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Theorem lgslem3 20942
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 10249 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  x.  B
)  e.  ZZ )
21ad2ant2r 728 . . 3  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  ( A  x.  B )  e.  ZZ )
3 zcn 10212 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  CC )
4 zcn 10212 . . . . . 6  |-  ( B  e.  ZZ  ->  B  e.  CC )
5 absmul 12019 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  x.  B )
)  =  ( ( abs `  A )  x.  ( abs `  B
) ) )
63, 4, 5syl2an 464 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( abs `  ( A  x.  B )
)  =  ( ( abs `  A )  x.  ( abs `  B
) ) )
76ad2ant2r 728 . . . 4  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  ( abs `  ( A  x.  B ) )  =  ( ( abs `  A
)  x.  ( abs `  B ) ) )
8 abscl 12003 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
9 absge0 12012 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  0  <_  ( abs `  A
) )
108, 9jca 519 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
113, 10syl 16 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  (
( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
1211adantr 452 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
13 1re 9016 . . . . . . . . 9  |-  1  e.  RR
1413a1i 11 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  1  e.  RR )
15 abscl 12003 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  ( abs `  B )  e.  RR )
16 absge0 12012 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  0  <_  ( abs `  B
) )
1715, 16jca 519 . . . . . . . . . 10  |-  ( B  e.  CC  ->  (
( abs `  B
)  e.  RR  /\  0  <_  ( abs `  B
) ) )
184, 17syl 16 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  (
( abs `  B
)  e.  RR  /\  0  <_  ( abs `  B
) ) )
1918adantl 453 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( abs `  B
)  e.  RR  /\  0  <_  ( abs `  B
) ) )
20 lemul12a 9793 . . . . . . . 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 1185 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( abs `  A )  <_  1  /\  ( abs `  B
)  <_  1 )  ->  ( ( abs `  A )  x.  ( abs `  B ) )  <_  ( 1  x.  1 ) ) )
2221imp 419 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( abs `  A )  <_  1  /\  ( abs `  B
)  <_  1 ) )  ->  ( ( abs `  A )  x.  ( abs `  B
) )  <_  (
1  x.  1 ) )
2322an4s 800 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  (
( abs `  A
)  x.  ( abs `  B ) )  <_ 
( 1  x.  1 ) )
24 1t1e1 10051 . . . . 5  |-  ( 1  x.  1 )  =  1
2523, 24syl6breq 4185 . . . 4  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  (
( abs `  A
)  x.  ( abs `  B ) )  <_ 
1 )
267, 25eqbrtrd 4166 . . 3  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  ( abs `  ( A  x.  B ) )  <_ 
1 )
272, 26jca 519 . 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 5661 . . . . 5  |-  ( x  =  A  ->  ( abs `  x )  =  ( abs `  A
) )
2928breq1d 4156 . . . 4  |-  ( x  =  A  ->  (
( abs `  x
)  <_  1  <->  ( abs `  A )  <_  1
) )
30 lgslem2.z . . . 4  |-  Z  =  { x  e.  ZZ  |  ( abs `  x
)  <_  1 }
3129, 30elrab2 3030 . . 3  |-  ( A  e.  Z  <->  ( A  e.  ZZ  /\  ( abs `  A )  <_  1
) )
32 fveq2 5661 . . . . 5  |-  ( x  =  B  ->  ( abs `  x )  =  ( abs `  B
) )
3332breq1d 4156 . . . 4  |-  ( x  =  B  ->  (
( abs `  x
)  <_  1  <->  ( abs `  B )  <_  1
) )
3433, 30elrab2 3030 . . 3  |-  ( B  e.  Z  <->  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )
3531, 34anbi12i 679 . 2  |-  ( ( A  e.  Z  /\  B  e.  Z )  <->  ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) ) )
36 fveq2 5661 . . . 4  |-  ( x  =  ( A  x.  B )  ->  ( abs `  x )  =  ( abs `  ( A  x.  B )
) )
3736breq1d 4156 . . 3  |-  ( x  =  ( A  x.  B )  ->  (
( abs `  x
)  <_  1  <->  ( abs `  ( A  x.  B
) )  <_  1
) )
3837, 30elrab2 3030 . 2  |-  ( ( A  x.  B )  e.  Z  <->  ( ( A  x.  B )  e.  ZZ  /\  ( abs `  ( A  x.  B
) )  <_  1
) )
3927, 35, 383imtr4i 258 1  |-  ( ( A  e.  Z  /\  B  e.  Z )  ->  ( A  x.  B
)  e.  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2646   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   CCcc 8914   RRcr 8915   0cc0 8916   1c1 8917    x. cmul 8921    <_ cle 9047   ZZcz 10207   abscabs 11959
This theorem is referenced by:  lgsfcl2  20946  lgscllem  20947  lgsdirprm  20973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961
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