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Theorem lgslem3 16326
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 9709 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A x. B ) e. ZZ )
21ad2ant2r 513 . . 3 |- ( ( ( A e. ZZ /\ ( abs ` A ) <_ 1 ) /\ ( B e. ZZ /\ ( abs ` B ) <_ 1 ) ) -> ( A x. B ) e. ZZ )
3 zcn 9660 . . . . . 6 |- ( A e. ZZ -> A e. CC )
4 zcn 9660 . . . . . 6 |- ( B e. ZZ -> B e. CC )
5 absmul 11857 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A x. B ) ) = ( ( abs ` A ) x. ( abs ` B ) ) )
63, 4, 5syl2an 289 . . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( abs ` ( A x. B ) ) = ( ( abs ` A ) x. ( abs ` B ) ) )
76ad2ant2r 513 . . . 4 |- ( ( ( A e. ZZ /\ ( abs ` A ) <_ 1 ) /\ ( B e. ZZ /\ ( abs ` B ) <_ 1 ) ) -> ( abs ` ( A x. B ) ) = ( ( abs ` A ) x. ( abs ` B ) ) )
8 abscl 11839 . . . . . . . . . . 11 |- ( A e. CC -> ( abs ` A ) e. RR )
9 absge0 11848 . . . . . . . . . . 11 |- ( A e. CC -> 0 <_ ( abs ` A ) )
108, 9jca 306 . . . . . . . . . 10 |- ( A e. CC -> ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) )
113, 10syl 14 . . . . . . . . 9 |- ( A e. ZZ -> ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) )
1211adantr 276 . . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) )
13 1red 8347 . . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> 1 e. RR )
14 abscl 11839 . . . . . . . . . . 11 |- ( B e. CC -> ( abs ` B ) e. RR )
15 absge0 11848 . . . . . . . . . . 11 |- ( B e. CC -> 0 <_ ( abs ` B ) )
1614, 15jca 306 . . . . . . . . . 10 |- ( B e. CC -> ( ( abs ` B ) e. RR /\ 0 <_ ( abs ` B ) ) )
174, 16syl 14 . . . . . . . . 9 |- ( B e. ZZ -> ( ( abs ` B ) e. RR /\ 0 <_ ( abs ` B ) ) )
1817adantl 277 . . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( abs ` B ) e. RR /\ 0 <_ ( abs ` B ) ) )
19 lemul12a 9200 . . . . . . . 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 1279 . . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( abs ` A ) <_ 1 /\ ( abs ` B ) <_ 1 ) -> ( ( abs ` A ) x. ( abs ` B ) ) <_ ( 1 x. 1 ) ) )
2120imp 124 . . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( abs ` A ) <_ 1 /\ ( abs ` B ) <_ 1 ) ) -> ( ( abs ` A ) x. ( abs ` B ) ) <_ ( 1 x. 1 ) )
2221an4s 596 . . . . 5 |- ( ( ( A e. ZZ /\ ( abs ` A ) <_ 1 ) /\ ( B e. ZZ /\ ( abs ` B ) <_ 1 ) ) -> ( ( abs ` A ) x. ( abs ` B ) ) <_ ( 1 x. 1 ) )
23 1t1e1 9466 . . . . 5 |- ( 1 x. 1 ) = 1
2422, 23breqtrdi 4172 . . . 4 |- ( ( ( A e. ZZ /\ ( abs ` A ) <_ 1 ) /\ ( B e. ZZ /\ ( abs ` B ) <_ 1 ) ) -> ( ( abs ` A ) x. ( abs ` B ) ) <_ 1 )
257, 24eqbrtrd 4153 . . 3 |- ( ( ( A e. ZZ /\ ( abs ` A ) <_ 1 ) /\ ( B e. ZZ /\ ( abs ` B ) <_ 1 ) ) -> ( abs ` ( A x. B ) ) <_ 1 )
262, 25jca 306 . 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 5698 . . . . 5 |- ( x = A -> ( abs ` x ) = ( abs ` A ) )
2827breq1d 4141 . . . 4 |- ( x = A -> ( ( abs ` x ) <_ 1 <-> ( abs ` A ) <_ 1 ) )
29 lgslem2.z . . . 4 |- Z = { x e. ZZ | ( abs ` x ) <_ 1 }
3028, 29elrab2 2985 . . 3 |- ( A e. Z <-> ( A e. ZZ /\ ( abs ` A ) <_ 1 ) )
31 fveq2 5698 . . . . 5 |- ( x = B -> ( abs ` x ) = ( abs ` B ) )
3231breq1d 4141 . . . 4 |- ( x = B -> ( ( abs ` x ) <_ 1 <-> ( abs ` B ) <_ 1 ) )
3332, 29elrab2 2985 . . 3 |- ( B e. Z <-> ( B e. ZZ /\ ( abs ` B ) <_ 1 ) )
3430, 33anbi12i 464 . 2 |- ( ( A e. Z /\ B e. Z ) <-> ( ( A e. ZZ /\ ( abs ` A ) <_ 1 ) /\ ( B e. ZZ /\ ( abs ` B ) <_ 1 ) ) )
35 fveq2 5698 . . . 4 |- ( x = ( A x. B ) -> ( abs ` x ) = ( abs ` ( A x. B ) ) )
3635breq1d 4141 . . 3 |- ( x = ( A x. B ) -> ( ( abs ` x ) <_ 1 <-> ( abs ` ( A x. B ) ) <_ 1 ) )
3736, 29elrab2 2985 . 2 |- ( ( A x. B ) e. Z <-> ( ( A x. B ) e. ZZ /\ ( abs ` ( A x. B ) ) <_ 1 ) )
3826, 34, 373imtr4i 201 1 |- ( ( A e. Z /\ B e. Z ) -> ( A x. B ) e. Z )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {crab 2532   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   RRcr 8184   0cc0 8185   1c1 8186   x. cmul 8190   <_ cle 8367   ZZcz 9655   abscabs 11785
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-rp 10072  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787
This theorem is used by:  lgsfcl2  16330  lgscllem  16331  lgsdirprm  16358
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