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Theorem gcdcllem3 13005
Description: Lemma for gcdn0cl 13006, gcddvds 13007 and dvdslegcd 13008. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
gcdcllem2.1  |-  S  =  { z  e.  ZZ  |  A. n  e.  { M ,  N }
z  ||  n }
gcdcllem2.2  |-  R  =  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N
) }
Assertion
Ref Expression
gcdcllem3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( sup ( R ,  RR ,  <  )  e.  NN  /\  ( sup ( R ,  RR ,  <  )  ||  M  /\  sup ( R ,  RR ,  <  ) 
||  N )  /\  ( ( K  e.  ZZ  /\  K  ||  M  /\  K  ||  N
)  ->  K  <_  sup ( R ,  RR ,  <  ) ) ) )
Distinct variable groups:    z, K    z, n, M    n, N, z
Allowed substitution hints:    R( z, n)    S( z, n)    K( n)

Proof of Theorem gcdcllem3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gcdcllem2.2 . . . . 5  |-  R  =  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N
) }
2 ssrab2 3420 . . . . 5  |-  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N ) } 
C_  ZZ
31, 2eqsstri 3370 . . . 4  |-  R  C_  ZZ
4 prssi 3946 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  { M ,  N }  C_  ZZ )
54adantr 452 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  { M ,  N }  C_  ZZ )
6 neorian 2685 . . . . . . . 8  |-  ( ( M  =/=  0  \/  N  =/=  0 )  <->  -.  ( M  =  0  /\  N  =  0 ) )
7 prid1g 3902 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  { M ,  N } )
8 neeq1 2606 . . . . . . . . . . . 12  |-  ( n  =  M  ->  (
n  =/=  0  <->  M  =/=  0 ) )
98rspcev 3044 . . . . . . . . . . 11  |-  ( ( M  e.  { M ,  N }  /\  M  =/=  0 )  ->  E. n  e.  { M ,  N } n  =/=  0
)
107, 9sylan 458 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  ->  E. n  e.  { M ,  N } n  =/=  0 )
1110adantlr 696 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  E. n  e.  { M ,  N } n  =/=  0
)
12 prid2g 3903 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  { M ,  N } )
13 neeq1 2606 . . . . . . . . . . . 12  |-  ( n  =  N  ->  (
n  =/=  0  <->  N  =/=  0 ) )
1413rspcev 3044 . . . . . . . . . . 11  |-  ( ( N  e.  { M ,  N }  /\  N  =/=  0 )  ->  E. n  e.  { M ,  N } n  =/=  0
)
1512, 14sylan 458 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  ->  E. n  e.  { M ,  N } n  =/=  0 )
1615adantll 695 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  E. n  e.  { M ,  N } n  =/=  0
)
1711, 16jaodan 761 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  \/  N  =/=  0 ) )  ->  E. n  e.  { M ,  N } n  =/=  0 )
186, 17sylan2br 463 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  E. n  e.  { M ,  N }
n  =/=  0 )
19 gcdcllem2.1 . . . . . . . 8  |-  S  =  { z  e.  ZZ  |  A. n  e.  { M ,  N }
z  ||  n }
2019gcdcllem1 13003 . . . . . . 7  |-  ( ( { M ,  N }  C_  ZZ  /\  E. n  e.  { M ,  N } n  =/=  0 )  ->  ( S  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  S  y  <_  x
) )
215, 18, 20syl2anc 643 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( S  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  S  y  <_  x
) )
2219, 1gcdcllem2 13004 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  R  =  S )
23 neeq1 2606 . . . . . . . . 9  |-  ( R  =  S  ->  ( R  =/=  (/)  <->  S  =/=  (/) ) )
24 raleq 2896 . . . . . . . . . 10  |-  ( R  =  S  ->  ( A. y  e.  R  y  <_  x  <->  A. y  e.  S  y  <_  x ) )
2524rexbidv 2718 . . . . . . . . 9  |-  ( R  =  S  ->  ( E. x  e.  ZZ  A. y  e.  R  y  <_  x  <->  E. x  e.  ZZ  A. y  e.  S  y  <_  x
) )
2623, 25anbi12d 692 . . . . . . . 8  |-  ( R  =  S  ->  (
( R  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x )  <->  ( S  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  S  y  <_  x
) ) )
2722, 26syl 16 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( R  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x
)  <->  ( S  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  S  y  <_  x
) ) )
2827adantr 452 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( R  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x
)  <->  ( S  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  S  y  <_  x
) ) )
2921, 28mpbird 224 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( R  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x
) )
30 suprzcl2 10558 . . . . . 6  |-  ( ( R  C_  ZZ  /\  R  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x
)  ->  sup ( R ,  RR ,  <  )  e.  R )
313, 30mp3an1 1266 . . . . 5  |-  ( ( R  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x )  ->  sup ( R ,  RR ,  <  )  e.  R )
3229, 31syl 16 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  sup ( R ,  RR ,  <  )  e.  R )
333, 32sseldi 3338 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  sup ( R ,  RR ,  <  )  e.  ZZ )
343a1i 11 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  R  C_  ZZ )
3529simprd 450 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  E. x  e.  ZZ  A. y  e.  R  y  <_  x )
36 1dvds 12856 . . . . . . 7  |-  ( M  e.  ZZ  ->  1  ||  M )
37 1dvds 12856 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  ||  N )
3836, 37anim12i 550 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 1  ||  M  /\  1  ||  N ) )
39 1z 10303 . . . . . . 7  |-  1  e.  ZZ
40 breq1 4207 . . . . . . . . 9  |-  ( z  =  1  ->  (
z  ||  M  <->  1  ||  M ) )
41 breq1 4207 . . . . . . . . 9  |-  ( z  =  1  ->  (
z  ||  N  <->  1  ||  N ) )
4240, 41anbi12d 692 . . . . . . . 8  |-  ( z  =  1  ->  (
( z  ||  M  /\  z  ||  N )  <-> 
( 1  ||  M  /\  1  ||  N ) ) )
4342, 1elrab2 3086 . . . . . . 7  |-  ( 1  e.  R  <->  ( 1  e.  ZZ  /\  (
1  ||  M  /\  1  ||  N ) ) )
4439, 43mpbiran 885 . . . . . 6  |-  ( 1  e.  R  <->  ( 1 
||  M  /\  1  ||  N ) )
4538, 44sylibr 204 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  1  e.  R )
4645adantr 452 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  1  e.  R
)
47 suprzub 10559 . . . 4  |-  ( ( R  C_  ZZ  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x  /\  1  e.  R
)  ->  1  <_  sup ( R ,  RR ,  <  ) )
4834, 35, 46, 47syl3anc 1184 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  1  <_  sup ( R ,  RR ,  <  ) )
49 elnnz1 10299 . . 3  |-  ( sup ( R ,  RR ,  <  )  e.  NN  <->  ( sup ( R ,  RR ,  <  )  e.  ZZ  /\  1  <_  sup ( R ,  RR ,  <  ) ) )
5033, 48, 49sylanbrc 646 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  sup ( R ,  RR ,  <  )  e.  NN )
51 breq1 4207 . . . . . 6  |-  ( x  =  sup ( R ,  RR ,  <  )  ->  ( x  ||  M 
<->  sup ( R ,  RR ,  <  )  ||  M ) )
52 breq1 4207 . . . . . 6  |-  ( x  =  sup ( R ,  RR ,  <  )  ->  ( x  ||  N 
<->  sup ( R ,  RR ,  <  )  ||  N ) )
5351, 52anbi12d 692 . . . . 5  |-  ( x  =  sup ( R ,  RR ,  <  )  ->  ( ( x 
||  M  /\  x  ||  N )  <->  ( sup ( R ,  RR ,  <  )  ||  M  /\  sup ( R ,  RR ,  <  )  ||  N
) ) )
54 breq1 4207 . . . . . . . 8  |-  ( z  =  x  ->  (
z  ||  M  <->  x  ||  M
) )
55 breq1 4207 . . . . . . . 8  |-  ( z  =  x  ->  (
z  ||  N  <->  x  ||  N
) )
5654, 55anbi12d 692 . . . . . . 7  |-  ( z  =  x  ->  (
( z  ||  M  /\  z  ||  N )  <-> 
( x  ||  M  /\  x  ||  N ) ) )
5756cbvrabv 2947 . . . . . 6  |-  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N ) }  =  { x  e.  ZZ  |  ( x 
||  M  /\  x  ||  N ) }
581, 57eqtri 2455 . . . . 5  |-  R  =  { x  e.  ZZ  |  ( x  ||  M  /\  x  ||  N
) }
5953, 58elrab2 3086 . . . 4  |-  ( sup ( R ,  RR ,  <  )  e.  R  <->  ( sup ( R ,  RR ,  <  )  e.  ZZ  /\  ( sup ( R ,  RR ,  <  )  ||  M  /\  sup ( R ,  RR ,  <  )  ||  N ) ) )
6032, 59sylib 189 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( sup ( R ,  RR ,  <  )  e.  ZZ  /\  ( sup ( R ,  RR ,  <  )  ||  M  /\  sup ( R ,  RR ,  <  ) 
||  N ) ) )
6160simprd 450 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( sup ( R ,  RR ,  <  )  ||  M  /\  sup ( R ,  RR ,  <  )  ||  N
) )
62 breq1 4207 . . . . . . 7  |-  ( z  =  K  ->  (
z  ||  M  <->  K  ||  M
) )
63 breq1 4207 . . . . . . 7  |-  ( z  =  K  ->  (
z  ||  N  <->  K  ||  N
) )
6462, 63anbi12d 692 . . . . . 6  |-  ( z  =  K  ->  (
( z  ||  M  /\  z  ||  N )  <-> 
( K  ||  M  /\  K  ||  N ) ) )
6564, 1elrab2 3086 . . . . 5  |-  ( K  e.  R  <->  ( K  e.  ZZ  /\  ( K 
||  M  /\  K  ||  N ) ) )
6665biimpri 198 . . . 4  |-  ( ( K  e.  ZZ  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  e.  R
)
67663impb 1149 . . 3  |-  ( ( K  e.  ZZ  /\  K  ||  M  /\  K  ||  N )  ->  K  e.  R )
68 suprzub 10559 . . . . 5  |-  ( ( R  C_  ZZ  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x  /\  K  e.  R
)  ->  K  <_  sup ( R ,  RR ,  <  ) )
69683expia 1155 . . . 4  |-  ( ( R  C_  ZZ  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x )  ->  ( K  e.  R  ->  K  <_  sup ( R ,  RR ,  <  ) ) )
703, 69mpan 652 . . 3  |-  ( E. x  e.  ZZ  A. y  e.  R  y  <_  x  ->  ( K  e.  R  ->  K  <_  sup ( R ,  RR ,  <  ) ) )
7135, 67, 70syl2im 36 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( K  e.  ZZ  /\  K  ||  M  /\  K  ||  N )  ->  K  <_  sup ( R ,  RR ,  <  ) ) )
7250, 61, 713jca 1134 1  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( sup ( R ,  RR ,  <  )  e.  NN  /\  ( sup ( R ,  RR ,  <  )  ||  M  /\  sup ( R ,  RR ,  <  ) 
||  N )  /\  ( ( K  e.  ZZ  /\  K  ||  M  /\  K  ||  N
)  ->  K  <_  sup ( R ,  RR ,  <  ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701    C_ wss 3312   (/)c0 3620   {cpr 3807   class class class wbr 4204   supcsup 7437   RRcr 8981   0cc0 8982   1c1 8983    < clt 9112    <_ cle 9113   NNcn 9992   ZZcz 10274    || cdivides 12844
This theorem is referenced by:  gcdn0cl  13006  gcddvds  13007  dvdslegcd  13008
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  df-dvds 12845
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