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Theorem gcdcllem3 12968
Description: Lemma for gcdn0cl 12969, gcddvds 12970 and dvdslegcd 12971. (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 3388 . . . . 5  |-  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N ) } 
C_  ZZ
31, 2eqsstri 3338 . . . 4  |-  R  C_  ZZ
4 prssi 3914 . . . . . . . 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 2654 . . . . . . . 8  |-  ( ( M  =/=  0  \/  N  =/=  0 )  <->  -.  ( M  =  0  /\  N  =  0 ) )
7 prid1g 3870 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  { M ,  N } )
8 neeq1 2575 . . . . . . . . . . . 12  |-  ( n  =  M  ->  (
n  =/=  0  <->  M  =/=  0 ) )
98rspcev 3012 . . . . . . . . . . 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 3871 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  { M ,  N } )
13 neeq1 2575 . . . . . . . . . . . 12  |-  ( n  =  N  ->  (
n  =/=  0  <->  N  =/=  0 ) )
1413rspcev 3012 . . . . . . . . . . 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 12966 . . . . . . 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 12967 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  R  =  S )
23 neeq1 2575 . . . . . . . . 9  |-  ( R  =  S  ->  ( R  =/=  (/)  <->  S  =/=  (/) ) )
24 raleq 2864 . . . . . . . . . 10  |-  ( R  =  S  ->  ( A. y  e.  R  y  <_  x  <->  A. y  e.  S  y  <_  x ) )
2524rexbidv 2687 . . . . . . . . 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 10522 . . . . . 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 3306 . . 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 12819 . . . . . . 7  |-  ( M  e.  ZZ  ->  1  ||  M )
37 1dvds 12819 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  ||  N )
3836, 37anim12i 550 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 1  ||  M  /\  1  ||  N ) )
39 1z 10267 . . . . . . 7  |-  1  e.  ZZ
40 breq1 4175 . . . . . . . . 9  |-  ( z  =  1  ->  (
z  ||  M  <->  1  ||  M ) )
41 breq1 4175 . . . . . . . . 9  |-  ( z  =  1  ->  (
z  ||  N  <->  1  ||  N ) )
4240, 41anbi12d 692 . . . . . . . 8  |-  ( z  =  1  ->  (
( z  ||  M  /\  z  ||  N )  <-> 
( 1  ||  M  /\  1  ||  N ) ) )
4342, 1elrab2 3054 . . . . . . 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 10523 . . . 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 10263 . . 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 4175 . . . . . 6  |-  ( x  =  sup ( R ,  RR ,  <  )  ->  ( x  ||  M 
<->  sup ( R ,  RR ,  <  )  ||  M ) )
52 breq1 4175 . . . . . 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 4175 . . . . . . . 8  |-  ( z  =  x  ->  (
z  ||  M  <->  x  ||  M
) )
55 breq1 4175 . . . . . . . 8  |-  ( z  =  x  ->  (
z  ||  N  <->  x  ||  N
) )
5654, 55anbi12d 692 . . . . . . 7  |-  ( z  =  x  ->  (
( z  ||  M  /\  z  ||  N )  <-> 
( x  ||  M  /\  x  ||  N ) ) )
5756cbvrabv 2915 . . . . . 6  |-  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N ) }  =  { x  e.  ZZ  |  ( x 
||  M  /\  x  ||  N ) }
581, 57eqtri 2424 . . . . 5  |-  R  =  { x  e.  ZZ  |  ( x  ||  M  /\  x  ||  N
) }
5953, 58elrab2 3054 . . . 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 4175 . . . . . . 7  |-  ( z  =  K  ->  (
z  ||  M  <->  K  ||  M
) )
63 breq1 4175 . . . . . . 7  |-  ( z  =  K  ->  (
z  ||  N  <->  K  ||  N
) )
6462, 63anbi12d 692 . . . . . 6  |-  ( z  =  K  ->  (
( z  ||  M  /\  z  ||  N )  <-> 
( K  ||  M  /\  K  ||  N ) ) )
6564, 1elrab2 3054 . . . . 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 10523 . . . . 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 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    C_ wss 3280   (/)c0 3588   {cpr 3775   class class class wbr 4172   supcsup 7403   RRcr 8945   0cc0 8946   1c1 8947    < clt 9076    <_ cle 9077   NNcn 9956   ZZcz 10238    || cdivides 12807
This theorem is referenced by:  gcdn0cl  12969  gcddvds  12970  dvdslegcd  12971
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808
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