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Theorem archirng 28184
Description: Property of Archimedean ordered groups, framing positive  Y between multiples of  X. (Contributed by Thierry Arnoux, 12-Apr-2018.)
Hypotheses
Ref Expression
archirng.b  |-  B  =  ( Base `  W
)
archirng.0  |-  .0.  =  ( 0g `  W )
archirng.i  |-  .<  =  ( lt `  W )
archirng.l  |-  .<_  =  ( le `  W )
archirng.x  |-  .x.  =  (.g
`  W )
archirng.1  |-  ( ph  ->  W  e. oGrp )
archirng.2  |-  ( ph  ->  W  e. Archi )
archirng.3  |-  ( ph  ->  X  e.  B )
archirng.4  |-  ( ph  ->  Y  e.  B )
archirng.5  |-  ( ph  ->  .0.  .<  X )
archirng.6  |-  ( ph  ->  .0.  .<  Y )
Assertion
Ref Expression
archirng  |-  ( ph  ->  E. n  e.  NN0  ( ( n  .x.  X )  .<  Y  /\  Y  .<_  ( ( n  +  1 )  .x.  X ) ) )
Distinct variable groups:    n, X    n, Y    ph, n    .0. , n    .<_ , n    .< , n    .x. , n
Allowed substitution hints:    B( n)    W( n)

Proof of Theorem archirng
Dummy variables  x  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6285 . . . 4  |-  ( m  =  0  ->  (
m  .x.  X )  =  ( 0  .x. 
X ) )
21breq2d 4407 . . 3  |-  ( m  =  0  ->  ( Y  .<_  ( m  .x.  X )  <->  Y  .<_  ( 0  .x.  X ) ) )
3 oveq1 6285 . . . 4  |-  ( m  =  n  ->  (
m  .x.  X )  =  ( n  .x.  X ) )
43breq2d 4407 . . 3  |-  ( m  =  n  ->  ( Y  .<_  ( m  .x.  X )  <->  Y  .<_  ( n  .x.  X ) ) )
5 oveq1 6285 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
m  .x.  X )  =  ( ( n  +  1 )  .x.  X ) )
65breq2d 4407 . . 3  |-  ( m  =  ( n  + 
1 )  ->  ( Y  .<_  ( m  .x.  X )  <->  Y  .<_  ( ( n  +  1 )  .x.  X ) ) )
7 archirng.6 . . . . 5  |-  ( ph  ->  .0.  .<  Y )
8 archirng.1 . . . . . . 7  |-  ( ph  ->  W  e. oGrp )
9 isogrp 28144 . . . . . . . 8  |-  ( W  e. oGrp 
<->  ( W  e.  Grp  /\  W  e. oMnd ) )
109simprbi 462 . . . . . . 7  |-  ( W  e. oGrp  ->  W  e. oMnd )
11 omndtos 28147 . . . . . . 7  |-  ( W  e. oMnd  ->  W  e. Toset )
128, 10, 113syl 18 . . . . . 6  |-  ( ph  ->  W  e. Toset )
13 ogrpgrp 28145 . . . . . . . 8  |-  ( W  e. oGrp  ->  W  e.  Grp )
148, 13syl 17 . . . . . . 7  |-  ( ph  ->  W  e.  Grp )
15 archirng.b . . . . . . . 8  |-  B  =  ( Base `  W
)
16 archirng.0 . . . . . . . 8  |-  .0.  =  ( 0g `  W )
1715, 16grpidcl 16402 . . . . . . 7  |-  ( W  e.  Grp  ->  .0.  e.  B )
1814, 17syl 17 . . . . . 6  |-  ( ph  ->  .0.  e.  B )
19 archirng.4 . . . . . 6  |-  ( ph  ->  Y  e.  B )
20 archirng.l . . . . . . 7  |-  .<_  =  ( le `  W )
21 archirng.i . . . . . . 7  |-  .<  =  ( lt `  W )
2215, 20, 21tltnle 28102 . . . . . 6  |-  ( ( W  e. Toset  /\  .0.  e.  B  /\  Y  e.  B
)  ->  (  .0.  .<  Y 
<->  -.  Y  .<_  .0.  )
)
2312, 18, 19, 22syl3anc 1230 . . . . 5  |-  ( ph  ->  (  .0.  .<  Y  <->  -.  Y  .<_  .0.  ) )
247, 23mpbid 210 . . . 4  |-  ( ph  ->  -.  Y  .<_  .0.  )
25 archirng.3 . . . . . 6  |-  ( ph  ->  X  e.  B )
26 archirng.x . . . . . . 7  |-  .x.  =  (.g
`  W )
2715, 16, 26mulg0 16471 . . . . . 6  |-  ( X  e.  B  ->  (
0  .x.  X )  =  .0.  )
2825, 27syl 17 . . . . 5  |-  ( ph  ->  ( 0  .x.  X
)  =  .0.  )
2928breq2d 4407 . . . 4  |-  ( ph  ->  ( Y  .<_  ( 0 
.x.  X )  <->  Y  .<_  .0.  ) )
3024, 29mtbird 299 . . 3  |-  ( ph  ->  -.  Y  .<_  ( 0 
.x.  X ) )
3125, 19jca 530 . . . 4  |-  ( ph  ->  ( X  e.  B  /\  Y  e.  B
) )
32 omndmnd 28146 . . . . . 6  |-  ( W  e. oMnd  ->  W  e.  Mnd )
338, 10, 323syl 18 . . . . 5  |-  ( ph  ->  W  e.  Mnd )
34 archirng.2 . . . . 5  |-  ( ph  ->  W  e. Archi )
3515, 16, 26, 20, 21isarchi2 28181 . . . . . 6  |-  ( ( W  e. Toset  /\  W  e. 
Mnd )  ->  ( W  e. Archi  <->  A. x  e.  B  A. y  e.  B  (  .0.  .<  x  ->  E. m  e.  NN  y  .<_  ( m  .x.  x
) ) ) )
3635biimpa 482 . . . . 5  |-  ( ( ( W  e. Toset  /\  W  e.  Mnd )  /\  W  e. Archi )  ->  A. x  e.  B  A. y  e.  B  (  .0.  .<  x  ->  E. m  e.  NN  y  .<_  ( m  .x.  x ) ) )
3712, 33, 34, 36syl21anc 1229 . . . 4  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  (  .0.  .<  x  ->  E. m  e.  NN  y  .<_  ( m  .x.  x
) ) )
38 archirng.5 . . . 4  |-  ( ph  ->  .0.  .<  X )
39 breq2 4399 . . . . . 6  |-  ( x  =  X  ->  (  .0.  .<  x  <->  .0.  .<  X ) )
40 oveq2 6286 . . . . . . . 8  |-  ( x  =  X  ->  (
m  .x.  x )  =  ( m  .x.  X ) )
4140breq2d 4407 . . . . . . 7  |-  ( x  =  X  ->  (
y  .<_  ( m  .x.  x )  <->  y  .<_  ( m  .x.  X ) ) )
4241rexbidv 2918 . . . . . 6  |-  ( x  =  X  ->  ( E. m  e.  NN  y  .<_  ( m  .x.  x )  <->  E. m  e.  NN  y  .<_  ( m 
.x.  X ) ) )
4339, 42imbi12d 318 . . . . 5  |-  ( x  =  X  ->  (
(  .0.  .<  x  ->  E. m  e.  NN  y  .<_  ( m  .x.  x ) )  <->  (  .0.  .<  X  ->  E. m  e.  NN  y  .<_  ( m  .x.  X ) ) ) )
44 breq1 4398 . . . . . . 7  |-  ( y  =  Y  ->  (
y  .<_  ( m  .x.  X )  <->  Y  .<_  ( m  .x.  X ) ) )
4544rexbidv 2918 . . . . . 6  |-  ( y  =  Y  ->  ( E. m  e.  NN  y  .<_  ( m  .x.  X )  <->  E. m  e.  NN  Y  .<_  ( m 
.x.  X ) ) )
4645imbi2d 314 . . . . 5  |-  ( y  =  Y  ->  (
(  .0.  .<  X  ->  E. m  e.  NN  y  .<_  ( m  .x.  X ) )  <->  (  .0.  .<  X  ->  E. m  e.  NN  Y  .<_  ( m  .x.  X ) ) ) )
4743, 46rspc2v 3169 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( A. x  e.  B  A. y  e.  B  (  .0.  .<  x  ->  E. m  e.  NN  y  .<_  ( m  .x.  x ) )  -> 
(  .0.  .<  X  ->  E. m  e.  NN  Y  .<_  ( m  .x.  X ) ) ) )
4831, 37, 38, 47syl3c 60 . . 3  |-  ( ph  ->  E. m  e.  NN  Y  .<_  ( m  .x.  X ) )
492, 4, 6, 30, 48nn0min 28063 . 2  |-  ( ph  ->  E. n  e.  NN0  ( -.  Y  .<_  ( n  .x.  X )  /\  Y  .<_  ( ( n  +  1 ) 
.x.  X ) ) )
5012adantr 463 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  W  e. Toset )
5114adantr 463 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  W  e.  Grp )
52 simpr 459 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  n  e.  NN0 )
5352nn0zd 11006 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  n  e.  ZZ )
5425adantr 463 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  X  e.  B )
5515, 26mulgcl 16483 . . . . . 6  |-  ( ( W  e.  Grp  /\  n  e.  ZZ  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
5651, 53, 54, 55syl3anc 1230 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( n  .x.  X )  e.  B
)
5719adantr 463 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  Y  e.  B )
5815, 20, 21tltnle 28102 . . . . 5  |-  ( ( W  e. Toset  /\  (
n  .x.  X )  e.  B  /\  Y  e.  B )  ->  (
( n  .x.  X
)  .<  Y  <->  -.  Y  .<_  ( n  .x.  X
) ) )
5950, 56, 57, 58syl3anc 1230 . . . 4  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
n  .x.  X )  .<  Y  <->  -.  Y  .<_  ( n  .x.  X ) ) )
6059anbi1d 703 . . 3  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
( n  .x.  X
)  .<  Y  /\  Y  .<_  ( ( n  + 
1 )  .x.  X
) )  <->  ( -.  Y  .<_  ( n  .x.  X )  /\  Y  .<_  ( ( n  + 
1 )  .x.  X
) ) ) )
6160rexbidva 2915 . 2  |-  ( ph  ->  ( E. n  e. 
NN0  ( ( n 
.x.  X )  .<  Y  /\  Y  .<_  ( ( n  +  1 ) 
.x.  X ) )  <->  E. n  e.  NN0  ( -.  Y  .<_  ( n  .x.  X )  /\  Y  .<_  ( ( n  +  1 ) 
.x.  X ) ) ) )
6249, 61mpbird 232 1  |-  ( ph  ->  E. n  e.  NN0  ( ( n  .x.  X )  .<  Y  /\  Y  .<_  ( ( n  +  1 )  .x.  X ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   0cc0 9522   1c1 9523    + caddc 9525   NNcn 10576   NN0cn0 10836   ZZcz 10905   Basecbs 14841   lecple 14916   0gc0g 15054   ltcplt 15894  Tosetctos 15987   Mndcmnd 16243   Grpcgrp 16377  .gcmg 16380  oMndcomnd 28139  oGrpcogrp 28140  Archicarchi 28173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-seq 12152  df-0g 15056  df-preset 15881  df-poset 15899  df-plt 15912  df-toset 15988  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381  df-minusg 16382  df-mulg 16384  df-omnd 28141  df-ogrp 28142  df-inftm 28174  df-archi 28175
This theorem is referenced by:  archirngz  28185  archiabllem1a  28187
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