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Theorem lgslem3 21074
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 10316 . . . 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 10279 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  CC )
4 zcn 10279 . . . . . 6  |-  ( B  e.  ZZ  ->  B  e.  CC )
5 absmul 12091 . . . . . 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 12075 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
9 absge0 12084 . . . . . . . . . . 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 9082 . . . . . . . . 9  |-  1  e.  RR
1413a1i 11 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  1  e.  RR )
15 abscl 12075 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  ( abs `  B )  e.  RR )
16 absge0 12084 . . . . . . . . . . 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 9860 . . . . . . . 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 10118 . . . . 5  |-  ( 1  x.  1 )  =  1
2523, 24syl6breq 4243 . . . 4  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  (
( abs `  A
)  x.  ( abs `  B ) )  <_ 
1 )
267, 25eqbrtrd 4224 . . 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 5720 . . . . 5  |-  ( x  =  A  ->  ( abs `  x )  =  ( abs `  A
) )
2928breq1d 4214 . . . 4  |-  ( x  =  A  ->  (
( abs `  x
)  <_  1  <->  ( abs `  A )  <_  1
) )
30 lgslem2.z . . . 4  |-  Z  =  { x  e.  ZZ  |  ( abs `  x
)  <_  1 }
3129, 30elrab2 3086 . . 3  |-  ( A  e.  Z  <->  ( A  e.  ZZ  /\  ( abs `  A )  <_  1
) )
32 fveq2 5720 . . . . 5  |-  ( x  =  B  ->  ( abs `  x )  =  ( abs `  B
) )
3332breq1d 4214 . . . 4  |-  ( x  =  B  ->  (
( abs `  x
)  <_  1  <->  ( abs `  B )  <_  1
) )
3433, 30elrab2 3086 . . 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 5720 . . . 4  |-  ( x  =  ( A  x.  B )  ->  ( abs `  x )  =  ( abs `  ( A  x.  B )
) )
3736breq1d 4214 . . 3  |-  ( x  =  ( A  x.  B )  ->  (
( abs `  x
)  <_  1  <->  ( abs `  ( A  x.  B
) )  <_  1
) )
3837, 30elrab2 3086 . 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 1652    e. wcel 1725   {crab 2701   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    <_ cle 9113   ZZcz 10274   abscabs 12031
This theorem is referenced by:  lgsfcl2  21078  lgscllem  21079  lgsdirprm  21105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033
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