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Theorem gcdcllem3 12692
Description: Lemma for gcdn0cl 12693, gcddvds 12694 and dvdslegcd 12695. (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 3258 . . . . 5  |-  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N ) } 
C_  ZZ
31, 2eqsstri 3208 . . . 4  |-  R  C_  ZZ
4 prssi 3771 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  { M ,  N }  C_  ZZ )
54adantr 451 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  { M ,  N }  C_  ZZ )
6 neorian 2533 . . . . . . . 8  |-  ( ( M  =/=  0  \/  N  =/=  0 )  <->  -.  ( M  =  0  /\  N  =  0 ) )
7 prid1g 3732 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  { M ,  N } )
8 neeq1 2454 . . . . . . . . . . . 12  |-  ( n  =  M  ->  (
n  =/=  0  <->  M  =/=  0 ) )
98rspcev 2884 . . . . . . . . . . 11  |-  ( ( M  e.  { M ,  N }  /\  M  =/=  0 )  ->  E. n  e.  { M ,  N } n  =/=  0
)
107, 9sylan 457 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  ->  E. n  e.  { M ,  N } n  =/=  0 )
1110adantlr 695 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  E. n  e.  { M ,  N } n  =/=  0
)
12 prid2g 3733 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  { M ,  N } )
13 neeq1 2454 . . . . . . . . . . . 12  |-  ( n  =  N  ->  (
n  =/=  0  <->  N  =/=  0 ) )
1413rspcev 2884 . . . . . . . . . . 11  |-  ( ( N  e.  { M ,  N }  /\  N  =/=  0 )  ->  E. n  e.  { M ,  N } n  =/=  0
)
1512, 14sylan 457 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  ->  E. n  e.  { M ,  N } n  =/=  0 )
1615adantll 694 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  E. n  e.  { M ,  N } n  =/=  0
)
1711, 16jaodan 760 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  \/  N  =/=  0 ) )  ->  E. n  e.  { M ,  N } n  =/=  0 )
186, 17sylan2br 462 . . . . . . 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 12690 . . . . . . 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 642 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( S  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  S  y  <_  x
) )
2219, 1gcdcllem2 12691 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  R  =  S )
23 neeq1 2454 . . . . . . . . 9  |-  ( R  =  S  ->  ( R  =/=  (/)  <->  S  =/=  (/) ) )
24 raleq 2736 . . . . . . . . . 10  |-  ( R  =  S  ->  ( A. y  e.  R  y  <_  x  <->  A. y  e.  S  y  <_  x ) )
2524rexbidv 2564 . . . . . . . . 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 691 . . . . . . . 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 15 . . . . . . 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 451 . . . . . 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 223 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( R  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x
) )
30 suprzcl2 10308 . . . . . 6  |-  ( ( R  C_  ZZ  /\  R  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x
)  ->  sup ( R ,  RR ,  <  )  e.  R )
313, 30mp3an1 1264 . . . . 5  |-  ( ( R  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x )  ->  sup ( R ,  RR ,  <  )  e.  R )
3229, 31syl 15 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  sup ( R ,  RR ,  <  )  e.  R )
333, 32sseldi 3178 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  sup ( R ,  RR ,  <  )  e.  ZZ )
343a1i 10 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  R  C_  ZZ )
3529simprd 449 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  E. x  e.  ZZ  A. y  e.  R  y  <_  x )
36 1dvds 12543 . . . . . . 7  |-  ( M  e.  ZZ  ->  1  ||  M )
37 1dvds 12543 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  ||  N )
3836, 37anim12i 549 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 1  ||  M  /\  1  ||  N ) )
39 1z 10053 . . . . . . 7  |-  1  e.  ZZ
40 breq1 4026 . . . . . . . . 9  |-  ( z  =  1  ->  (
z  ||  M  <->  1  ||  M ) )
41 breq1 4026 . . . . . . . . 9  |-  ( z  =  1  ->  (
z  ||  N  <->  1  ||  N ) )
4240, 41anbi12d 691 . . . . . . . 8  |-  ( z  =  1  ->  (
( z  ||  M  /\  z  ||  N )  <-> 
( 1  ||  M  /\  1  ||  N ) ) )
4342, 1elrab2 2925 . . . . . . 7  |-  ( 1  e.  R  <->  ( 1  e.  ZZ  /\  (
1  ||  M  /\  1  ||  N ) ) )
4439, 43mpbiran 884 . . . . . 6  |-  ( 1  e.  R  <->  ( 1 
||  M  /\  1  ||  N ) )
4538, 44sylibr 203 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  1  e.  R )
4645adantr 451 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  1  e.  R
)
47 suprzub 10309 . . . 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 1182 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  1  <_  sup ( R ,  RR ,  <  ) )
49 elnnz1 10049 . . 3  |-  ( sup ( R ,  RR ,  <  )  e.  NN  <->  ( sup ( R ,  RR ,  <  )  e.  ZZ  /\  1  <_  sup ( R ,  RR ,  <  ) ) )
5033, 48, 49sylanbrc 645 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  sup ( R ,  RR ,  <  )  e.  NN )
51 breq1 4026 . . . . . 6  |-  ( x  =  sup ( R ,  RR ,  <  )  ->  ( x  ||  M 
<->  sup ( R ,  RR ,  <  )  ||  M ) )
52 breq1 4026 . . . . . 6  |-  ( x  =  sup ( R ,  RR ,  <  )  ->  ( x  ||  N 
<->  sup ( R ,  RR ,  <  )  ||  N ) )
5351, 52anbi12d 691 . . . . 5  |-  ( x  =  sup ( R ,  RR ,  <  )  ->  ( ( x 
||  M  /\  x  ||  N )  <->  ( sup ( R ,  RR ,  <  )  ||  M  /\  sup ( R ,  RR ,  <  )  ||  N
) ) )
54 breq1 4026 . . . . . . . 8  |-  ( z  =  x  ->  (
z  ||  M  <->  x  ||  M
) )
55 breq1 4026 . . . . . . . 8  |-  ( z  =  x  ->  (
z  ||  N  <->  x  ||  N
) )
5654, 55anbi12d 691 . . . . . . 7  |-  ( z  =  x  ->  (
( z  ||  M  /\  z  ||  N )  <-> 
( x  ||  M  /\  x  ||  N ) ) )
5756cbvrabv 2787 . . . . . 6  |-  { z  e.  ZZ  |  ( z  ||  M  /\  z  ||  N ) }  =  { x  e.  ZZ  |  ( x 
||  M  /\  x  ||  N ) }
581, 57eqtri 2303 . . . . 5  |-  R  =  { x  e.  ZZ  |  ( x  ||  M  /\  x  ||  N
) }
5953, 58elrab2 2925 . . . 4  |-  ( sup ( R ,  RR ,  <  )  e.  R  <->  ( sup ( R ,  RR ,  <  )  e.  ZZ  /\  ( sup ( R ,  RR ,  <  )  ||  M  /\  sup ( R ,  RR ,  <  )  ||  N ) ) )
6032, 59sylib 188 . . 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 449 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( sup ( R ,  RR ,  <  )  ||  M  /\  sup ( R ,  RR ,  <  )  ||  N
) )
62 breq1 4026 . . . . . . 7  |-  ( z  =  K  ->  (
z  ||  M  <->  K  ||  M
) )
63 breq1 4026 . . . . . . 7  |-  ( z  =  K  ->  (
z  ||  N  <->  K  ||  N
) )
6462, 63anbi12d 691 . . . . . 6  |-  ( z  =  K  ->  (
( z  ||  M  /\  z  ||  N )  <-> 
( K  ||  M  /\  K  ||  N ) ) )
6564, 1elrab2 2925 . . . . 5  |-  ( K  e.  R  <->  ( K  e.  ZZ  /\  ( K 
||  M  /\  K  ||  N ) ) )
6665biimpri 197 . . . 4  |-  ( ( K  e.  ZZ  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  e.  R
)
67663impb 1147 . . 3  |-  ( ( K  e.  ZZ  /\  K  ||  M  /\  K  ||  N )  ->  K  e.  R )
68 suprzub 10309 . . . . 5  |-  ( ( R  C_  ZZ  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x  /\  K  e.  R
)  ->  K  <_  sup ( R ,  RR ,  <  ) )
69683expia 1153 . . . 4  |-  ( ( R  C_  ZZ  /\  E. x  e.  ZZ  A. y  e.  R  y  <_  x )  ->  ( K  e.  R  ->  K  <_  sup ( R ,  RR ,  <  ) ) )
703, 69mpan 651 . . 3  |-  ( E. x  e.  ZZ  A. y  e.  R  y  <_  x  ->  ( K  e.  R  ->  K  <_  sup ( R ,  RR ,  <  ) ) )
7135, 67, 70syl2im 34 . 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 1132 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    C_ wss 3152   (/)c0 3455   {cpr 3641   class class class wbr 4023   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    < clt 8867    <_ cle 8868   NNcn 9746   ZZcz 10024    || cdivides 12531
This theorem is referenced by:  gcdn0cl  12693  gcddvds  12694  dvdslegcd  12695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532
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