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Theorem archirng 26227
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 6119 . . . 4  |-  ( m  =  0  ->  (
m  .x.  X )  =  ( 0  .x. 
X ) )
21breq2d 4325 . . 3  |-  ( m  =  0  ->  ( Y  .<_  ( m  .x.  X )  <->  Y  .<_  ( 0  .x.  X ) ) )
3 oveq1 6119 . . . 4  |-  ( m  =  n  ->  (
m  .x.  X )  =  ( n  .x.  X ) )
43breq2d 4325 . . 3  |-  ( m  =  n  ->  ( Y  .<_  ( m  .x.  X )  <->  Y  .<_  ( n  .x.  X ) ) )
5 oveq1 6119 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (
m  .x.  X )  =  ( ( n  +  1 )  .x.  X ) )
65breq2d 4325 . . 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 26187 . . . . . . . 8  |-  ( W  e. oGrp 
<->  ( W  e.  Grp  /\  W  e. oMnd ) )
109simprbi 464 . . . . . . 7  |-  ( W  e. oGrp  ->  W  e. oMnd )
11 omndtos 26190 . . . . . . 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 15587 . . . . . . 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 26145 . . . . . 6  |-  ( ( W  e. Toset  /\  .0.  e.  B  /\  Y  e.  B
)  ->  (  .0.  .<  Y 
<->  -.  Y  .<_  .0.  )
)
2312, 18, 19, 22syl3anc 1218 . . . . 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 15653 . . . . . 6  |-  ( X  e.  B  ->  (
0  .x.  X )  =  .0.  )
2825, 27syl 16 . . . . 5  |-  ( ph  ->  ( 0  .x.  X
)  =  .0.  )
2928breq2d 4325 . . . 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 26189 . . . . . 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 26224 . . . . . 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 1217 . . . 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 4317 . . . . . . . 8  |-  ( x  =  X  ->  (  .0.  .<  x  <->  .0.  .<  X ) )
40 oveq2 6120 . . . . . . . . . 10  |-  ( x  =  X  ->  (
m  .x.  x )  =  ( m  .x.  X ) )
4140breq2d 4325 . . . . . . . . 9  |-  ( x  =  X  ->  (
y  .<_  ( m  .x.  x )  <->  y  .<_  ( m  .x.  X ) ) )
4241rexbidv 2757 . . . . . . . 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 4316 . . . . . . . . 9  |-  ( y  =  Y  ->  (
y  .<_  ( m  .x.  X )  <->  Y  .<_  ( m  .x.  X ) ) )
4544rexbidv 2757 . . . . . . . 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 3100 . . . . . 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 1217 . . 3  |-  ( ph  ->  E. m  e.  NN  Y  .<_  ( m  .x.  X ) )
512, 4, 6, 30, 50nn0min 26112 . 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 10766 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  n  e.  ZZ )
5625adantr 465 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  X  e.  B )
5715, 26mulgcl 15665 . . . . . 6  |-  ( ( W  e.  Grp  /\  n  e.  ZZ  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
5853, 55, 56, 57syl3anc 1218 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( n  .x.  X )  e.  B
)
5919adantr 465 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  Y  e.  B )
6015, 20, 21tltnle 26145 . . . . 5  |-  ( ( W  e. Toset  /\  (
n  .x.  X )  e.  B  /\  Y  e.  B )  ->  (
( n  .x.  X
)  .<  Y  <->  -.  Y  .<_  ( n  .x.  X
) ) )
6152, 58, 59, 60syl3anc 1218 . . . 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 2753 . 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 1369    e. wcel 1756   A.wral 2736   E.wrex 2737   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   0cc0 9303   1c1 9304    + caddc 9306   NNcn 10343   NN0cn0 10600   ZZcz 10667   Basecbs 14195   lecple 14266   0gc0g 14399   ltcplt 15132  Tosetctos 15224   Mndcmnd 15430   Grpcgrp 15431  .gcmg 15435  oMndcomnd 26182  oGrpcogrp 26183  Archicarchi 26216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-seq 11828  df-0g 14401  df-poset 15137  df-plt 15149  df-toset 15225  df-mnd 15436  df-grp 15566  df-minusg 15567  df-mulg 15569  df-omnd 26184  df-ogrp 26185  df-inftm 26217  df-archi 26218
This theorem is referenced by:  archirngz  26228  archiabllem1a  26230
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