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Theorem archirng 27382
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 6284 . . . 4  |-  ( m  =  0  ->  (
m  .x.  X )  =  ( 0  .x. 
X ) )
21breq2d 4454 . . 3  |-  ( m  =  0  ->  ( Y  .<_  ( m  .x.  X )  <->  Y  .<_  ( 0  .x.  X ) ) )
3 oveq1 6284 . . . 4  |-  ( m  =  n  ->  (
m  .x.  X )  =  ( n  .x.  X ) )
43breq2d 4454 . . 3  |-  ( m  =  n  ->  ( Y  .<_  ( m  .x.  X )  <->  Y  .<_  ( n  .x.  X ) ) )
5 oveq1 6284 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
m  .x.  X )  =  ( ( n  +  1 )  .x.  X ) )
65breq2d 4454 . . 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 27342 . . . . . . . 8  |-  ( W  e. oGrp 
<->  ( W  e.  Grp  /\  W  e. oMnd ) )
109simprbi 464 . . . . . . 7  |-  ( W  e. oGrp  ->  W  e. oMnd )
11 omndtos 27345 . . . . . . 7  |-  ( W  e. oMnd  ->  W  e. Toset )
128, 10, 113syl 20 . . . . . 6  |-  ( ph  ->  W  e. Toset )
139simplbi 460 . . . . . . . 8  |-  ( W  e. oGrp  ->  W  e.  Grp )
148, 13syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  Grp )
15 archirng.b . . . . . . . 8  |-  B  =  ( Base `  W
)
16 archirng.0 . . . . . . . 8  |-  .0.  =  ( 0g `  W )
1715, 16grpidcl 15874 . . . . . . 7  |-  ( W  e.  Grp  ->  .0.  e.  B )
1814, 17syl 16 . . . . . 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 27300 . . . . . 6  |-  ( ( W  e. Toset  /\  .0.  e.  B  /\  Y  e.  B
)  ->  (  .0.  .<  Y 
<->  -.  Y  .<_  .0.  )
)
2312, 18, 19, 22syl3anc 1223 . . . . 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 15942 . . . . . 6  |-  ( X  e.  B  ->  (
0  .x.  X )  =  .0.  )
2825, 27syl 16 . . . . 5  |-  ( ph  ->  ( 0  .x.  X
)  =  .0.  )
2928breq2d 4454 . . . 4  |-  ( ph  ->  ( Y  .<_  ( 0 
.x.  X )  <->  Y  .<_  .0.  ) )
3024, 29mtbird 301 . . 3  |-  ( ph  ->  -.  Y  .<_  ( 0 
.x.  X ) )
3125, 19jca 532 . . . 4  |-  ( ph  ->  ( X  e.  B  /\  Y  e.  B
) )
32 omndmnd 27344 . . . . . 6  |-  ( W  e. oMnd  ->  W  e.  Mnd )
338, 10, 323syl 20 . . . . 5  |-  ( ph  ->  W  e.  Mnd )
34 archirng.2 . . . . 5  |-  ( ph  ->  W  e. Archi )
3515, 16, 26, 20, 21isarchi2 27379 . . . . . 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 484 . . . . 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 1222 . . . 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 4446 . . . . . . . 8  |-  ( x  =  X  ->  (  .0.  .<  x  <->  .0.  .<  X ) )
40 oveq2 6285 . . . . . . . . . 10  |-  ( x  =  X  ->  (
m  .x.  x )  =  ( m  .x.  X ) )
4140breq2d 4454 . . . . . . . . 9  |-  ( x  =  X  ->  (
y  .<_  ( m  .x.  x )  <->  y  .<_  ( m  .x.  X ) ) )
4241rexbidv 2968 . . . . . . . 8  |-  ( x  =  X  ->  ( E. m  e.  NN  y  .<_  ( m  .x.  x )  <->  E. m  e.  NN  y  .<_  ( m 
.x.  X ) ) )
4339, 42imbi12d 320 . . . . . . 7  |-  ( x  =  X  ->  (
(  .0.  .<  x  ->  E. m  e.  NN  y  .<_  ( m  .x.  x ) )  <->  (  .0.  .<  X  ->  E. m  e.  NN  y  .<_  ( m  .x.  X ) ) ) )
44 breq1 4445 . . . . . . . . 9  |-  ( y  =  Y  ->  (
y  .<_  ( m  .x.  X )  <->  Y  .<_  ( m  .x.  X ) ) )
4544rexbidv 2968 . . . . . . . 8  |-  ( y  =  Y  ->  ( E. m  e.  NN  y  .<_  ( m  .x.  X )  <->  E. m  e.  NN  Y  .<_  ( m 
.x.  X ) ) )
4645imbi2d 316 . . . . . . 7  |-  ( y  =  Y  ->  (
(  .0.  .<  X  ->  E. m  e.  NN  y  .<_  ( m  .x.  X ) )  <->  (  .0.  .<  X  ->  E. m  e.  NN  Y  .<_  ( m  .x.  X ) ) ) )
4743, 46rspc2v 3218 . . . . . 6  |-  ( ( 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 ) ) ) )
4847imp 429 . . . . 5  |-  ( ( ( 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 ) ) )
4948imp 429 . . . 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 ) )
5031, 37, 38, 49syl21anc 1222 . . 3  |-  ( ph  ->  E. m  e.  NN  Y  .<_  ( m  .x.  X ) )
512, 4, 6, 30, 50nn0min 27267 . 2  |-  ( ph  ->  E. n  e.  NN0  ( -.  Y  .<_  ( n  .x.  X )  /\  Y  .<_  ( ( n  +  1 ) 
.x.  X ) ) )
5212adantr 465 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  W  e. Toset )
5314adantr 465 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  W  e.  Grp )
54 simpr 461 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  n  e.  NN0 )
5554nn0zd 10955 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  n  e.  ZZ )
5625adantr 465 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  X  e.  B )
5715, 26mulgcl 15954 . . . . . 6  |-  ( ( W  e.  Grp  /\  n  e.  ZZ  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
5853, 55, 56, 57syl3anc 1223 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( n  .x.  X )  e.  B
)
5919adantr 465 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  Y  e.  B )
6015, 20, 21tltnle 27300 . . . . 5  |-  ( ( W  e. Toset  /\  (
n  .x.  X )  e.  B  /\  Y  e.  B )  ->  (
( n  .x.  X
)  .<  Y  <->  -.  Y  .<_  ( n  .x.  X
) ) )
6152, 58, 59, 60syl3anc 1223 . . . 4  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
n  .x.  X )  .<  Y  <->  -.  Y  .<_  ( n  .x.  X ) ) )
6261anbi1d 704 . . 3  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
( n  .x.  X
)  .<  Y  /\  Y  .<_  ( ( n  + 
1 )  .x.  X
) )  <->  ( -.  Y  .<_  ( n  .x.  X )  /\  Y  .<_  ( ( n  + 
1 )  .x.  X
) ) ) )
6362rexbidva 2965 . 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 ) ) ) )
6451, 63mpbird 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 369    = wceq 1374    e. wcel 1762   A.wral 2809   E.wrex 2810   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   0cc0 9483   1c1 9484    + caddc 9486   NNcn 10527   NN0cn0 10786   ZZcz 10855   Basecbs 14481   lecple 14553   0gc0g 14686   ltcplt 15419  Tosetctos 15511   Mndcmnd 15717   Grpcgrp 15718  .gcmg 15722  oMndcomnd 27337  oGrpcogrp 27338  Archicarchi 27371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-seq 12066  df-0g 14688  df-poset 15424  df-plt 15436  df-toset 15512  df-mnd 15723  df-grp 15853  df-minusg 15854  df-mulg 15856  df-omnd 27339  df-ogrp 27340  df-inftm 27372  df-archi 27373
This theorem is referenced by:  archirngz  27383  archiabllem1a  27385
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