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Theorem lgslem3 24230
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 10998 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  x.  B
)  e.  ZZ )
21ad2ant2r 752 . . 3  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  ( A  x.  B )  e.  ZZ )
3 zcn 10955 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  CC )
4 zcn 10955 . . . . . 6  |-  ( B  e.  ZZ  ->  B  e.  CC )
5 absmul 13363 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  x.  B )
)  =  ( ( abs `  A )  x.  ( abs `  B
) ) )
63, 4, 5syl2an 480 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( abs `  ( A  x.  B )
)  =  ( ( abs `  A )  x.  ( abs `  B
) ) )
76ad2ant2r 752 . . . 4  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  ( abs `  ( A  x.  B ) )  =  ( ( abs `  A
)  x.  ( abs `  B ) ) )
8 abscl 13347 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
9 absge0 13356 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  0  <_  ( abs `  A
) )
108, 9jca 535 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
113, 10syl 17 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  (
( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
1211adantr 467 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( abs `  A
)  e.  RR  /\  0  <_  ( abs `  A
) ) )
13 1red 9671 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  1  e.  RR )
14 abscl 13347 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  ( abs `  B )  e.  RR )
15 absge0 13356 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  0  <_  ( abs `  B
) )
1614, 15jca 535 . . . . . . . . . 10  |-  ( B  e.  CC  ->  (
( abs `  B
)  e.  RR  /\  0  <_  ( abs `  B
) ) )
174, 16syl 17 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  (
( abs `  B
)  e.  RR  /\  0  <_  ( abs `  B
) ) )
1817adantl 468 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( abs `  B
)  e.  RR  /\  0  <_  ( abs `  B
) ) )
19 lemul12a 10476 . . . . . . . 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 ) ) )
2012, 13, 18, 13, 19syl22anc 1266 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( abs `  A )  <_  1  /\  ( abs `  B
)  <_  1 )  ->  ( ( abs `  A )  x.  ( abs `  B ) )  <_  ( 1  x.  1 ) ) )
2120imp 431 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( abs `  A )  <_  1  /\  ( abs `  B
)  <_  1 ) )  ->  ( ( abs `  A )  x.  ( abs `  B
) )  <_  (
1  x.  1 ) )
2221an4s 834 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  (
( abs `  A
)  x.  ( abs `  B ) )  <_ 
( 1  x.  1 ) )
23 1t1e1 10770 . . . . 5  |-  ( 1  x.  1 )  =  1
2422, 23syl6breq 4469 . . . 4  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  (
( abs `  A
)  x.  ( abs `  B ) )  <_ 
1 )
257, 24eqbrtrd 4450 . . 3  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  ( abs `  ( A  x.  B ) )  <_ 
1 )
262, 25jca 535 . 2  |-  ( ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )  ->  (
( A  x.  B
)  e.  ZZ  /\  ( abs `  ( A  x.  B ) )  <_  1 ) )
27 fveq2 5887 . . . . 5  |-  ( x  =  A  ->  ( abs `  x )  =  ( abs `  A
) )
2827breq1d 4439 . . . 4  |-  ( x  =  A  ->  (
( abs `  x
)  <_  1  <->  ( abs `  A )  <_  1
) )
29 lgslem2.z . . . 4  |-  Z  =  { x  e.  ZZ  |  ( abs `  x
)  <_  1 }
3028, 29elrab2 3236 . . 3  |-  ( A  e.  Z  <->  ( A  e.  ZZ  /\  ( abs `  A )  <_  1
) )
31 fveq2 5887 . . . . 5  |-  ( x  =  B  ->  ( abs `  x )  =  ( abs `  B
) )
3231breq1d 4439 . . . 4  |-  ( x  =  B  ->  (
( abs `  x
)  <_  1  <->  ( abs `  B )  <_  1
) )
3332, 29elrab2 3236 . . 3  |-  ( B  e.  Z  <->  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) )
3430, 33anbi12i 702 . 2  |-  ( ( A  e.  Z  /\  B  e.  Z )  <->  ( ( A  e.  ZZ  /\  ( abs `  A
)  <_  1 )  /\  ( B  e.  ZZ  /\  ( abs `  B )  <_  1
) ) )
35 fveq2 5887 . . . 4  |-  ( x  =  ( A  x.  B )  ->  ( abs `  x )  =  ( abs `  ( A  x.  B )
) )
3635breq1d 4439 . . 3  |-  ( x  =  ( A  x.  B )  ->  (
( abs `  x
)  <_  1  <->  ( abs `  ( A  x.  B
) )  <_  1
) )
3736, 29elrab2 3236 . 2  |-  ( ( A  x.  B )  e.  Z  <->  ( ( A  x.  B )  e.  ZZ  /\  ( abs `  ( A  x.  B
) )  <_  1
) )
3826, 34, 373imtr4i 270 1  |-  ( ( A  e.  Z  /\  B  e.  Z )  ->  ( A  x.  B
)  e.  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1438    e. wcel 1873   {crab 2780   class class class wbr 4429   ` cfv 5607  (class class class)co 6311   CCcc 9550   RRcr 9551   0cc0 9552   1c1 9553    x. cmul 9557    <_ cle 9689   ZZcz 10950   abscabs 13303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-8 1875  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-sep 4552  ax-nul 4561  ax-pow 4608  ax-pr 4666  ax-un 6603  ax-cnex 9608  ax-resscn 9609  ax-1cn 9610  ax-icn 9611  ax-addcl 9612  ax-addrcl 9613  ax-mulcl 9614  ax-mulrcl 9615  ax-mulcom 9616  ax-addass 9617  ax-mulass 9618  ax-distr 9619  ax-i2m1 9620  ax-1ne0 9621  ax-1rid 9622  ax-rnegex 9623  ax-rrecex 9624  ax-cnre 9625  ax-pre-lttri 9626  ax-pre-lttrn 9627  ax-pre-ltadd 9628  ax-pre-mulgt0 9629  ax-pre-sup 9630
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3087  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3918  df-pw 3989  df-sn 4005  df-pr 4007  df-tp 4009  df-op 4011  df-uni 4226  df-iun 4307  df-br 4430  df-opab 4489  df-mpt 4490  df-tr 4525  df-eprel 4770  df-id 4774  df-po 4780  df-so 4781  df-fr 4818  df-we 4820  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-pred 5405  df-ord 5451  df-on 5452  df-lim 5453  df-suc 5454  df-iota 5571  df-fun 5609  df-fn 5610  df-f 5611  df-f1 5612  df-fo 5613  df-f1o 5614  df-fv 5615  df-riota 6273  df-ov 6314  df-oprab 6315  df-mpt2 6316  df-om 6713  df-2nd 6814  df-wrecs 7045  df-recs 7107  df-rdg 7145  df-er 7380  df-en 7587  df-dom 7588  df-sdom 7589  df-sup 7971  df-pnf 9690  df-mnf 9691  df-xr 9692  df-ltxr 9693  df-le 9694  df-sub 9875  df-neg 9876  df-div 10283  df-nn 10623  df-2 10681  df-3 10682  df-n0 10883  df-z 10951  df-uz 11173  df-rp 11316  df-seq 12226  df-exp 12285  df-cj 13168  df-re 13169  df-im 13170  df-sqrt 13304  df-abs 13305
This theorem is referenced by:  lgsfcl2  24234  lgscllem  24235  lgsdirprm  24261
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