  1  e.  RR )  /\  ( ( ( abs `  B )  e.  RR  /\  0  <_  ( abs `  B ) )  /\  1  e.  RR )
)  ->  ( (
( abs `  A
)  <_  1  /\  ( abs `  B )  <_  1 )  -> 
( ( abs `  A
)  x.  ( abs `  B ) )  <_ 
( 1  x.  1 ) ) )
2112, 14, 19, 14, 20syl22anc 1183 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( abs `  A )  <_  1  /\  ( abs `  B
)  <_  1 )  ->  ( ( abs `  A )  x.  ( abs `  B ) )  <_  ( 1  x.  1 ) ) )
2221imp 418 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( abs `  A )  <_  1  /\  ( abs `  B
)  <_  1 ) )  ->  ( ( abs `  A )  x.  ( abs `  B
) )  <_  (
1  x.  1 ) )
2322an4s 799 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  (
( abs `  A
)  x.  ( abs `  B ) )  <_ 
( 1  x.  1 ) )
24 1t1e1 9870 . . . . 5  |-  ( 1  x.  1 )  =  1
2523, 24syl6breq 4062 . . . 4  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  (
( abs `  A
)  x.  ( abs `  B ) )  <_ 
1 )
267, 25eqbrtrd 4043 . . 3  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  ( abs `  ( A  x.  B ) )  <_ 
1 )
272, 26jca 518 . 2  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  (
( A  x.  B
)  e.  ZZ  /\  ( abs `  ( A  x.  B ) )  <_  1 ) )
28 fveq2 5525 . . . . 5  |-  ( x  =  A  ->  ( abs `  x )  =  ( abs `  A
) )
2928breq1d 4033 . . . 4  |-  ( x  =  A  ->  (
( abs `  x
)  <_  1  <->  ( abs `  A )  <_  1
) )
30 lgslem2.z . . . 4  |-  Z  =  { x  e.  ZZ  |  ( abs `  x
)  <_  1 }
3129, 30elrab2 2925 . . 3  |-  ( A  e.  Z  <->  ( A  e.  ZZ  /\  ( abs `  A )  <_  1
) )
32 fveq2 5525 . . . . 5  |-  ( x  =  B  ->  ( abs `  x )  =  ( abs `  B
) )
3332breq1d 4033 . . . 4  |-  ( x  =  B  ->  (
( abs `  x
)  <_  1  <->  ( abs `  B )  <_  1
) )
3433, 30elrab2 2925 . . 3  |-  ( B  e.  Z  <->  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )
3531, 34anbi12i 678 . 2  |-  ( ( A  e.  Z  /\  B  e.  Z )  <->  ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) ) )
36 fveq2 5525 . . . 4  |-  ( x  =  ( A  x.  B )  ->  ( abs `  x )  =  ( abs `  ( A  x.  B )
) )
3736breq1d 4033 . . 3  |-  ( x  =  ( A  x.  B )  ->  (
( abs `  x
)  <_  1  <->  ( abs `  ( A  x.  B
) )  <_  1
) )
3837, 30elrab2 2925 . 2  |-  ( ( A  x.  B )  e.  Z  <->  ( ( A  x.  B )  e.  ZZ  /\  ( abs `  ( A  x.  B
) )  <_  1
) )
3927, 35, 383imtr4i 257 1  |-  ( ( A  e.  Z  /\  B  e.  Z )  ->  ( A  x.  B
)  e.  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    <_ cle 8868   ZZcz 10024   abscabs 11719
This theorem is referenced by:  lgsfcl2  20541  lgscllem  20542  lgsdirprm  20568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
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